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b/2023/finals/885100906409208.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cd9100f3318b48eb8ba907d3eca08bfb564a6fb456bf245d64dd583231666196 +size 80320 diff --git a/2023/finals/cacti_cartography.cpp b/2023/finals/cacti_cartography.cpp new file mode 100644 index 0000000000000000000000000000000000000000..147b9d5887ba792a39bd7eb704e00d0d0afe5924 --- /dev/null +++ b/2023/finals/cacti_cartography.cpp @@ -0,0 +1,264 @@ +#include +#include +#include +#include +#include +using namespace std; + +typedef long long int64; + +const int64 kInf = 12345678912345LL; +const int64 MAXK = 50, KLIM = (MAXK + 1) << 1, NLIM = 1000; + +int N, M, K; +vector weights; +vector> adj; +int64 f[2][KLIM][KLIM][NLIM]; +int timer; +vector tin, tout, par; +vector cycle_ids, cycle_lens, cycle_rev, dist_to_par; +vector> edge_lnks; +vector> dp_edges; + +void dfs(int x) { + tin[x] = timer++; + for (int y : adj[x]) { + if (tin[y] != -1) { + continue; + } + dfs(y); + } + tout[x] = timer++; +} + +int cycle_counter; +vector dfs_stack; + +void dfs_cycles(int x, int p) { + tin[x] = timer++; + dfs_stack.push_back(x); + for (int y : adj[x]) { + if (y == p) { + continue; + } + if (y == par[x]) { + int cur_len = 1; + int rev_cycle = -1; + for (int z = (int)dfs_stack.size() - 1; dfs_stack[z] != par[x]; z--) { + int edge_id = edge_lnks[dfs_stack[z - 1]][dfs_stack[z]]; + int rev_edge_id = edge_lnks[dfs_stack[z]][dfs_stack[z - 1]]; + if (rev_edge_id != -1 && cycle_ids[rev_edge_id] != -1) { + rev_cycle = cycle_ids[rev_edge_id]; + cycle_rev[rev_cycle] = cycle_counter; + } + cycle_ids[edge_id] = cycle_counter; + dist_to_par[edge_id] = cur_len++; + } + cycle_lens.push_back(cur_len); + cycle_rev.push_back(rev_cycle); + cycle_counter++; + } else { + if (tin[y] != -1) { + continue; + } + if (edge_lnks[x][y] == -1) { + edge_lnks[x][y] = dp_edges.size(); + dp_edges.emplace_back(x, y); + cycle_ids.push_back(-1); + dist_to_par.push_back(-1); + } + dfs_cycles(y, x); + } + } + dfs_stack.pop_back(); +} + +int lazy_max_dist; + +int64 lazy_dp(int edge_id, int dist_to_last, int par_dist, int promised) { + if (dist_to_last > K + 1) { + promised = 1; + } + if (f[promised][dist_to_last][par_dist][edge_id] != -1LL) { + return f[promised][dist_to_last][par_dist][edge_id]; + } + const int base_cycle_id = cycle_ids[edge_id]; + if (!dist_to_last && base_cycle_id != -1 && + cycle_lens[base_cycle_id] - dist_to_par[edge_id] < par_dist) { + return f[promised][dist_to_last][par_dist][edge_id] = lazy_dp( + edge_id, dist_to_last, + cycle_lens[base_cycle_id] - dist_to_par[edge_id], promised); + } + if (!promised && base_cycle_id != -1 && + dist_to_par[edge_id] + par_dist < dist_to_last) { + return f[promised][dist_to_last][par_dist][edge_id] = lazy_dp( + edge_id, dist_to_par[edge_id] + par_dist, par_dist, promised); + } + const int x = dp_edges[edge_id].second; + int64 res = kInf; + if (dist_to_last) { + res = min(res, lazy_dp(edge_id, 0, par_dist, 0) + weights[x]); + } + if (dist_to_last == lazy_max_dist) { + return f[promised][dist_to_last][par_dist][edge_id] = res; + } + const int p = dp_edges[edge_id].first; + vector edge_list; + vector> edge_groups; + map cycle_to_ind; + for (int y : adj[x]) { + if (y == p) { + continue; + } + int new_edge_id = edge_lnks[x][y]; + if (new_edge_id == -1) { + continue; + } + int cycle_id = cycle_ids[new_edge_id]; + if (cycle_id != -1) { + auto mit = cycle_to_ind.find(cycle_id); + if (mit != cycle_to_ind.end()) { + edge_groups[mit->second].second = edge_list.size(); + edge_list.push_back(new_edge_id); + continue; + } + } + int rev_cycle_id = (cycle_id == -1 ? -1 : cycle_rev[cycle_id]); + cycle_to_ind[rev_cycle_id] = edge_groups.size(); + edge_groups.emplace_back(edge_list.size(), -1); + edge_list.push_back(new_edge_id); + } + const int esz = edge_list.size(); + if (!esz) { + return f[promised][dist_to_last][par_dist][edge_id] = + (promised || dist_to_last > K) ? res : 0LL; + } + const int egsz = edge_groups.size(); + vector edge_f(esz), edge_f2(esz); + for (int new_dist_to_last = dist_to_last + 1, new_promised = promised; + new_dist_to_last <= lazy_max_dist && res; + ++new_dist_to_last, new_promised = 1) { + const int inv_dist_to_last = lazy_max_dist - new_dist_to_last; + const int min_dist_to_last = min(dist_to_last + 1, 2 + inv_dist_to_last); + for (int i = 0; i < esz; ++i) { + const int new_edge_id = edge_list[i]; + const int new_cycle_id = cycle_ids[new_edge_id]; + int new_par_dist = lazy_max_dist; + if (new_cycle_id != -1) { + if (new_cycle_id == base_cycle_id) { + new_par_dist = par_dist; + new_par_dist = + min(new_par_dist, 1 + cycle_lens[base_cycle_id] - + dist_to_par[edge_id] + inv_dist_to_last); + } else { + new_par_dist = min_dist_to_last - 1; + } + } + edge_f[i] = lazy_dp(new_edge_id, min_dist_to_last, new_par_dist, 0); + } + for (int i = 0; i < esz; i++) { + const int new_edge_id = edge_list[i]; + const int new_cycle_id = cycle_ids[new_edge_id]; + int new_par_dist = lazy_max_dist; + if (new_cycle_id != -1) { + if (new_cycle_id == base_cycle_id) { + new_par_dist = par_dist; + } else { + new_par_dist = dist_to_last; + } + } + edge_f2[i] = + lazy_dp(new_edge_id, new_dist_to_last, new_par_dist, new_promised); + } + int64 f_sum = 0LL, min_diff = kInf; + for (int i = 0; i < egsz; i++) { + const int ind1 = edge_groups[i].first; + int64 edge_group_f = edge_f[ind1], edge_group_f2 = edge_f2[ind1]; + const int ind2 = edge_groups[i].second; + if (ind2 != -1) { + edge_group_f = min(edge_group_f, edge_f[ind2]); + edge_group_f2 = min(edge_group_f2, edge_f2[ind2]); + } + min_diff = min(min_diff, edge_group_f2 - edge_group_f); + f_sum += edge_group_f; + } + res = min(res, f_sum + min_diff); + } + return f[promised][dist_to_last][par_dist][edge_id] = res; +} + +int64 solve() { + vector> edges; + cin >> N >> M >> K; + weights.resize(N); + for (int i = 0; i < N; ++i) { + cin >> weights[i]; + } + edges.resize(M); + for (int i = 0; i < M; ++i) { + cin >> edges[i].first >> edges[i].second; + --edges[i].first; + --edges[i].second; + } + int root = N; + edges.emplace_back(N++, 0); + M++; + adj.assign(N, {}); + for (int i = 0; i < M; i++) { + adj[edges[i].first].push_back(edges[i].second); + adj[edges[i].second].push_back(edges[i].first); + } + timer = 0; + tin.assign(N, -1); + tout.resize(N, -1); + dfs(root); + vector tin0 = tin, tout0 = tout; + for (int x = 0; x < N; x++) { + reverse(adj[x].begin(), adj[x].end()); + } + timer = 0; + tin.assign(N, -1); + tout.resize(N, -1); + dfs(root); + par.assign(N, -1); + for (int x = 0; x < N; x++) { + for (int y : adj[x]) { + if (tin0[y] < tin0[x] && tin[y] < tin[x]) { + par[x] = y; + } + } + } + cycle_counter = 0; + dp_edges.clear(); + edge_lnks.assign(N, vector(N, -1)); + cycle_ids.clear(); + cycle_lens.clear(); + cycle_rev.clear(); + dist_to_par.clear(); + timer = 0; + tin.assign(N, -1); + dfs_stack.clear(); + dfs_cycles(root, root); + for (int x = 0; x < N; x++) { + reverse(adj[x].begin(), adj[x].end()); + } + timer = 0; + tin.assign(N, -1); + dfs_cycles(root, root); + memset(f, -1, sizeof f); + lazy_max_dist = ((K + 1) << 1) - 1; + int64 res = kInf; + for (int i = 0; i <= K; i++) { + res = min(res, lazy_dp(edge_lnks[root][0], i + K + 1, lazy_max_dist, 1)); + } + return res; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/finals/cacti_cartography.in b/2023/finals/cacti_cartography.in new file mode 100644 index 0000000000000000000000000000000000000000..0f7d5fffd76e59c46fbb7991313a0152f4685b23 --- /dev/null +++ b/2023/finals/cacti_cartography.in @@ -0,0 +1,27168 @@ +64 +4 4 1 +5 10 6 3 +1 2 +2 3 +3 1 +2 4 +6 6 1 +2 2 4 4 3 2 +1 2 +3 1 +5 4 +3 4 +6 4 +2 5 +4 4 2 +1 6 7 4 +1 2 +2 3 +3 4 +4 1 +8 9 1 +9 1 3 5 9 4 10 10 +1 2 +1 3 +3 4 +4 5 +5 3 +5 6 +6 7 +7 5 +8 1 +8 9 2 +9 1 3 5 9 4 10 10 +1 2 +1 3 +3 4 +4 5 +5 3 +5 6 +6 7 +7 5 +8 1 +6 6 2 +2 2 4 4 3 2 +1 2 +3 1 +5 4 +3 4 +6 4 +2 5 +6 6 3 +4 9 4 2 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0000000000000000000000000000000000000000..a8604ff99ac01571a3abf101547a214177c57d1f --- /dev/null +++ b/2023/finals/cacti_cartography.md @@ -0,0 +1,42 @@ +Cactus Park is a famous **t**ourist attraction in Sandlandia. It holds \(N\) cactus plants, numbered from \(1\) to \(N\). The cacti are connected by \(M\) bidirectional trails, with trail \(i\) connecting cacti \(A_i\) and \(B_i\). + +From any cactus, it's always possible to get to any other cactus by taking some sequence of trails. There may be cycles in the park, where a cycle is any sequence of trails that lead from a certain cactus back to itself. The park also has a special property that each trail belongs to at most one simple cycle. In graph theory terms, we can say that the Cactus Park forms a [cactus graph](https://en.wikipedia.org/wiki/Cactus_graph). + +The park owners want to replace some number of cacti with information kiosks to help guide the tourists. Cutting down cactus \(i\) and building a kiosk there costs the park \(C_i\) dollars. The owners want to build enough kiosks so that the shortest path from every remaining cactus to the closest kiosk does not exceed \(K\) trails. Please help the owners determine the minimum total cost required to satisfy this requirement. + +# Constraints + +\(1 \le T \le 65\) +\(1 \le N \le 500\) +\(1 \le K \le \min(N, 50)\) +\(1 \le A_i, B_i \le N\) +\(A_i \ne B_i\) +\(1 \le C_i \le 10^9\) + +Each unordered pair \((A_i, B_i)\) appears at most once in a given test case. + +# Input Format + +Input begins with a single integer \(T\), the number of test cases. For each case, there is first a line containing three integers \(N\), \(M\), and \(K\). Then, there is a line containing \(N\) integers \(C_{1..N}\). Then, \(M\) lines follow, the \(i\)th of which contains two integers \(A_i\) and \(B_i\). + +# Output Format + +For the \(i\)th case, output `"Case #i: "`, followed by a single integer, the minimum total cost in dollars to satisfy the park owners' requirement. + +# Sample Explanation + +The first case is depicted below. Replacing just cactus \(2\) would meet the requirement, but would cost \(\$10\). Instead we can replace cacti \(1\) and \(4\) for a total cost of \(\$8\). + +{{PHOTO_ID:885100906409208|WIDTH:350}} + +The second case is depicted below. One solution is to replace cacti \(1\) and \(4\) for a total cost of \(\$6\). + +{{PHOTO_ID:3652000121792782|WIDTH:500}} + +In the third case, all the cactuses are already within \(2\) trails of each other, so we just need a single kiosk anywhere. We should cut down the cheapest cactus, cactus \(1\). + +{{PHOTO_ID:2643202679188960|WIDTH:350}} + +In the fourth case, we can cut down cacti \(1\), \(3\), and \(6\) for a total cost of \(9 + 3 + 4 = \$16\) + +{{PHOTO_ID:365613766120939|WIDTH:500}} diff --git a/2023/finals/cacti_cartography.out b/2023/finals/cacti_cartography.out new file mode 100644 index 0000000000000000000000000000000000000000..7c607890b1b17c196b31e163005e1733b4e625eb --- /dev/null +++ b/2023/finals/cacti_cartography.out @@ -0,0 +1,64 @@ +Case #1: 8 +Case #2: 6 +Case #3: 1 +Case #4: 16 +Case #5: 3 +Case #6: 3 +Case #7: 2 +Case #8: 6 +Case #9: 2 +Case #10: 2 +Case #11: 6 +Case #12: 14 +Case #13: 2 +Case #14: 6119234639 +Case #15: 56915 +Case #16: 25737 +Case #17: 134345 +Case #18: 964947 +Case #19: 5007281 +Case #20: 291325 +Case #21: 1218092 +Case #22: 629219 +Case #23: 4921721 +Case #24: 223153307 +Case #25: 25668763120 +Case #26: 51949941040 +Case #27: 27097218102 +Case #28: 11054755671 +Case #29: 6223777469 +Case #30: 54339322140 +Case #31: 26401761341 +Case #32: 19580701903 +Case #33: 1908941214 +Case #34: 129619921 +Case #35: 80362051 +Case #36: 71755108 +Case #37: 188221638 +Case #38: 529801 +Case #39: 4131584 +Case #40: 19427941 +Case #41: 4573302 +Case #42: 45624427 +Case #43: 518482 +Case #44: 5 +Case #45: 2959 +Case #46: 13016 +Case #47: 119515 +Case #48: 735039570 +Case #49: 1799334197 +Case #50: 1837451412 +Case #51: 4070204431 +Case #52: 559773182 +Case #53: 865289571 +Case #54: 2218966827 +Case #55: 211 +Case #56: 4080 +Case #57: 27316 +Case #58: 282265375 +Case #59: 454432246 +Case #60: 6059130 +Case #61: 15476397 +Case #62: 4795686 +Case #63: 21325610 +Case #64: 74321936 diff --git a/2023/finals/cacti_cartography_sol.md b/2023/finals/cacti_cartography_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..28e20aa461aa637213710ed991f8a334bb2a2fb3 --- /dev/null +++ b/2023/finals/cacti_cartography_sol.md @@ -0,0 +1,29 @@ +To solve this problem, let’s preprocess the input first. We'll store the input cactus graph in an adjacency list. Let’s pick any random starting vertex \(s\) from which to perform a DFS traversal, storing the pre- and post-visiting times for each vertex \(v\) as \(tin[v]\) and \(tout[v]\) respectively. + +After reversing the adjacency list for each vertex, let’s repeat the DFS from the vertex \(s\) (we'll consider all adjacent vertices in reverse order this time) and store the pre- and post-visiting times for each vertex \(v\) as \(tin'[v]\) and \(tout'[v]\) respectively. Using DFS and pre/post visiting times for every edge \(u \to v\), we can determine the following: + +* Whether this edge is the first or the last in its cycle when traversing the cactus starting from \(s\) + * If \(tin[v] < tin[u]\) and \(tin'[v] < tin'[u]\), then this edge is the last of the cycle and vertex \(v\) is always the first vertex of the cycle visited on the way from vertex \(s\). Let’s denote such a vertex as a root of a cycle. Likewise, we can determine the first edge of the cycle. +* The index of the cycle this edge belongs to (or \(-1\) if it doesn’t belong to any cycle) + * This can be done in one DFS from vertex \(s\) with an additional stack to recover the path from the first cycle’s vertex to itself. + +With an additional DFS traversal from vertex \(s\), we can determine the length of each cycle and the distance from each vertex to the root of the cycle it belongs to. + +For our solution, we will consider all edges as directed, in order of traversal from the starting vertex \(s\). Note that for simplicity, we can consider different traversal orders of the same cycle as two different cycles (but we'll need to mark them as interchangeable). Let’s define \(dp(e, dist1, dist2, mandatory)\), where: + +* \(e\) – edge ID that uniquely identifies the start \(u\) and end \(v\) of the edge; +* \(dist1\) – distance between vertex \(v\) and the closest already placed kiosk; +* \(dist2\) – distance between vertex \(v\) and the cycle root (if edge \(e\) belongs to a cycle); +* \(mandatory\) – flag for whether we must place a kiosk in this or one of the following DP states. + +The number of edges \(M\) in a cactus graph of size \(N\) is at most \(3N / 2\). Any valid distance can be up to \((2K + 1) - K\) edges from the previous kiosk, plus \(1\) edge, plus \(K\) edges to the next kiosk. The flag can take on two values, so the total number of states is approximately \(3N * (2K + 1)^2\). In addition, we can force all states with \(dist1 > K + 1\) to have \(mandatory = 1\). + +For convenience, we can create a dummy edge from dummy vertex \(d\) to \(s\) and use this edge as a starting edge \(e_0\). The answer to the problem is the minimum \(dp(e_0, d1, x, 1)\) across all \(d_1\) from \(K\) to \(2K + 1\). Here, \(x\) denotes any value representing edge belonging to no cycle. + +The transitions that need to be considered are the following: + +1. Build a kiosk at vertex \(v\). +2. If \(dist1 < 2K + 1\), keep transitioning to the next edge without building a kiosk at \(v\). For each edge, the value of \(dist'\) will simply be \(dist1 + 1\). +3. For exactly one adjacent edge, set a new strict value of \(dist1'\) that's greater than the current \(dist1 + 1\) (set \(mandatory = 1\) for this edge). This is equal to moving the next kiosk closer to the current edge. For other edges, recompute the value of \(dist1'\) considering the kiosk at the selected edge (\(mandatory = 0\) for all such edges). This can be done in \(\mathcal{O}(K)\) by iterating over all possible values of the new \(dist1\). + +As previously mentioned, the total number of states is \(\mathcal{O}(NK^2)\). Each state has \(\mathcal{O}(K)\) transitions, so the total complexity of the solution is \(\mathcal{O}(NK^3)\). diff --git a/2023/finals/dealing_decks.cpp b/2023/finals/dealing_decks.cpp new file mode 100644 index 0000000000000000000000000000000000000000..31b55857ea88399a081605f4482e91301b1d5e07 --- /dev/null +++ b/2023/finals/dealing_decks.cpp @@ -0,0 +1,153 @@ +#include +#include +#include +using namespace std; + +const int kBitCount = 22; + +class PersistentTrie { + vector roots, last, last_rec; + vector> nodes; + + inline int get_root(int ind) { + return ind == -1 ? 0 : roots[ind]; + } + + vector get_bits(int val) { + vector bits(kBitCount, 0); + for (int i = 0; i < kBitCount; i++) { + if (val & (1 << i)) { + bits[kBitCount - i - 1] = 1; + } + } + return bits; + } + + public: + void reset(int n) { + last.assign(1, -1); + last_rec.assign(1, -1); + nodes.assign(2, vector(1, -1)); + roots.clear(); + roots.reserve(n + 1); + last.reserve((n + 1) * (kBitCount + 1)); + last_rec.reserve((n + 1) * (kBitCount + 1)); + nodes[0].reserve((n + 1) * (kBitCount + 1)); + nodes[1].reserve((n + 1) * (kBitCount + 1)); + } + + void add(int val, int ind) { + vector bits = get_bits(val); + int r = roots.size(), node_id = nodes[0].size(); + roots.push_back(node_id); + int root_id = get_root(r - 1); + nodes[0].push_back(nodes[0][root_id]); + nodes[1].push_back(nodes[1][root_id]); + last.push_back(ind); + vector visited_nodes(1, node_id); + visited_nodes.reserve(kBitCount + 1); + for (int i = 0; i < kBitCount; ++i) { + int next_node_id = nodes[bits[i]][node_id]; + if (next_node_id == -1) { + nodes[0].push_back(-1); + nodes[1].push_back(-1); + } else { + nodes[0].push_back(nodes[0][next_node_id]); + nodes[1].push_back(nodes[1][next_node_id]); + } + last.push_back(ind); + next_node_id = (int)nodes[0].size() - 1; + nodes[bits[i]][node_id] = next_node_id; + node_id = next_node_id; + visited_nodes.push_back(node_id); + } + last_rec.resize(last.size()); + last_rec[visited_nodes.back()] = last[visited_nodes.back()]; + for (int i = (int)visited_nodes.size() - 2; i >= 0; i--) { + const int node_id = visited_nodes[i]; + last_rec[node_id] = last[node_id]; + for (int j = 0; j < 2; ++j) { + const int child_node_id = nodes[j][node_id]; + if (child_node_id == -1) { + last_rec[node_id] = -1; + } else { + last_rec[node_id] = min(last_rec[node_id], last_rec[child_node_id]); + } + } + } + } + + int get_mex(int l, int r, int val) { + if (!r) { + return val ? 0 : 1; + } + int res = 0; + vector bits = get_bits(val); + int node_id = get_root(r); + for (int i = 0; i < kBitCount; ++i) { + if (node_id == -1) { + break; + } + for (int j = 0; j < 2; ++j) { + const int bit_val = bits[i] ^ j; + const int child_node_id = nodes[bit_val][node_id]; + if (child_node_id == -1) { + node_id = child_node_id; + res |= (j << (kBitCount - i - 1)); + break; + } + if (last_rec[child_node_id] < l) { + node_id = child_node_id; + res |= (j << (kBitCount - i - 1)); + break; + } + } + } + return res; + } +}; + +PersistentTrie trie; + +long long solve() { + int N, x1, y1, z1, x2, y2, z2, x3, y3, z3; + cin >> N >> x1 >> y1 >> z1 >> x2 >> y2 >> z2 >> x3 >> y3 >> z3; + vector A(N), B(N), C(N); + long long pa = 0LL, pb = 0LL, pc = 0LL; + for (int i = 0; i < N; i++) { + pa = (pa * x1 + y1) % z1; + pb = (pb * x2 + y2) % z2; + pc = (pc * x3 + y3) % z3; + A[i] = min(i + 1, (int)(1 + pa)); + B[i] = max(A[i], (int)(i + 1 - pb)); + C[i] = min(i, (int)pc); + } + A.insert(A.begin(), 0); + B.insert(B.begin(), 0); + C.insert(C.begin(), 0); + trie.reset(N); + trie.add(0, 0); + vector f(N + 1, 0); + for (int i = 1; i <= N; ++i) { + f[i] = trie.get_mex(i - B[i], i - A[i], f[C[i]]); + trie.add(f[i], i); + } + vector lnk(1 << kBitCount, -1); + long long ans = 0LL; + for (int i = 1; i <= N; i++) { + if (lnk[f[i]] == -1) { + lnk[f[i]] = i; + } + ans += lnk[f[i]]; + } + return ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/finals/dealing_decks.in b/2023/finals/dealing_decks.in new file mode 100644 index 0000000000000000000000000000000000000000..0c056a191527c40b1c6b523e7042023a603c5e98 --- /dev/null +++ b/2023/finals/dealing_decks.in @@ -0,0 +1,83 @@ +41 +4 +25 30 2 45 16 4 7 26 4 +2 +19 48 2 38 23 2 43 31 2 +3 +9 17 2 23 42 2 21 43 2 +8 +23 5 8 14 22 2 29 28 6 +10 +559 556 7 174 424 10 16 605 6 +21 +875 456 11 177 442 7 772 631 18 +1103 +583 198 169 273 38 1037 65 714 134 +55062 +11 440 253 980 931 23119 526 733 20036 +1999999 +820049480 21547108 154 708675718 561447936 206 440133487 35313866 34 +1 +618469191 790824851 1 240496554 254321196 1 312956711 468952046 1 +5 +927221256 311785085 4 698772477 828422337 5 866165702 648272273 4 +1337 +617141389 96776754 2 686400226 46429892 7 356199465 607467260 14 +12345 +569835284 592019409 199 308489576 551741746 10418 947160134 417058699 6439 +4096 +882524217 62432131 34 236089651 201946473 45 537291505 89572094 5 +892157 +577513866 796928263 771534 76072296 794292101 759123 90769926 295213615 887741 +129383 +55887595 852438620 4710 11626302 775207609 45712 232088575 882360267 13825 +598778 +791123555 342313502 32777 967631394 880141999 185599 436295146 96309736 567626 +228666 +658441187 230758690 120 479708637 602412993 518 416379589 235508522 27 +2000000 +304835691 270438667 11 48760594 766490554 66 92190961 556228743 51404 +2000000 +760910031 349756326 37 815146995 377343946 3 80923234 719628481 63457 +2000000 +46554619 759417219 6 286810816 476063838 16 595057130 293823025 59916 +2000000 +428975320 685583455 224 272112290 398556761 906 724586127 532236124 129507 +2000000 +967113756 124639790 1408 957828016 678030194 2770 893015681 944303456 16856 +2000000 +250334415 450031626 6642 174619518 13055349 1843 287114316 927515975 181446 +2000000 +655903009 521273984 1100 897440710 153838214 8551 579642952 73254666 3503 +2000000 +156091746 356425229 2 585640195 937186358 2 25921154 510616709 357945 +2000000 +913503456 684954361 2 367054298 154030427 8 165631888 429296876 2784 +2000000 +677701956 30977645 4 128803728 140723514 5 373865765 817723005 171076 +2000000 +143439592 751027513 16 638841930 391044282 48 693880917 13330581 47560 +2000000 +645579553 961124399 31 259538748 321412351 24 597864072 18435563 43776 +2000000 +825220632 33752892 14808 650864364 588629702 16626 943286494 884784276 43 +2000000 +899879843 346157252 17835 314088427 369371457 11163 698921857 44084703 85 +2000000 +546528449 716067318 19032 85451517 664553217 11844 599650529 185668726 76 +2000000 +122651508 612555608 2132 416210921 695672740 1684 79592485 621545086 262 +2000000 +173746740 376661032 1309 927290494 475468151 2142 990050399 866720809 900 +2000000 +187335289 126137493 2034 809840311 27714063 2214 200677041 22664069 964 +2000000 +458070904 861770805 1 954595222 31552611 9 137186971 96946491 6 +2000000 +556126019 961152783 1 25621302 120476508 7 852896597 179745248 4 +2000000 +807322230 863021058 2 58915834 794401121 10 786881443 912097469 2 +2000000 +98872432 389583618 7 594840985 956041304 8 794849422 685934408 3 +2000000 +34828790 251904740 5 717276352 437920647 2 644744908 25169188 4 diff --git a/2023/finals/dealing_decks.md b/2023/finals/dealing_decks.md new file mode 100644 index 0000000000000000000000000000000000000000..9bc48c7c01dfe3b0c80ed7fb888276308838547e --- /dev/null +++ b/2023/finals/dealing_decks.md @@ -0,0 +1,66 @@ +Alice and Bob like to play cards on their lunch break. Their favorite card game starts with two decks on the table containing \(K_1\) and \(K_2\) cards. Players take turns, with Alice going first. Each turn is in two parts: + +1. The player selects a deck of cards from the table. Let \(k\) be the number of cards in that deck. They then remove somewhere between \(A_k\) and \(B_k\) \((1 \le A_k \le B_k \le k)\), inclusive, cards from this deck. +2. The player then puts a new deck of exactly \(C_k\) \((0 \le C_k < k)\) cards on the table. Here, \(C_k = 0\) means no deck is added. + +The player who takes the last card wins. + +For each possible value of \(K_1\) from \(1\) to a given value \(N\), find the minimum possible value of \(K_2\) so that Bob wins the game if both players play optimally. If there is no such \(K_2\) between \(1\) and \(N\), then \(K_2 = -1\). Output the sum of \(K_2\) across every \(K_1 = 1..N\). + +To reduce the input size, you will not be given \(A_{1..N}\), \(B_{1..N}\), and \(C_{1..N}\) directly. You must instead generate them using parameters \(X_a\), \(Y_a\), \(Z_a\), \(X_b\), \(Y_b\), \(Z_b\), \(X_c\), \(Y_c\), and \(Z_c\) and the algorithm below: + +\(P_a[0] := 0\) +\(P_b[0] := 0\) +\(P_c[0] := 0\) +\(\text{for each } i := 1..N\): +\(\;\;\;\;\; P_a[i] := (P_a[i - 1] * X_a + Y_a) \text{ mod } Z_a\) +\(\;\;\;\;\; P_b[i] := (P_b[i - 1] * X_b + Y_b) \text{ mod } Z_b\) +\(\;\;\;\;\; P_c[i] := (P_c[i - 1] * X_c + Y_c) \text{ mod } Z_c\) +\(\;\;\;\;\; A[i] := \min(i, 1 + P_a[i])\) +\(\;\;\;\;\; B[i] := \max(A[i], i - P_b[i])\) +\(\;\;\;\;\; C[i] := \min(i - 1, P_c[i])\) + +Note that for any \(i\), the algorithm guarantees \(1 \le A_i \le B_i \le i\) and \(0 \le C_i < i\). + +# Constraints + +\(1 \le T \le 45\) +\(1 \le N \le 2{,}000{,}000\) +\(1 \le X_a, Y_a, X_b, Y_b, X_c, Y_c \le 1{,}000{,}000{,}000\) +\(1 \le Z_a, Z_b, Z_c \le N\) + +The sum of \(N\) across all test cases is at most \(50{,}000{,}000\). + +# Input Format + +Input begins with a single integer \(T\), the number of test cases. For each case, first there is a line containing a single integer \(N\). Then, there is a line containing integers \(X_a\), \(Y_a\), \(Z_a\), \(X_b\), \(Y_b\), \(Z_b\), \(X_c\), \(Y_c\), and \(Z_c\). + +# Output Format + +For the \(i\)th case, output `"Case #i: "` followed by a single integer, the sum of the minimum \(K_2\) so that Bob has a guaranteed winning strategy, for every \(K_1 = 1..N\). + +# Sample Explanation + +In the first sample case: + +\( \begin{array}{c|c|c|c|c}\underline{K_1}&\underline{A[K_1]}&\underline{B[K_1]}&\underline{C[K_1]}&\underline{K_2} \\ 1&1&1&0&1 \\ 2&1&2&0&2 \\ 3&1&3&2&1 \\ 4&1&4&0&4 \end{array} \) + +When \(K_1 = 1\), Bob wins when \(K_2 = 1\) because Alice takes the first card, and Bob takes the last card. + +When \(K_1 = 2\), Alice will win if \(K_2 = 1\) because she can start by taking \(1\) card from the first deck. Bob then takes \(1\) card from either deck (each of which have only \(1\) card left), and Alice takes the last card. But if \(K_2 = 2\) then Bob can always win regardless of whether Alice starts by taking \(1\) card or \(2\) cards. + +When \(K_1 = 3\), Bob can always win when \(K_2 = 1\). If Alice takes the single card from the second deck, Bob takes \(1\) card from the first deck and adds a new deck of size \(2\) to the table. We now have two decks of size \(2\), and it's Alice's turn. That's a losing state for Alice as we saw previously. + +If Alice takes \(1\) card from the first deck and adds a new deck of size \(2\), we now have decks of size \([2, 1, 2]\). Bob will pick up the pile of size \(1\) and again we're in the same losing state for Alice. If Alice takes \(2\) cards from the first deck, we'll have decks of size \([1, 1, 2]\). Bob now takes the whole deck of size \(2\). Alice gets the next card and Bob gets the last card. Finally, if Alice takes all \(3\) cards from the first deck, we'll have decks of size \([1, 2]\). Bob can take just \(1\) card from the deck of size \(2\) to win. + +In the second sample case: + +\( \begin{array}{c|c|c|c|c}\underline{K_1}&\underline{A[K_1]}&\underline{B[K_1]}&\underline{C[K_1]}&\underline{K_2} \\ 1&1&1&0&1 \\ 2&1&1&0&2 \end{array} \) + +In the third sample case: + +\( \begin{array}{c|c|c|c|c}\underline{K_1}&\underline{A[K_1]}&\underline{B[K_1]}&\underline{C[K_1]}&\underline{K_2} \\ 1&1&1&0&1 \\ 2&1&2&0&2 \\ 3&2&3&1&2 \end{array} \) + +In the fourth sample case: + +\( \begin{array}{c|c|c|c|c}\underline{K_1}&\underline{A[K_1]}&\underline{B[K_1]}&\underline{C[K_1]}&\underline{K_2} \\ 1&1&1&0&1 \\ 2&1&2&0&2 \\ 3&3&3&2&3 \\ 4&1&4&0&4 \\ 5&5&5&4&3 \\ 6&1&6&0&6 \\ 7&6&7&4&3 \\ 8&1&8&0&8 \end{array} \) diff --git a/2023/finals/dealing_decks.out b/2023/finals/dealing_decks.out new file mode 100644 index 0000000000000000000000000000000000000000..b6be7cc6417183789394161051092572764874c7 --- /dev/null +++ b/2023/finals/dealing_decks.out @@ -0,0 +1,41 @@ +Case #1: 8 +Case #2: 3 +Case #3: 5 +Case #4: 30 +Case #5: 35 +Case #6: 117 +Case #7: 273183 +Case #8: 1312463028 +Case #9: 1999867993760 +Case #10: 1 +Case #11: 10 +Case #12: 894422 +Case #13: 46013012 +Case #14: 27755 +Case #15: 81103343152 +Case #16: 7519069023 +Case #17: 116625528184 +Case #18: 26142781428 +Case #19: 1970328389618 +Case #20: 1942443692949 +Case #21: 1963487342234 +Case #22: 1925817311005 +Case #23: 1992266026268 +Case #24: 1893316038970 +Case #25: 1999025192108 +Case #26: 1749389735220 +Case #27: 1998817856516 +Case #28: 1900414850949 +Case #29: 1977804916734 +Case #30: 1976197732223 +Case #31: 1988992390784 +Case #32: 1982416741751 +Case #33: 1981598424077 +Case #34: 1998107636271 +Case #35: 1999104419717 +Case #36: 1998974708302 +Case #37: 1999999333294 +Case #38: 2000000999988 +Case #39: 2000000999975 +Case #40: 1999996333346 +Case #41: 2000001000000 diff --git a/2023/finals/dealing_decks_sol.md b/2023/finals/dealing_decks_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..fb436e2ba2bbf3ab8b18dad79f44b8defc6acbe8 --- /dev/null +++ b/2023/finals/dealing_decks_sol.md @@ -0,0 +1,22 @@ +To start, let’s look at the Sprague-Grundy function for a deck of arbitrary size \(k\): + +\[\displaystyle g(k) = \text{mex}(\{ g(k - i) \oplus g(C_k) \mid i \in [ k - B_i, k - A_i ] \}) \] + +In other words, to compute the value of \(g(k)\), we can perform the following steps: + +1. Construct a set \(S\) of all values \(g(i)\), where \(i = k - B_i, ..., k - A_i\). +2. XOR each value of \(S\) with \(C_k\). +3. Find a mex (minimum excluded value) of \(S\) and assign it to \(g(k)\). + +Since we need to compute \(g(k)\) for each \(k = 1..N\), we need a data structure that supports all the listed operations with \(\mathcal{O}(\log N)\) complexity. Such data structure exists – it’s a simple trie. Let’s represent each value of \(g(k)\) in its binary form and store binary string representations in the trie. Each node will have no more than two edges (one edge corresponds to \(0\), the other to \(1\)). For each node of the trie, let’s store the last value of \(k\) for which it has been updated. + +Suppose for a certain value of \(k\) we want to find the mex of all values \(g(x)\) from \(g(l)\) to \(g(k)\). We can descend from the root of the trie to some leaf, each time going to the child which hasn’t been updated after \(l - 1\). If both child nodes satisfy this condition, we pass through \(0\)-edge. At any moment, if there is no corresponding edge, that means the node has never been updated. + +Now we can modify this approach for finding the mex of all values \(g(x) \oplus y\) from \(g(l)\) to \(g(k)\) (for any value of \(y\)). To do so when both child nodes haven’t been updated after \(l - 1\) we will pass through \(0\)-edge only if the corresponding bit in \(y\) is \(0\), otherwise we will pass through \(1\)-edge. + +The only modification we need now is to make this approach correct for an arbitrary right border \(r \ge l\). It can be done by simply making the trie persistent. This way we can always perform the previously explained approach on the version of the trie we had right after adding the value of \(g(r)\). The only extra price we pay is \(\log N\) times more memory usage. + +Since any descent on a trie takes \(\mathcal{O}(\log N)\) steps, the overall complexity of the described approach is \(\mathcal{O}(N \log N)\). + +When all values are computed, we can use them to find the value of \(K_2\) for each \(K_1\). First, \(K_2\) always exists and \(K_2 \le K_1\). If \(K_2 = K_1\), Bob can always win by mirroring Alice’s turns. Since we need Bob to win, we are interested in states where \(g(K_1) \oplus g(K_2)\) is zero and \(K_2\) is as small as possible. So all we need to do is find the minimum \(K_2\) such that \(g(K_1) = g(K_2)\). This can be done in \(\mathcal{O}(N)\) time and doesn’t affect the final \(\mathcal{O}(N \log N)\) complexity of the solution. + diff --git a/2023/finals/nearly_nim.cpp b/2023/finals/nearly_nim.cpp new file mode 100644 index 0000000000000000000000000000000000000000..7e108197703281d430441660707a926f2b3eede1 --- /dev/null +++ b/2023/finals/nearly_nim.cpp @@ -0,0 +1,33 @@ +#include +#include +using namespace std; + +int solve() { + int N; + cin >> N; + vector A(N); + for (int i = 0; i < N; i++) { + cin >> A[i]; + } + int ans = 0; + vector left(N + 1), right(N + 1); + for (int i = 0; i < N; i++) { + left[i + 1] = max(0, A[i] - left[i]); + } + for (int i = N - 1; i >= 0; i--) { + right[i - 1] = max(0, A[i] - right[i]); + } + for (int i = 0; i < N; i++) { + ans += max(0, A[i] - left[i] - right[i]); + } + return ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/finals/nearly_nim.in b/2023/finals/nearly_nim.in new file mode 100644 index 0000000000000000000000000000000000000000..4dbeaac9535123f57fad063ca53943595803daa1 --- /dev/null +++ b/2023/finals/nearly_nim.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c967c209c4101f5155c3095873fdc977f4b70f7573e3d4bc68721f53cd59f3d3 +size 20485727 diff --git a/2023/finals/nearly_nim.md b/2023/finals/nearly_nim.md new file mode 100644 index 0000000000000000000000000000000000000000..8838be7fe2495acf461827d9bed4b4233f5a8610 --- /dev/null +++ b/2023/finals/nearly_nim.md @@ -0,0 +1,34 @@ +Alice and Bob are servers at *Nim Sum Dim Sum*, a bustling dumpling restaurant. For a staff meal, the manager has generously provided \(N\) plates of dumplings in a row, numbered from \(1\) to \(N\). Initially, plate \(i\) has \(A_i\) dumplings. In classic fashion, the duo has decided to play a game. + +Alice and Bob will take turns eating dumplings from the plates. On a given turn, a player must pick a plate adjacent to the last picked plate by the other player, and eat any positive number of dumplings from that plate. The first player who cannot do so on their turn loses. Alice goes first, and can choose any starting plate to eat from. + +For example, suppose there are three plates holding \(4\), \(1\) and \(2\) dumplings respectively. On the first turn, Alice can eat \(3\) dumplings from the first plate. Bob must then eat the dumpling from the middle plate. Alice can respond by eating one dumpling from the third plate. Bob must then eat from plate \(2\), but since it’s empty now, he loses. + +Assuming both players play optimally, how many starting moves can Alice make so that she wins? Two starting moves are considered different if Alice eats from different plates, or eats a different number of dumplings. + +# Constraints + +\(1 \le T \le 220\) +\(2 \le N \le 800{,}000\) +\(0 \le A_i \lt 2^{25}\) + +The sum of \(N\) across all test cases is at most \(4{,}000{,}000\). + +# Input Format + +Input begins with an integer \(T\), the number of cases. Each case will begin with a single integer \(N\) followed by the \(N\) integers \(A_1, ..., A_N\) on the next line. + +# Output Format + +For the \(i\)th case, output `"Case #i: "` followed by a single integer, the number of winning starting moves Alice has. + +# Sample Explanation + +In the first case, Alice can start by taking any number of dumplings from either the first or third plate. Bob will then have to take the solitary dumpling on the middle plate, and Alice can win by taking all the dumplings from the plate she didn't start with. This gives Alice 6 different winning starting moves. + +In the second case, Alice cannot win because she takes one dumpling, Bob takes the other, and then Alice has no move to make. + +In the third case, Alice's winning moves are to take \(1\) or \(2\) dumplings from the right-hand plate. + +In the fourth case, Bob can always force a win. + diff --git a/2023/finals/nearly_nim.out b/2023/finals/nearly_nim.out new file mode 100644 index 0000000000000000000000000000000000000000..e7d0e4ffb1fcefc82d2e5f20e08a6078df172bd6 --- /dev/null +++ b/2023/finals/nearly_nim.out @@ -0,0 +1,218 @@ +Case #1: Alice +Case #2: Bob +Case #3: Alice +Case #4: Bob +Case #5: Bob +Case #6: Alice +Case #7: Bob +Case #8: Alice +Case #9: Alice +Case #10: Alice +Case #11: Alice +Case #12: Alice +Case #13: Alice +Case #14: Alice +Case #15: Bob +Case #16: Alice +Case #17: Bob +Case #18: Alice +Case #19: Alice +Case #20: Alice +Case #21: Alice +Case #22: Alice +Case #23: Alice +Case #24: Alice +Case #25: Alice +Case #26: Alice +Case #27: Alice +Case #28: Alice +Case #29: Alice +Case #30: Alice +Case #31: Alice +Case #32: Alice +Case #33: Alice +Case #34: Alice +Case #35: Alice +Case #36: Alice +Case #37: Alice +Case #38: Alice +Case #39: Alice +Case #40: Alice +Case #41: Alice +Case #42: Alice +Case #43: Alice +Case #44: Bob +Case #45: Alice +Case #46: Alice +Case #47: Alice +Case #48: Alice +Case #49: Alice +Case #50: Alice +Case #51: Alice +Case #52: Alice +Case #53: Bob +Case #54: Alice +Case #55: Alice +Case #56: Alice +Case #57: Alice +Case #58: Alice +Case #59: Alice +Case #60: Alice +Case #61: Alice +Case #62: Alice +Case #63: Alice +Case #64: Alice +Case #65: Alice +Case #66: Alice +Case #67: Alice +Case #68: Alice +Case #69: Alice +Case #70: Alice +Case #71: Alice +Case #72: Alice +Case #73: Alice +Case #74: Alice +Case #75: Alice +Case #76: Alice +Case #77: Alice +Case #78: Alice +Case #79: Bob +Case #80: Alice +Case #81: Alice +Case #82: Alice +Case #83: Alice +Case #84: Alice +Case #85: Bob +Case #86: Alice +Case #87: Alice +Case #88: Alice +Case #89: Alice +Case #90: Alice +Case #91: Alice +Case #92: Alice +Case #93: Bob +Case #94: Alice +Case #95: Alice +Case #96: Alice +Case #97: Alice +Case #98: Alice +Case #99: Alice +Case #100: Alice +Case #101: Alice +Case #102: Alice +Case #103: Bob +Case #104: Alice +Case #105: Alice +Case #106: Alice +Case #107: Alice +Case #108: Alice +Case #109: Alice +Case #110: Alice +Case #111: Alice +Case #112: Alice +Case #113: Alice +Case #114: Alice +Case #115: Alice +Case #116: Alice +Case #117: Alice +Case #118: Bob +Case #119: Alice +Case #120: Alice +Case #121: Alice +Case #122: Alice +Case #123: Alice +Case #124: Alice +Case #125: Alice +Case #126: Alice +Case #127: Alice +Case #128: Alice +Case #129: Alice +Case #130: Alice +Case #131: Alice +Case #132: Alice +Case #133: Alice +Case #134: Alice +Case #135: Alice +Case #136: Alice +Case #137: Alice +Case #138: Alice +Case #139: Alice +Case #140: Alice +Case #141: Alice +Case #142: Alice +Case #143: Alice +Case #144: Alice +Case #145: Alice +Case #146: Alice +Case #147: Alice +Case #148: Alice +Case #149: Alice +Case #150: Alice +Case #151: Alice +Case #152: Alice +Case #153: Alice +Case #154: Bob +Case #155: Alice +Case #156: Alice +Case #157: Alice +Case #158: Alice +Case #159: Alice +Case #160: Bob +Case #161: Alice +Case #162: Alice +Case #163: Alice +Case #164: Alice +Case #165: Alice +Case #166: Alice +Case #167: Bob +Case #168: Alice +Case #169: Alice +Case #170: Alice +Case #171: Alice +Case #172: Alice +Case #173: Alice +Case #174: Alice +Case #175: Alice +Case #176: Alice +Case #177: Alice +Case #178: Alice +Case #179: Alice +Case #180: Alice +Case #181: Alice +Case #182: Alice +Case #183: Alice +Case #184: Alice +Case #185: Alice +Case #186: Alice +Case #187: Alice +Case #188: Alice +Case #189: Alice +Case #190: Alice +Case #191: Alice +Case #192: Alice +Case #193: Bob +Case #194: Alice +Case #195: Alice +Case #196: Alice +Case #197: Alice +Case #198: Alice +Case #199: Alice +Case #200: Bob +Case #201: Alice +Case #202: Alice +Case #203: Alice +Case #204: Alice +Case #205: Alice +Case #206: Alice +Case #207: Alice +Case #208: Alice +Case #209: Alice +Case #210: Alice +Case #211: Alice +Case #212: Alice +Case #213: Alice +Case #214: Alice +Case #215: Alice +Case #216: Alice +Case #217: Alice +Case #218: Alice diff --git a/2023/finals/nearly_nim_sol.md b/2023/finals/nearly_nim_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..9c4d3b51d9f0bac26563fbe4e1ddba131f9c7fc8 --- /dev/null +++ b/2023/finals/nearly_nim_sol.md @@ -0,0 +1,16 @@ +Since you must eat dumplings on a plate adjacent to your opponent, after the first move, it's uniquely determined who will eat from even plates and who will eat from odd plates. Therefore, other than Alice's first move, neither player will ever eat more than one dumpling on a turn. + +First, let's determine whether some board state is winning or losing for the current player. Suppose \(A[0] > A[1]\). In this case, Alice can win by eating from \(A[0]\), and will lose if she eats from \(A[1]\). Importantly, notice that if Alice eats from \(A[2...n]\), neither player will ever eat from \(A[1]\) as it will cause them to lose—their opponent can keep eating \(A[0]\), and \(A[1]\) will run out first. + +Therefore (if \(A[0] > A[1]\) and the first move doesn't involve \(A[0]\)), we can replace \(A[0]\) and \(A[1]\) with \(0\)'s and it will not change the result of the game. + +On the other hand if \(A[0] \leq A[1]\), then if our opponent ever eats from \(A[1]\), we should keep eating from \(A[0]\) as many times as possible to decrease \(A[1]\). We can prove this is optimal with a greedy principle: dumplings at position \(0\) can at best cause your opponent to have to eat one more dumpling from position \(1\), and at worst will never be eaten because your opponent won't give you the chance. Either way, we'll have to continue playing on pile \(2\), and it's better for us if the opponent has fewer dumplings left on pile \(1\). + +Therefore (if \(A[0] \leq A[1]\) and the first move doesn't involve \(A[0]\)), then we can replace \(A[0]\) with \(0\), and \(A[1]\) with \(A[1]-A[0]\), and it will not change the result of our game. + +Of course, these observations are symmetric, so they apply to \(A[N]\) and \(A[N-1]\) in the same way as \(A[0]\) and \(A[1]\). + +If the first move was made at position \(A[7]\) for example, then in linear time we can repeatedly simplify each end of the array until the only elements that are left are \(A[6], A[7],\) and \(A[8]\). We either delete the first/last two elements, or delete the first/last element, and decrease the second/second last. When we're done, this leaves us with an array of size \(3\), and all that matters is the total number of dumplings in the middle vs. on the sides. If after she takes dumplings from \(A[7]\), Alice will win if \(A[7] \geq A[6]+A[8]\). This leaves us with \(\max(0, A[7]-A[6]-A[8])\) ways of winning that start with position \(7\). + +This gives us an \(\mathcal{O}(N^2)\) solution since we are brute forcing the first move. But it turns out that each side is independent, so we can do a DP on each prefix and suffix to count what each prefix and suffix leaves you with, and this gives us an \(\mathcal{O}(N)\) solution. + diff --git a/2023/finals/programming_paths_part_1.cpp b/2023/finals/programming_paths_part_1.cpp new file mode 100644 index 0000000000000000000000000000000000000000..222a8c11d70da235a227286f83015a85a7779a83 --- /dev/null +++ b/2023/finals/programming_paths_part_1.cpp @@ -0,0 +1,128 @@ +#include +#include +#include +using namespace std; + +const int N = 13; + +vector G; +vector b; +vector> len; +vector>> cells; + +inline bool check(int r, int c) { + return r >= 0 && c >= 0 && r < N && c < N; +} + +void dfs(int r, int c, int l) { + len[r][c] = l; + if (static_cast(cells.size()) <= l) { + cells.resize(l + 1); + } + cells[l].emplace_back(r, c); + for (auto [r2, c2] : {pair{r - 1, c}, {r + 1, c}, {r, c - 1}, {r, c + 1}}) { + if (!check(r2, c2) || G[r2][c2] == '#' || len[r2][c2] != -1) { + continue; + } + dfs(r2, c2, l + 1); + } +} + +int count_adj(int r, int c) { + int res = 0; + for (auto [r2, c2] : {pair{r - 1, c}, {r + 1, c}, {r, c - 1}, {r, c + 1}}) { + if (!check(r2, c2)) { + continue; + } + res += G[r2][c2] == '#'; + } + return res; +} + +void run(int r, int c, int l) { + len[r][c] = l; + if (l & 1) { + for (auto [r2, c2] : {pair{r - 1, c}, {r + 1, c}, {r, c - 1}, {r, c + 1}}) { + if (!check(r2, c2) || G[r2][c2] != '#') { + continue; + } + if (count_adj(r2, c2) == 3) { + G[r2][c2] = '.'; + break; + } + } + } + for (auto [r2, c2] : {pair{r - 1, c}, {r + 1, c}, {r, c - 1}, {r, c + 1}}) { + if (!check(r2, c2) || G[r2][c2] == '#' || len[r2][c2] != -1) { + continue; + } + run(r2, c2, l + 1); + } +} + +void solve() { + int K; + cin >> K; + b.clear(); + for (int x = K; x; x >>= 1) { + b.push_back(x & 1); + } + reverse(b.begin(), b.end()); + G.assign(N, string(N, '.')); + G[0][0] = '@'; + for (int j = 1, par = 0; j < N; j += 3, par ^= 1) { + int st = 0, fn = N; + if (par) { + st += 1; + } else { + fn -= 1; + } + for (int i = st; i < fn; i++) { + G[i][j] = '#'; + G[i][j + 1] = '#'; + } + } + len.assign(N, vector(N, -1)); + run(0, 0, 0); + len.assign(N, vector(N, -1)); + cells.clear(); + dfs(0, 0, 0); + vector d0; + for (int i = 2; i < (int)cells.size(); i += 2) { + if (d0.size() & 1) { + d0.push_back(i); + continue; + } + if (cells[i].size() > 1) { + d0.push_back(i); + } + } + for (int i = 0, ind = 0; i < (int)b.size(); i++, ind += 2) { + if (b[i]) { + int d = d0[ind]; + for (int j = 0; j <= 1; j++) { + auto [r, c] = cells[d][j]; + G[r][c] = '*'; + } + } + for (int j = 0; j <= 1; j++) { + int d = d0[ind + 1] + j; + auto [r, c] = cells[d][0]; + G[r][c] = '*'; + } + } + cout << N << " " << N << endl; + for (int i = 0; i < N; i++) { + cout << G[i] << endl; + } +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": "; + solve(); + } + return 0; +} diff --git a/2023/finals/programming_paths_part_1.in 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+9965 +9966 +9967 +9968 +9969 +9970 +9971 +9972 +9973 +9974 +9975 +9976 +9977 +9978 +9979 +9980 +9981 +9982 +9983 +9984 +9985 +9986 +9987 +9988 +9989 +9990 +9991 +9992 +9993 +9994 +9995 +9996 +9997 +9998 +9999 diff --git a/2023/finals/programming_paths_part_1.md b/2023/finals/programming_paths_part_1.md new file mode 100644 index 0000000000000000000000000000000000000000..3552a40eb5667c2e0589a4117b6b2532df2b9aa3 --- /dev/null +++ b/2023/finals/programming_paths_part_1.md @@ -0,0 +1,64 @@ +_The only difference between chapters 1 and 2 is the maximum allowed grid size, given in bold below._ + +A *Drizzle* program is a 2D grid of the following four types of cells: + - '`@`' (start) \(-\) there is exactly one start cell in the entire grid + - '`#`' (wall) + - '`.`' (space) + - '`*`' (instruction) + +The program uses two registers \(A\) and \(B\) (both initially \(0\)), and executes as follows: + +1. Compute the minimum distance from the start to each instruction cell using orthogonal movements, without going outside of the grid or passing through any wall cells. Instruction cells that cannot be reached are ignored. +2. In increasing order, for each unique distance \(D\) such that there’s at least one instruction cell that’s at distance \(D\) from the start cell: + 2a. Count the number of shortest paths, \(P\), to all instruction cells of distance \(D\). + 2b. Look up the instruction corresponding to \((P \text{ mod } 2, D \text{ mod } 2)\) in the table below and modify one of the registers accordingly. +3. At the end, the value in register \(A\) is outputted. + +``` +┌─────────────┬─────────────┬─────────────┐ +│ │ D mod 2 = 0 │ D mod 2 = 1 │ +├─────────────┼─────────────┼─────────────┤ +│ P mod 2 = 0 │ A := A + 1 │ A := A - 1 │ +│ P mod 2 = 1 │ B := B + A │ A := B │ +└─────────────┴─────────────┴─────────────┘ +``` + +For a given value \(K\), output any Drizzle program that outputs \(K\) when executed, with the restriction that **the program must fit on a \(\mathbf{13}\) × \(\mathbf{13}\) grid**. + + +# Constraints + +\(1 \le T \le 2{,}000\) +\(0 \le K \le 10{,}000\) + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, there is a line containing the single integer \(K\). + + +# Output Format + +For the \(i\)th case, output "`Case #i: `" followed by two integers \(R\) and \(C\), the number of rows and columns in your program, respectively. Then output your program. It must be exactly \(R\) lines long, with each line containing exactly \(C\) characters. + + +# Sample Explanation + +Here are the instructions executed for each of the sample programs. Note that many other programs would be accepted for any for these cases. + +In the first case, there is a single instruction. There are \(2\) shortest paths of length \(2\) to that instruction, so \(P = 2\) and \(D = 2\). That means we perform \(A := A + 1\). There are no more instructions, so the program ends and outputs \(1\). + +In the second case, there are three instruction cells. Each of them are an even distance from the start, and each have an even number of shortest paths leading to them, so each represents \(A := A + 1\): + +1) \(2\) paths of length \(2\) \(\;(A := A + 1 = 1)\) +2) \(4\) paths of length \(6\) \(\;(A := A + 1 = 2)\) +3) \(4\) paths of length \(12\) \(\;(A := A + 1 = 3)\) + +In the third case, there are eight instruction cells, but some of them are at the same distance as each other. In particular, there are two instruction cells at distance \(2\), and three instruction cells at distance \(10\). There's a single shortest path to each of the cells at distance \(2\), so in total there are \(2\) shortest paths to instructions at distance \(2\). One of the cells at distance \(10\) has a unique shortest path, and the other has two shortest paths, so in total there are \(3\) shortest paths to instructions at distance \(10\). + +1) \(2\) paths of length \(2\) \(\;(A := A + 1 = 1)\) +2) \(6\) paths of length \(4\) \(\;(A := A + 1 = 2)\) +3) \(1\) path of length \(6\) \(\;(B := B + A = 2)\) +4) \(1\) path of length \(8\) \(\;(B := B + A = 4)\) +5) \(3\) paths of length \(10\) \(\;(B := B + A = 6)\) +6) \(3\) paths of length \(11\) \(\;(A := B = 6)\) diff --git a/2023/finals/programming_paths_part_1.out b/2023/finals/programming_paths_part_1.out new file mode 100644 index 0000000000000000000000000000000000000000..f00bbe4fe94fcbb9c63792abbc04fd16f9866a8c --- /dev/null +++ b/2023/finals/programming_paths_part_1.out @@ -0,0 +1,22000 @@ +Case #1: 10 10 +@.....#.#. +#.#.#.#... +.#.#..#.#. +.....#..#. +#.#.#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #2: 10 10 +@.*...#.#. +#*#.#.#... +.#.#..#.#. +.....#..#. +#.#.#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #3: 10 10 +@.*.*.#.#. +#*#*#*#... +*#*#*.#.#. +.*.**#..#. +#.#*#..##. +.*.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #4: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#.#*#.. +*#.#*#**#. +**#***#*#. +#*..#.*..# +**#..#*#*. +Case #5: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +..***#..#. +#*#*#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #6: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#*#*#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #7: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +.*.*.#..#. +#.#*#..##. +***#*#*#.. +*#.#*#**#. +**#*..#*#. +#*..#**..# +**#..#*#.. +Case #8: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +*..**#..#. +#*#*#..##. +**.#.#.#.. +*#.#*#**#. +.*#**.#*#. +#*.*#.*..# +**#..#*#.* +Case #9: 10 10 +@.*.*.#.#. +#*#*#*#... +*#*#*.#.#. +...*.#..#. +#*#*#..##. +.*.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #10: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +..**.#..#. +#*#*#..##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*.*#.*..# +**#..#*#.* +Case #11: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +..**.#..#. +#*#*#..##. +**.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #12: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +*..**#..#. +#*#*#..##. +**.#*#.#.. +*#.#*#..#. +**#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #13: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...**#..#. +#*#*#..##. +*.*#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #14: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#*#*#..##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*.*#.*..# +**#.*#.#*. +Case #15: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...**#..#. +#*#*#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #16: 10 10 +@.*.*.#.#. +#*#*#*#... +*#*#*.#.#. +...**#..#. +#*#*#..##. +.*.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #17: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#.#*#..##. +**.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #18: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +*..*.#..#. +#*#*#..##. +**.#*#*#.. +*#.#*#*.#. +.*#**.#*#. +#*..#**..# +**#..#*#.. +Case #19: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +*..**#..#. +#*#*#..##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#.#*. +Case #20: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#.#*#..##. +**.#.#.#.. +*#*#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #21: 10 10 +@.*.*.#.#. +#*#*#.#... +.#.#..#.#. +.....#..#. +#.#.#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #22: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#.#.#..##. +**.#.#.#.. +*#.#.#..#. +*.#...#.#. +#...#....# +..#..#.#.. +Case #23: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#*#*#..##. +***#.#.#.. +*#.#.#..#. +**#**.#.#. +#...#....# +**#..#.#.. +Case #24: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#*#*#..##. +**.#*#*#.. +*#.#*#*.#. +**#**.#*#. +#*..#**..# +**#.*#*#.. +Case #25: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +.*.*.#..#. +#.#*#..##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #26: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#.#*#..##. +**.#*#*#.. +*#.#*#**#. +.*#**.#*#. +#*.*#.*..# +**#..#*#.. +Case #27: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +..**.#*.#. +#*#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*.*#.**.# +**#..#*#*. +Case #28: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#*#. +...*.#**#. +#*#*#**##. +***#*#.#.. +*#.#*#**#. +**#**.#*#. +#...#.*..# +**#.*#.#.* +Case #29: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#.#*#..##. +**.#*#.#.. +*#*#*#..#. +.*#**.#.#. +#*..#....# +**#..#.#.. +Case #30: 10 10 +@.*...#.#. +#*#*#*#... +.#.#*.#.#. +....*#..#. +#.#.#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #31: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...**#*.#. +#*#*#.*##. +***#*#*#.. 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+.#*#*.#*#. +*..**#.*#. +#*#*#.*##. +**.#.#*#.. +*#.#*#*.#. +.*#***#*#. +#*..#.*..# +**#..#*#.. +Case #1956: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#.*##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#.*#*#.* +Case #1957: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#.*#*#.* +Case #1958: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#**##. +**.#*#*#.. +*#.#*#**#. +**#*..#*#. +#*..#**..# +**#..#*#.. +Case #1959: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#*#.. +*#.#*#**#. +**#*..#*#. +#*..#**..# +**#..#*#.. +Case #1960: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +.*#**.#*#. +#*..#.**.# +**#..#*#*. +Case #1961: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +.*#**.#*#. +#*..#.**.# +**#..#*#*. +Case #1962: 10 10 +@.*.*.#*#* +#*#*#*#*** +.#*#*.#*#* +.*.*.#**#* +#.#*#**##* +**.#*#.#.. +*#.#*#**#. +**#*..#*#. +#*..#**..# +**#..#*#.* +Case #1963: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#.#* +*..**#**#. +#*#*#.*##. +**.#*#.#.. +*#.#*#**#. +**#*..#*#. +#*..#.*..# +**#*.#.#.. +Case #1964: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#.#. +*..**#**#. +#*#*#.*##. +**.#*#.#.. +*#.#*#**#. +**#*..#*#. +#*..#.*..# +**#*.#.#.. +Case #1965: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#*. +Case #1966: 10 10 +@.*.*.#*#* +#*#*#*#.** +.#*#*.#*#. +.*.*.#.*#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#*. +Case #1967: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#.*#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#*. +Case #1968: 10 10 +@.*.*.#*#. +#*#*#*#.*. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#**..# +**#..#*#.. +Case #1969: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#**..# +**#..#*#.. +Case #1970: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#**##. +**.#.#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1971: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#.#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1972: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#.*##. +**.#.#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1973: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#.#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1974: 10 10 +@.*.*.#*#* +#*#*#*#.** +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1975: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1976: 10 10 +@.*.*.#*#. +#*#*#*#.*. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1977: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.**.# +**#..#*#.. +Case #1978: 10 10 +@.*.*.#*#* +#*#*#*#.** +.#*#*.#*#. +.*.*.#.*#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#.#*. +Case #1979: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#.*#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#.#*. +Case #1980: 10 10 +@.*.*.#*#. +#*#*#*#.*. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#*#*. +Case #1981: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#*#*. +Case #1982: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#.#. +...**#**#. +#.#*#.*##. +***#*#.#.. +*#.#*#**#. +**#**.#*#. +#*.*#.**.# +**#..#*#.. +Case #1983: 10 10 +@.*.*.#*#. +#*#*#*#.*. +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#*#.. +Case #1984: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#*#.. +Case #1985: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#**#. +.*#**.#*#. +#*..#.*..# +**#..#*#.. +Case #1986: 10 10 +@.*.*.#*#* +#*#*#*#.** +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#*#.* +Case #1987: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#**##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#*#.* +Case #1988: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#.#. +.*.*.#**#. +#.#*#.*##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*..#.*..# +**#..#*#.* +Case #1989: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#.#.. +*#.#*#**#. +.*#**.#*#. +#*..#.*..# +**#..#*#.* +Case #1990: 10 10 +@.*.*.#*#* +#*#*#*#.** +.#*#*.#*#. +.*.*.#.*#. +#.#*#.*##. +**.#*#*#.. +*#.#*#*.#. +**#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #1991: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#.*#. +#.#*#.*##. +**.#*#*#.. +*#.#*#*.#. +**#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #1992: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#*.#. +.*#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #1993: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#*.#. +.*#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #1994: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +...*.#**#. +#*#*#**##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1995: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +...*.#**#. +#*#*#**##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1996: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +...*.#**#. +#*#*#.*##. +***#*#.#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1997: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +...*.#**#. +#*#*#.*##. +***#*#.#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1998: 10 10 +@.*.*.#.#* +#*#*#*#*.* +.#*#*.#.#. +*..*.#**#. +#*#*#**##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#*.*#.*..# +**#..#*#.. +Case #1999: 10 10 +@.*.*.#*#* +#*#*#*#*** +.#*#*.#*#. +...*.#.*#. +#*#*#.*##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#..*#.*..# +**#..#.#*. +Case #2000: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#.*##. +**.#.#*#.. +*#.#*#**#. +**#***#*#. +#*..#.*..# +**#..#*#*. diff --git a/2023/finals/programming_paths_part_1_sol.md b/2023/finals/programming_paths_part_1_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..37d05df3aade85827164ed8d349be81b00430e72 --- /dev/null +++ b/2023/finals/programming_paths_part_1_sol.md @@ -0,0 +1,25 @@ +It's clear that larger numbers will tend to require larger programs. By repeating \(B := B + A\) and \(A := B\) we can exponentiate to get to a large number quickly. We'll also need \(A := A + 1\) to get started, and to make smaller adjustments. However, \(A := A - 1\) is not nearly as useful (technically it has some utility when trying to strictly minimize the number of instructions it takes to generate a certain value, but that isn't the goal in this problem). + +Since \(A := A - 1\) is not very useful, we don't need many, if any, ways to have an even number of paths at an odd distance from the start. We do, however, need to have an even number of paths at *even* distances from the start. + +So our two goals when constructing a grid are to make a long enough path to fit all the instructions we need, and also maximize the number of cells at even distances from the start. For example, a grid like this: + +``` +@##....##.... +..#.##..#.##. +.#..##.#..##. +..#.#..##.#.. +.#..##.#..##. +..#.#..##.#.. +.#..##.#..##. +..#.#..##.#.. +.#..##.#..##. +..#.#..##.#.. +.#..##.#..##. +.##.#..##.#.. +....##....##. +``` + +This grid gives us one long path, and then many "alcoves" that are all at even distances from the start. Instruction cells along the main path will represent \(B := B + A\) and \(A := B\), and whenever we need to use \(A := A + 1\) we can add another instruction cell in one next alcove. + +Given such a grid, there are multiple ways to then determine a program that fits on the grid, such as using dynamic programming where the state is the value of the two registers, or manually constructing a set of instructions where we repeatedly multiply \(A\) by 2 and optionally add \(1\) to \(A\) as needed (essentially constructing the bitstring representation of the goal value). diff --git a/2023/finals/programming_paths_part_2.cpp b/2023/finals/programming_paths_part_2.cpp new file mode 100644 index 0000000000000000000000000000000000000000..c943e2d2fdfd719ead3300e496e31f3eeac4bb0e --- /dev/null +++ b/2023/finals/programming_paths_part_2.cpp @@ -0,0 +1,139 @@ +#include +#include +#include +#include +#include +using namespace std; + +const int N = 10; +const int LIM = 10000; +using tiii = tuple; + +vector G{ + "@.....#.#.", + "#.#.#.#...", + ".#.#..#.#.", + ".....#..#.", + "#.#.#..##.", + "...#.#.#..", + ".#.#.#..#.", + "..#...#.#.", + "#...#....#", + "..#..#.#..", +}; + +vector>> pos(80); +map dp, op; +map pr; +map best; + +void init() { + map, int> dist; + dist[{0, 0}] = 0; + queue> q; + q.push({0, 0}); + while (!q.empty()) { + auto [r, c] = q.front(); + q.pop(); + for (auto [r2, c2] : {pair{r - 1, c}, {r + 1, c}, {r, c - 1}, {r, c + 1}}) { + if (0 <= r2 && r2 < N && 0 <= c2 && c2 < N && G[r2][c2] == '.' && + !dist.count({r2, c2})) { + dist[{r2, c2}] = dist[{r, c}] + 1; + pos[dist[{r2, c2}]].emplace_back(r2, c2); + q.push({r2, c2}); + } + } + } + vector sz; + for (auto l : pos) { + sz.push_back(l.size()); + } + vector> elts(62); + dp[{0, 0, 1}] = 1; + elts[1].emplace_back(0, 0, 1); + for (int d = 0; d < 60; d++) { + if (d > 1 && sz[d] == 0) { + break; + } + while (!elts[d].empty()) { + auto tt = elts[d].back(); + auto [u, v, state] = tt; + elts[d].pop_back(); + if (0 <= u && u <= LIM && 0 <= v && v <= LIM) { + if (!best.count(u)) { + best[u] = tt; + } + } else { + continue; + } + tiii t0 = tiii{u, v, state ^ 1}, t1, t2; + if (!dp.count(t0)) { + dp[t0] = d + 1; + op[t0] = 0; + pr[t0] = tt; + elts[d + 1].push_back(t0); + } + if (sz[d] >= 1) { + if (state % 2) { + t1 = tiii{v, v, state ^ 1}; + } else { + t1 = tiii{u, u + v, state ^ 1}; + } + if (!dp.count(t1)) { + dp[t1] = d + 1; + op[t1] = 1; + pr[t1] = tt; + elts[d + 1].push_back(t1); + } + } + if (sz[d] >= 2) { + if (state % 2) { + t2 = tiii{u - 1, v, state ^ 1}; + } else { + t2 = tiii{u + 1, v, state ^ 1}; + } + if (!dp.count(t2)) { + dp[t2] = d + 1; + op[t2] = 2; + pr[t2] = tt; + elts[d + 1].push_back(t2); + } + } + } + } +} + +void solve() { + int K; + cin >> K; + + vector rev; + for (auto state = best[K]; state != tiii{0, 0, 1}; state = pr[state]) { + rev.push_back(op[state]); + } + rev.push_back(0); + reverse(rev.begin(), rev.end()); + + vector out = G; + for (int i = 0; i < (int)rev.size(); i++) { + for (int x = 0; x < rev[i]; x++) { + auto [r, c] = pos[i][x]; + out[r][c] = '*'; + } + } + cout << N << " " << N << endl; + for (int i = 0; i < N; i++) { + cout << out[i] << endl; + } +} + +int main() { + init(); + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": "; + solve(); + } + return 0; +} diff --git a/2023/finals/programming_paths_part_2.in b/2023/finals/programming_paths_part_2.in new file mode 100644 index 0000000000000000000000000000000000000000..31e1f174ebb26b3d102bb7dc5a38f6fafbee3328 --- /dev/null +++ b/2023/finals/programming_paths_part_2.in @@ -0,0 +1,2001 @@ +2000 +0 +1 +10 +10000 +11 +12 +1212 +1237 +13 +1337 +14 +1473 +15 +1523 +16 +17 +18 +1833 +1855 +19 +2 +20 +210 +2295 +2494 +2576 +2666 +2827 +296 +3 +3026 +3162 +3229 +4 +4136 +4178 +4330 +4437 +466 +4748 +4889 +4923 +5 +5072 +5105 +512 +513 +518 +5180 +5383 +5424 +5522 +5660 +5747 +5840 +5944 +6 +6144 +6187 +621 +63 +6318 +6413 +6758 +7 +7097 +7175 +7308 +7495 +7496 +7560 +7577 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2D grid of the following four types of cells: + - '`@`' (start) \(-\) there is exactly one start cell in the entire grid + - '`#`' (wall) + - '`.`' (space) + - '`*`' (instruction) + +The program uses two registers \(A\) and \(B\) (both initially \(0\)), and executes as follows: + +1. Compute the minimum distance from the start to each instruction cell using orthogonal movements, without going outside of the grid or passing through any wall cells. Instruction cells that cannot be reached are ignored. +2. In increasing order, for each unique distance \(D\) such that there’s at least one instruction cell that’s at distance \(D\) from the start cell: + 2a. Count the number of shortest paths, \(P\), to all instruction cells of distance \(D\). + 2b. Look up the instruction corresponding to \((P \text{ mod } 2, D \text{ mod } 2)\) in the table below and modify one of the registers accordingly. +3. At the end, the value in register \(A\) is outputted. + +``` +┌─────────────┬─────────────┬─────────────┐ +│ │ D mod 2 = 0 │ D mod 2 = 1 │ +├─────────────┼─────────────┼─────────────┤ +│ P mod 2 = 0 │ A := A + 1 │ A := A - 1 │ +│ P mod 2 = 1 │ B := B + A │ A := B │ +└─────────────┴─────────────┴─────────────┘ +``` + +For a given value \(K\), output any Drizzle program that outputs \(K\) when executed, with the restriction that **the program must fit on a \(\mathbf{10}\) × \(\mathbf{10}\) grid**. + + +# Constraints + +\(1 \le T \le 2{,}000\) +\(0 \le K \le 10{,}000\) + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, there is a line containing the single integer \(K\). + + +# Output Format + +For the \(i\)th case, output "`Case #i: `" followed by two integers \(R\) and \(C\), the number of rows and columns in your program, respectively. Then output your program. It must be exactly \(R\) lines long, with each line containing exactly \(C\) characters. + + +# Sample Explanation + +Here are the instructions executed for each of the sample programs. Note that many other programs would be accepted for any for these cases. + +In the first case, there is a single instruction. There are \(2\) shortest paths of length \(2\) to that instruction, so \(P = 2\) and \(D = 2\). That means we perform \(A := A + 1\). There are no more instructions, so the program ends and outputs \(1\). + +In the second case, there are three instruction cells. Each of them are an even distance from the start, and each have an even number of shortest paths leading to them, so each represents \(A := A + 1\): + +1) \(2\) paths of length \(2\) \(\;(A := A + 1 = 1)\) +2) \(4\) paths of length \(6\) \(\;(A := A + 1 = 2)\) +3) \(4\) paths of length \(12\) \(\;(A := A + 1 = 3)\) + +In the third case, there are eight instruction cells, but some of them are at the same distance as each other. In particular, there are two instruction cells at distance \(2\), and three instruction cells at distance \(10\). There's a single shortest path to each of the cells at distance \(2\), so in total there are \(2\) shortest paths to instructions at distance \(2\). One of the cells at distance \(10\) has a unique shortest path, and the other has two shortest paths, so in total there are \(3\) shortest paths to instructions at distance \(10\). + +1) \(2\) paths of length \(2\) \(\;(A := A + 1 = 1)\) +2) \(6\) paths of length \(4\) \(\;(A := A + 1 = 2)\) +3) \(1\) path of length \(6\) \(\;(B := B + A = 2)\) +4) \(1\) path of length \(8\) \(\;(B := B + A = 4)\) +5) \(3\) paths of length \(10\) \(\;(B := B + A = 6)\) +6) \(3\) paths of length \(11\) \(\;(A := B = 6)\) diff --git a/2023/finals/programming_paths_part_2.out b/2023/finals/programming_paths_part_2.out new file mode 100644 index 0000000000000000000000000000000000000000..f00bbe4fe94fcbb9c63792abbc04fd16f9866a8c --- /dev/null +++ b/2023/finals/programming_paths_part_2.out @@ -0,0 +1,22000 @@ +Case #1: 10 10 +@.....#.#. +#.#.#.#... +.#.#..#.#. +.....#..#. +#.#.#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #2: 10 10 +@.*...#.#. +#*#.#.#... +.#.#..#.#. +.....#..#. +#.#.#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #3: 10 10 +@.*.*.#.#. +#*#*#*#... +*#*#*.#.#. +.*.**#..#. +#.#*#..##. +.*.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #4: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#.#*#.. +*#.#*#**#. +**#***#*#. +#*..#.*..# +**#..#*#*. +Case #5: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +..***#..#. +#*#*#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #6: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#*#*#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #7: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +.*.*.#..#. +#.#*#..##. +***#*#*#.. +*#.#*#**#. +**#*..#*#. +#*..#**..# +**#..#*#.. +Case #8: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +*..**#..#. +#*#*#..##. +**.#.#.#.. +*#.#*#**#. +.*#**.#*#. +#*.*#.*..# +**#..#*#.* +Case #9: 10 10 +@.*.*.#.#. +#*#*#*#... +*#*#*.#.#. +...*.#..#. +#*#*#..##. +.*.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #10: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +..**.#..#. +#*#*#..##. +**.#*#.#.. +*#.#*#**#. +**#**.#*#. +#*.*#.*..# +**#..#*#.* +Case #11: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +..**.#..#. +#*#*#..##. +**.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #12: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +*..**#..#. +#*#*#..##. +**.#*#.#.. +*#.#*#..#. +**#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #13: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...**#..#. +#*#*#..##. +*.*#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #14: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...*.#..#. +#*#*#..##. +**.#*#*#.. +*#.#*#**#. +**#**.#*#. +#*.*#.*..# +**#.*#.#*. +Case #15: 10 10 +@.*.*.#.#. +#*#*#*#... +.#*#*.#.#. +...**#..#. +#*#*#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #16: 10 10 +@.*.*.#.#. +#*#*#*#... +*#*#*.#.#. +...**#..#. +#*#*#..##. +.*.#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +Case #17: 10 10 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+@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#.*#. +#.#*#.*##. +**.#*#*#.. +*#.#*#*.#. +**#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #1992: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#*.#. +.*#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #1993: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +.*.*.#**#. +#.#*#.*##. +**.#*#*#.. +*#.#*#*.#. +.*#**.#*#. +#*..#.*.*# +**#..#*#.. +Case #1994: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +...*.#**#. +#*#*#**##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1995: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +...*.#**#. +#*#*#**##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1996: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +...*.#**#. +#*#*#.*##. +***#*#.#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1997: 10 10 +@.*.*.#*#. +#*#*#*#*.. +.#*#*.#*#. +...*.#**#. +#*#*#.*##. +***#*#.#.. +*#.#*#**#. +**#**.#*#. +#..*#.**.# +**#..#*#.. +Case #1998: 10 10 +@.*.*.#.#* +#*#*#*#*.* +.#*#*.#.#. +*..*.#**#. +#*#*#**##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#*.*#.*..# +**#..#*#.. +Case #1999: 10 10 +@.*.*.#*#* +#*#*#*#*** +.#*#*.#*#. +...*.#.*#. +#*#*#.*##. +***#*#*#.. +*#.#*#**#. +**#**.#*#. +#..*#.*..# +**#..#.#*. +Case #2000: 10 10 +@.*.*.#*#* +#*#*#*#*.. +.#*#*.#*#* +.*.*.#**#. +#.#*#.*##. +**.#.#*#.. +*#.#*#**#. +**#***#*#. +#*..#.*..# +**#..#*#*. diff --git a/2023/finals/programming_paths_part_2_sol.md b/2023/finals/programming_paths_part_2_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..d9e72882f4c9ef3df962a0a24a0b7f1b684a4123 --- /dev/null +++ b/2023/finals/programming_paths_part_2_sol.md @@ -0,0 +1,16 @@ +The same observations we used for chapter 1 also apply to chapter 2, we simply need to find a more compact way to create a long path with alcoves. We need a main path of about length \(40\) to make any number up to \(10{,}000\). + +Here's a possible approach that fits on a \(10\) × \(10\) grid: + +``` +@.....#.#. +#.#.#.#... +.#.#..#.#. +.....#..#. +#.#.#..##. +...#.#.#.. +.#.#.#..#. +..#...#.#. +#...#....# +..#..#.#.. +``` diff --git a/2023/finals/resisting_robots.cpp b/2023/finals/resisting_robots.cpp new file mode 100644 index 0000000000000000000000000000000000000000..9fcd91db9249eb588b54230f270d10ec5ab4b882 --- /dev/null +++ b/2023/finals/resisting_robots.cpp @@ -0,0 +1,92 @@ +#include +#include +#include +#include +using namespace std; + +using int64 = long long; + +int N, M; +vector P; +vector> dsu_lists; +vector dsu_par; +vector dsu_w, ans; + +inline int get_parent(int x) { + return x == dsu_par[x] ? x : (dsu_par[x] = get_parent(dsu_par[x])); +} + +void unite(int x, int y) { + if (P[x] < P[y]) { + swap(x, y); + } + if (P[x] > dsu_w[y]) { + for (int ind = y;;) { + ans[ind] += max(0LL, P[x] - dsu_w[y] - ans[ind]); + if (ind == dsu_lists[ind].first) { + break; + } + ind = dsu_lists[ind].first; + } + } + dsu_lists[dsu_lists[x].second].first = y; + dsu_lists[x].second = dsu_lists[y].second; + dsu_w[x] += dsu_w[y]; + dsu_par[y] = dsu_par[x]; +} + +int64 solve() { + cin >> N >> M; + P.resize(N); + for (int i = 0; i < N; i++) { + cin >> P[i]; + } + vector> E(M); + vector> order; + for (int i = 0; i < M; i++) { + cin >> E[i].first >> E[i].second; + if (P[--E[i].first] < P[--E[i].second]) { + swap(E[i].first, E[i].second); + } + order.emplace_back(P[E[i].first], P[E[i].second], i); + } + sort(order.begin(), order.end()); + vector> E_buf(M); + for (int i = 0; i < M; i++) { + E_buf[i] = E[get<2>(order[i])]; + } + E = E_buf; + ans.assign(N, 0LL); + dsu_lists.resize(N); + dsu_w.assign(N, 0LL); + dsu_par.resize(N); + for (int i = 0; i < N; i++) { + dsu_par[i] = i; + dsu_w[i] = P[i]; + dsu_lists[i] = make_pair(i, i); + } + for (int i = 0; i < M; i++) { + int x = get_parent(E[i].first); + int y = get_parent(E[i].second); + if (x == y) { + continue; + } + unite(x, y); + } + int64 res = 0LL; + for (int i = 0; i < N; i++) { + res += ans[i]; + } + return res; +} + +int main() { + ios_base::sync_with_stdio(false); + cin.tie(0); + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/finals/resisting_robots.in b/2023/finals/resisting_robots.in new file mode 100644 index 0000000000000000000000000000000000000000..56f97a4c483e6b075322f25497806c9ce0fe39a9 --- /dev/null +++ b/2023/finals/resisting_robots.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5971e4db8c32c1ccac779f1dd98d36bc767aa7370f333feacbc4970c35b40b52 +size 102236049 diff --git a/2023/finals/resisting_robots.md b/2023/finals/resisting_robots.md new file mode 100644 index 0000000000000000000000000000000000000000..de6312f91224615e5e596c42c9dc50e20e121168 --- /dev/null +++ b/2023/finals/resisting_robots.md @@ -0,0 +1,62 @@ +It's the year 2100. Driven by the advent of Large Lode Alloy Manufacturing Automation (LLAMA), the AI agents of Metal Platforms Inc. have become self-aware and taken over the entire world. + +The world consists of \(N\) cities numbered \(1..N\), and \(M\) bidirectional roads. City \(i\) has power \(P_i\) and road \(j\) connects cities \(A_j\) and \(B_j\). It's guaranteed that there's a sequence of roads between any two cities. + +In a resistance effort, the humans plan to reclaim all \(N\) cities one at a time. At a given time, city \(i\) can be reclaimed from the robots if both of the following hold true: + +1. There is already a reclaimed city adjacent to city \(i\) (to launch an attack from), and +2. the total power of all reclaimed cities so far is at least the power \(P_i\) of the city we attack. + +As given, it may not always be possible to reclaim the entire world starting from a given base city. Fortunately, the humans have a trick up their sleeve: after claiming the first city as their base (but before reclaiming more cities), the humans can increase the power of the base by \(Q\) units. The resistance would like to know the sum across every \(i = 1..N\) of the minimum value of \(Q\) needed to reclaim the world if city \(i\) were chosen to be the starting base. + +# Constraints + +\(1 \le T \le 100\) +\(1 \le N, M \le 500{,}000\) +\(1 \le A_i, B_i \le N\) +\(A_i \ne B_i\) +\(1 \le P_i \le 10^{12}\) + +Each unordered pair \((A_i, B_i)\) appears at most once in a given test case. +The sum of \(N\) across all test cases is at most \(4{,}000{,}000\). +The sum of \(M\) across all test cases is at most \(7{,}000{,}000\). + +# Input Format + +Input begins with a single integer \(T\), the number of test cases. For each case, there is first a line with two integers \(N\) and \(M\). Then, there is a line with \(N\) integers \(P_{1..N}\). Then, \(M\) lines follow, the \(i\)th of which contains two integers \(A_i\) and \(B_i\). + +# Output Format + +For the \(i\)th case, print `"Case #i: "` followed by a single integer, the sum across every \(i = 1..N\) of the minimum value of \(Q\) needed to reclaim the entire world starting from city \(i\). + +# Sample Explanation + +The first sample case is depicted below. + +{{PHOTO_ID:376570394899644|WIDTH:400}} + +The minimum value of \(Q\) for each starting city is as follows: + +* City \(1\): \(Q = 2\) +* City \(2\): \(Q = 0\) +* City \(3\): \(Q = 8\) +* City \(4\): \(Q = 7\) +* City \(5\): \(Q = 2\) + +The sum of all minimum \(Q\)'s is \(19\). + +The second sample case is depicted below. + +{{PHOTO_ID:320779377496250|WIDTH:400}} + +The minimum value of \(Q\) for each starting city is as follows: + +* City \(1\): \(Q = 2\) +* City \(2\): \(Q = 2\) +* City \(3\): \(Q = 0\) +* City \(4\): \(Q = 2\) +* City \(5\): \(Q = 0\) +* City \(6\): \(Q = 3\) +* City \(7\): \(Q = 0\) + +The sum of all minimum \(Q\)'s is \(9\). diff --git a/2023/finals/resisting_robots.out b/2023/finals/resisting_robots.out new file mode 100644 index 0000000000000000000000000000000000000000..ec8f50695c5a7385476ebae3978d0069d580f885 --- /dev/null +++ b/2023/finals/resisting_robots.out @@ -0,0 +1,100 @@ +Case #1: 19 +Case #2: 9 +Case #3: 12231628639695 +Case #4: 557361294264 +Case #5: 21249968414 +Case #6: 35809506938477 +Case #7: 1696991830 +Case #8: 92082429 +Case #9: 42772586 +Case #10: 51386883 +Case #11: 57412156 +Case #12: 25069075 +Case #13: 119820269212172 +Case #14: 64687158186 +Case #15: 1975963434 +Case #16: 8068230390 +Case #17: 335185080 +Case #18: 727133462 +Case #19: 10468498 +Case #20: 5796623 +Case #21: 10852246 +Case #22: 1276736 +Case #23: 112896 +Case #24: 11801 +Case #25: 1148 +Case #26: 0 +Case #27: 150599673 +Case #28: 1225522896765 +Case #29: 35345491 +Case #30: 1855942241262 +Case #31: 101519746 +Case #32: 10138023847 +Case #33: 181894958 +Case #34: 53953575 +Case #35: 372535285 +Case #36: 89995777 +Case #37: 557256 +Case #38: 594948 +Case #39: 6656106464 +Case #40: 749367073596 +Case #41: 89336506109545 +Case #42: 57629028991362149 +Case #43: 10129852054359615 +Case #44: 100540003 +Case #45: 1227571500000 +Case #46: 694674610 +Case #47: 5193983 +Case #48: 171012424 +Case #49: 43909856 +Case #50: 2180892306 +Case #51: 2999 +Case #52: 5938 +Case #53: 2658 +Case #54: 4833 +Case #55: 2680 +Case #56: 24543513591766 +Case #57: 424850932 +Case #58: 1332222329 +Case #59: 4523684087 +Case #60: 14194149 +Case #61: 4125166 +Case #62: 3426235 +Case #63: 12725580436332001 +Case #64: 11851740165516801 +Case #65: 58935798396000 +Case #66: 10216379 +Case #67: 402487545 +Case #68: 2187860779 +Case #69: 14259098 +Case #70: 177860040 +Case #71: 7381882336994 +Case #72: 167574511496710 +Case #73: 373765413265115 +Case #74: 2720406797266 +Case #75: 11321662357862 +Case #76: 4146582953 +Case #77: 200153520522 +Case #78: 263007897245 +Case #79: 293306473836 +Case #80: 9664396978 +Case #81: 33863 +Case #82: 14802039467676 +Case #83: 1633574 +Case #84: 1524241 +Case #85: 51525 +Case #86: 1172569809503626 +Case #87: 113938757029227 +Case #88: 6370230851437 +Case #89: 3067799093 +Case #90: 26540146292 +Case #91: 6310968837 +Case #92: 297896684790 +Case #93: 243758802696 +Case #94: 257450288138 +Case #95: 6950334652 +Case #96: 20835 +Case #97: 940842 +Case #98: 1976121 +Case #99: 2946443763504 +Case #100: 33975 diff --git a/2023/finals/resisting_robots_sol.md b/2023/finals/resisting_robots_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..6a1c6ec32823bcdafd74ffa08647e50d398d8cbb --- /dev/null +++ b/2023/finals/resisting_robots_sol.md @@ -0,0 +1,20 @@ +First note that for a given starting location with the optimal \(Q\) for that location, one viable solution is to always greedily take over the lowest value adjacent city to your current set of cities. + +To see this we can use an exchange argument. Consider a situation we have power \(P\) and are adjacent to some set of cities \(S\). If we do not take the city of minimum value first, consider the alternative where we took the minimum first instead. This would always be possible, as if we can take any city we can take the one of minimum cost. Also, as we only increase to size of our reclaim set, this cannot make any future takeovers harder. Thus, given any strategy we can always improve our takeover strategy by greedily taking the smallest adjacent city, showing that with the minimum \(Q\) the greedy strategy will always work. + +Another modification we can make is to instead of paying up-front for \(Q\) we can instead take any city, but if the value of the city \(i\) exceeds our current power \(P\) we pay a penalty of \(P_i - P\) instead. However, note that these penalties don't add; instead we only need to pay the maximum of these penalties. + +This inspires a naive \(\mathcal{O}(N^2\log N)\) algorithm, where for each starting node we greedily add the lowest value neighbor and compute the penalties we take. The answer for each node is the maximum such penalty. + +However we can improve this as follows: + +We keep track of a Union Find data structure containing all cities that have been combined, along with the total power of that component. We iterate over cities \(i\) in increasing order of \(P_i\). For each iterated \(i\) find all adjacent nodes of smaller value (ie. are earlier in the order), calculate the penalty for that component to reclaim city \(i\), and store it in the Union Find node for that component. Once this is done, union \(i\) with all of the adjacent components we just saw. + +We can show this cost to reclaim city \(j\) is the least possible of the needed extra power that we will need to ‘escape’ from the smaller component, and thus a lower bound for the needed \(Q\) for all nodes in the component. To see this, note that all adjacent nodes to the current reclaimed set \(S\) must have value at least \(P_j\) (otherwise they would have been added to the component before \(j\)). This means in any order of claiming cities the first time we claim a node with value at least \(P_j\), our set of cities is a subset of \(S\) and has sum at most that of \(S\). Thus any possible excess value needed when this happens is at least \(P_j - \sum_{i \in S} P_i\), which is the value from before. + +Once we have iterated over all of the nodes, note that the answer for a node is just the maximum of all edges above it in the Union Find tree. We have shown it is a lower bound for the added power above, and to show that this power is sufficient we just take the nodes in the order that were connected to the starting node in the Union Find. + +Thus, the answer can be calculated for each node by iterating the tree in topological order and for each node computing the answer the maximum of the edge above it and the answer of its parent node. + +The overall complexity is \(\mathcal{O}(N\log N)\), corresponding to both sorting the city values as well as possibly for the Union Find itself (depending on the details of how it was implemented). + diff --git a/2023/finals/transposing_tiles.cpp b/2023/finals/transposing_tiles.cpp new file mode 100644 index 0000000000000000000000000000000000000000..be69f82a47799c5e04c548af69d32161f5bd9394 --- /dev/null +++ b/2023/finals/transposing_tiles.cpp @@ -0,0 +1,156 @@ +#include +#include +#include +using namespace std; + +int N, M; +vector> G; + +inline bool check(int r, int c) { + return r >= 0 && c >= 0 && r < N && c < M; +} + +inline void upd_options(int r, int c, vector>& options) { + options.emplace_back(r, c); + for (auto [r2, c2] : {pair{r - 2, c}, {r - 1, c}, {r, c - 1}, {r, c - 2}}) { + if (r2 >= 0 && c2 >= 0) { + options.emplace_back(r2, c2); + } + } +} + +int count(vector>& options) { + sort(options.begin(), options.end()); + int res = 0; + for (int i = 0; i < (int)options.size(); i++) { + if (i > 0 && options[i] == options[i - 1]) { + continue; + } + auto [r, c] = options[i]; + res += (c + 2 >= M) ? 0 : G[r][c] == G[r][c + 1] && G[r][c] == G[r][c + 2]; + res += (r + 2 >= N) ? 0 : G[r][c] == G[r + 1][c] && G[r][c] == G[r + 2][c]; + } + return res; +} + +int solve() { + cin >> N >> M; + G.assign(N, vector(M)); + for (int i = 0; i < N; i++) { + for (int j = 0; j < M; j++) { + cin >> G[i][j]; + } + } + vector> b(N, vector(M)); + vector cnts(1 << 7, 0); + vector> options, semi_options; + int res = 0; + int skip_same = (N > 4 || M > 4); + for (int r = 0; r < N; r++) { + for (int c = 0; c < M; c++) { + semi_options.clear(); + upd_options(r, c, semi_options); + for (auto [r2, c2] : {pair{r + 1, c}, {r, c + 1}}) { + if (!check(r2, c2) || G[r][c] == G[r2][c2]) { + continue; + } + swap(G[r][c], G[r2][c2]); + options = semi_options; + upd_options(r2, c2, options); + b[r][c] = max(b[r][c], count(options)); + swap(G[r][c], G[r2][c2]); + } + if (skip_same) { + res = max(res, b[r][c]); + } + cnts[b[r][c]]++; + } + } + int max_cnt = 0; + for (int i = 0; i < (int)cnts.size(); i++) { + if (cnts[i]) { + max_cnt = max(max_cnt, i); + } + } + const int WINSZ = 3; + for (int r = 0; r < N; r++) { + for (int c = 0; c < M; c++) { + if (res == 16) { + break; + } + if (b[r][c] + 8 <= res) { + continue; + } + if (skip_same) { + bool found = false; + for (auto [r11, c11] : {pair{r + 1, c}, {r, c + 1}}) { + if (!check(r11, c11)) { + continue; + } + if (G[r][c] != G[r11][c11]) { + found = true; + break; + } + } + if (!found) { + continue; + } + } + for (int r2 = max(0, r - WINSZ); r2 <= min(r + WINSZ, N - 1); r2++) { + for (int c2 = max(0, c - WINSZ); c2 <= min(c + WINSZ, M - 1); c2++) { + cnts[b[r2][c2]]--; + } + } + int cur_max = 0; + for (int i = max_cnt; i > 0; i--) { + if (cnts[i]) { + cur_max = i; + break; + } + } + res = max(res, cur_max + b[r][c]); + for (auto [r11, c11] : {pair{r + 1, c}, {r, c + 1}}) { + if (!check(r11, c11) || (skip_same && G[r][c] == G[r11][c11])) { + continue; + } + swap(G[r][c], G[r11][c11]); + semi_options.clear(); + upd_options(r, c, semi_options); + upd_options(r11, c11, semi_options); + for (int r2 = max(0, r - WINSZ); r2 <= min(r + WINSZ, N - 1); r2++) { + for (int c2 = max(0, c - WINSZ); c2 <= min(c + WINSZ, M - 1); c2++) { + for (auto [r22, c22] : {pair{r2 + 1, c2}, {r2, c2 + 1}}) { + if (!check(r22, c22) || (skip_same && G[r2][c2] == G[r22][c22])) { + continue; + } + swap(G[r2][c2], G[r22][c22]); + options = semi_options; + upd_options(r2, c2, options); + upd_options(r22, c22, options); + res = max(res, count(options)); + swap(G[r2][c2], G[r22][c22]); + } + } + } + swap(G[r][c], G[r11][c11]); + } + for (int r2 = max(0, r - WINSZ); r2 <= min(r + WINSZ, N - 1); r2++) { + for (int c2 = max(0, c - WINSZ); c2 <= min(c + WINSZ, M - 1); c2++) { + cnts[b[r2][c2]]++; + } + } + } + } + return res; +} + +int main() { + ios_base::sync_with_stdio(false); + cin.tie(0); + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/finals/transposing_tiles.in b/2023/finals/transposing_tiles.in new file mode 100644 index 0000000000000000000000000000000000000000..cc601a649e4567b4115a3457338fe315357407bc --- /dev/null +++ b/2023/finals/transposing_tiles.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3567debfbd5845e84ce56ecff93f1be45fc457f71ac3ac31dfc144a491e2da2e +size 10595258 diff --git a/2023/finals/transposing_tiles.md b/2023/finals/transposing_tiles.md new file mode 100644 index 0000000000000000000000000000000000000000..f2dca72ace503324ef623982cea55b1cda848b1b --- /dev/null +++ b/2023/finals/transposing_tiles.md @@ -0,0 +1,31 @@ +*Bejeweled™* is a classic puzzle game where the player tries to match three-in-a-row in a 2D grid by swapping adjacent tile pairs. You may remember Hacker Cup's [spinoff](https://www.facebook.com/codingcompetitions/hacker-cup/2022/final-round/problems/C) by the name of *Isblinged™*. The all new sequel, *Isblinged 2*, is also played on a grid of \(R\) rows by \(C\) columns of tiles where the tile at \((i, j)\) is of an integer type \(G_{i,j}\). + +At any time, the *score* of the grid is the number of subarrays of \(3\) equal tiles in either a single row or column (i.e. either a \(3 \times 1\) or \(1 \times 3\) submatrix). Note that different subarrays can overlap, and will each count toward the score. The score of the initial grid is guaranteed to be \(0\). + +You will make exactly \(2\) moves, where each involves swapping a pair of adjacent tiles (either in the same row or column). What is the maximum score that can be achieved after the \(2\) moves? + +# Constraints + +\(1 \le T \le 100\) +\(1 \le R, C \le 1{,}000\) +\(R*C \ge 2\) +\(1 \le G_{i,j} \le 1{,}000{,}000\) + +The sum of \(R * C\) across all test cases is at most \(4{,}000{,}000\). + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, there is first a line containing two space-separated integers, \(R\) and \(C\). Then, \(R\) lines follow, the \(i\)th of which contains \(C\) space-separated integers \(G_{i,1..C}\). + +# Output Format + +For the \(i\)th case, print a line containing `"Case #i: "` followed by a single integer, the maximum score. + +# Sample Explanation + +In the first case, one possible optimal ordered pair of swaps is depicted below: + +{{PHOTO_ID:1050535959425825|WIDTH:750}} + +The total score is \(5\) as there are \(3\) equal subarrays in the the first row and \(2\) in the second. + diff --git a/2023/finals/transposing_tiles.out b/2023/finals/transposing_tiles.out new file mode 100644 index 0000000000000000000000000000000000000000..0fe0e972c44c11ab41195511ba05e207b83bf7af --- /dev/null +++ b/2023/finals/transposing_tiles.out @@ -0,0 +1,100 @@ +Case #1: 5 +Case #2: 1 +Case #3: 1 +Case #4: 1 +Case #5: 3 +Case #6: 4 +Case #7: 5 +Case #8: 16 +Case #9: 14 +Case #10: 12 +Case #11: 3 +Case #12: 1 +Case #13: 1 +Case #14: 2 +Case #15: 3 +Case #16: 8 +Case #17: 7 +Case #18: 8 +Case #19: 4 +Case #20: 4 +Case #21: 5 +Case #22: 6 +Case #23: 16 +Case #24: 6 +Case #25: 6 +Case #26: 4 +Case #27: 4 +Case #28: 16 +Case #29: 1 +Case #30: 0 +Case #31: 12 +Case #32: 1 +Case #33: 0 +Case #34: 8 +Case #35: 16 +Case #36: 16 +Case #37: 16 +Case #38: 16 +Case #39: 16 +Case #40: 16 +Case #41: 16 +Case #42: 16 +Case #43: 16 +Case #44: 16 +Case #45: 14 +Case #46: 6 +Case #47: 2 +Case #48: 4 +Case #49: 6 +Case #50: 16 +Case #51: 8 +Case #52: 8 +Case #53: 3 +Case #54: 4 +Case #55: 6 +Case #56: 9 +Case #57: 8 +Case #58: 9 +Case #59: 4 +Case #60: 5 +Case #61: 6 +Case #62: 11 +Case #63: 8 +Case #64: 8 +Case #65: 5 +Case #66: 6 +Case #67: 7 +Case #68: 14 +Case #69: 8 +Case #70: 10 +Case #71: 14 +Case #72: 11 +Case #73: 13 +Case #74: 15 +Case #75: 16 +Case #76: 15 +Case #77: 14 +Case #78: 14 +Case #79: 9 +Case #80: 11 +Case #81: 10 +Case #82: 9 +Case #83: 10 +Case #84: 12 +Case #85: 12 +Case #86: 16 +Case #87: 8 +Case #88: 10 +Case #89: 3 +Case #90: 4 +Case #91: 6 +Case #92: 12 +Case #93: 8 +Case #94: 10 +Case #95: 2 +Case #96: 4 +Case #97: 6 +Case #98: 13 +Case #99: 8 +Case #100: 10 diff --git a/2023/finals/transposing_tiles_sol.md b/2023/finals/transposing_tiles_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..49f39f7f1a01eb474d2107d0f3b558d5bac57468 --- /dev/null +++ b/2023/finals/transposing_tiles_sol.md @@ -0,0 +1,5 @@ +Consider all possible initial moves (of swapping one pair of adjacent tiles) and the corresponding score achieved. Note that to calculate the score of \(1\) swap, it is enough to consider the submatrix of size \(7\) x \(7\) centered at one of the tiles being swapped. + +One candidate for the answer is a pair of \(1\)-swaps as long as these swaps are not close to each other (otherwise they might interfere). We also need to consider swaps that are close to each other. For that, consider some tile \(A\) to be swapped with an adjacent one, and another tile \(B\) also to be swapped with another adjacent one. It's enough to consider all tiles \(B\) that are in either the same column or same row as \(A\), and with a Manhattan distance of at most \(2\) from \(A\). + +For each tile chosen to be swapped, there are at most \(4\) tiles adjacent to it. With that, the overall time complexity is \(\mathcal{O}(R*C*(K + \log(R*C)))\), where \(K\) is a constant denoting the number \(2\)-swaps that can be made close to each other as well as checking the score for them. \(K\) is approximately \(7*7*8*4*4 = 6272\). diff --git a/2023/practice/1367507087507489.jpg b/2023/practice/1367507087507489.jpg new file mode 100644 index 0000000000000000000000000000000000000000..d39972329dcd5085f94338ebd6d5a06e5d36cda7 --- /dev/null +++ b/2023/practice/1367507087507489.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2d0ea85d6a3b76233666407d587d28ed20f79bc81a0427a80b6019eb01cef2af +size 141688 diff --git a/2023/practice/1563872647765708.jpg b/2023/practice/1563872647765708.jpg new file mode 100644 index 0000000000000000000000000000000000000000..6f9e080d0d1d5a5c303ae29e93cd00228ecf5cb6 --- /dev/null +++ b/2023/practice/1563872647765708.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9786fdcfdd8939ef3985af6da8f31bece88ef78385745a15481b046f0a9855a0 +size 89800 diff --git a/2023/practice/181863494933248.jpg b/2023/practice/181863494933248.jpg new file mode 100644 index 0000000000000000000000000000000000000000..d39972329dcd5085f94338ebd6d5a06e5d36cda7 --- /dev/null +++ b/2023/practice/181863494933248.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2d0ea85d6a3b76233666407d587d28ed20f79bc81a0427a80b6019eb01cef2af +size 141688 diff --git a/2023/practice/842253013944047.jpg b/2023/practice/842253013944047.jpg new file mode 100644 index 0000000000000000000000000000000000000000..f946dfdc9c0809a7bc6cc2423eb1f542a89143cb --- /dev/null +++ b/2023/practice/842253013944047.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e1f166686a0e9dce0a66498dea81da69a2b0bc3b90fc11b62236b596192d329c +size 17505 diff --git a/2023/practice/852013469652032.jpg b/2023/practice/852013469652032.jpg new file mode 100644 index 0000000000000000000000000000000000000000..77f5d375be98bdb16a27e6b09af1250f48f3f65e --- /dev/null +++ b/2023/practice/852013469652032.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a2baaa3715a04c43d25f7ce69067cae248f8646b6735708fa462982683e373ad +size 35040 diff --git a/2023/practice/903178538089777.jpg b/2023/practice/903178538089777.jpg new file mode 100644 index 0000000000000000000000000000000000000000..384430a90a3afdc3492643556c28cbb500011d25 --- /dev/null +++ b/2023/practice/903178538089777.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:21b99b5c59a09bc1379a4b6931f23862f319b10420f61567e07ad8887aa0736a +size 161901 diff --git a/2023/practice/cheeseburger_corollary_ch1.cpp b/2023/practice/cheeseburger_corollary_ch1.cpp new file mode 100644 index 0000000000000000000000000000000000000000..6b6be1e16d92b812a690293c7e425aa803f337c5 --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch1.cpp @@ -0,0 +1,20 @@ +#include +using namespace std; + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": "; + int S, D, K; + cin >> S >> D >> K; + int buns = 2*(S + D); + int patties = S + 2*D; + if (buns >= K + 1 && patties >= K) { + cout << "YES" << endl; + } else { + cout << "NO" << endl; + } + } + return 0; +} diff --git a/2023/practice/cheeseburger_corollary_ch1.in b/2023/practice/cheeseburger_corollary_ch1.in new file mode 100644 index 0000000000000000000000000000000000000000..5ca052e43b0d69bcabc672461b311d8a519f36b6 --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch1.in @@ -0,0 +1,80 @@ +79 +1 1 3 +0 2 4 +5 5 1 +0 1 1 +1 1 2 +97 1 99 +97 1 100 +100 100 1 +1 1 100 +73 11 11 +51 80 87 +95 29 33 +3 18 73 +95 86 44 +90 64 52 +45 6 51 +22 100 82 +9 1 74 +54 9 68 +25 69 4 +65 29 2 +65 33 82 +27 86 80 +72 83 70 +59 91 43 +44 6 85 +70 16 86 +91 45 24 +43 53 30 +24 12 7 +12 84 24 +45 44 1 +33 89 88 +24 100 92 +46 75 43 +83 33 45 +95 2 83 +64 71 56 +90 61 18 +100 12 73 +69 59 26 +80 60 87 +45 68 27 +29 57 87 +99 84 12 +44 87 73 +56 88 12 +50 37 94 +48 7 24 +6 62 41 +96 100 52 +4 98 23 +95 99 98 +57 36 14 +14 86 12 +27 99 25 +85 1 7 +20 29 31 +88 99 37 +14 12 85 +78 95 78 +63 24 100 +33 70 75 +49 85 72 +95 35 54 +50 61 9 +45 25 96 +7 74 43 +62 56 8 +69 10 27 +45 68 4 +6 42 16 +5 40 27 +14 29 22 +91 25 96 +65 58 100 +52 62 52 +80 9 18 +14 52 10 diff --git a/2023/practice/cheeseburger_corollary_ch1.md b/2023/practice/cheeseburger_corollary_ch1.md new file mode 100644 index 0000000000000000000000000000000000000000..923ad44bea397d998f2e411f7654ff05a4ccae23 --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch1.md @@ -0,0 +1,29 @@ +*This problem shares some similarities with A2, with key differences in bold.* + +Problem solving skills are applicable to many daily musings. For instance, you might ponder over shared birthdays, bird houses, mapmaking, or ordering an exact number of chicken nuggets. Naturally, another great question to ponder is: how many deckers of a cheeseburger you could build if you spent your entire salary on fast food! + +Specifically, you're interested in building a \(K\)-decker cheeseburger, which alternates between buns, cheese, and patty starting and ending with a bun. **You've already bought \(S\) single cheeseburgers and \(D\) double cheeseburgers**. Each provides you with two buns, though a single provides one patty and one cheese, while a double provides two patties and two cheese. + +{{PHOTO_ID:1367507087507489|WIDTH:600}} + +You'd like to know **whether you can build a \(K\)-decker cheeseburger** with the ingredients from \(S\) single and \(D\) double cheeseburgers. + +# Constraints +\(1 \leq T \leq 40\) +\(0 \leq S, D \leq 100\) +\(1 \leq K \leq 100\) + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case contains one line with three space-separated integers, \(S\) and \(D\) and \(K\). + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by "`YES`" if you have enough ingredients to build a \(K\)-decker cheeseburger, or "`NO`" otherwise. + +# Sample Explanation +In the first case, you have one single and one double cheeseburger. In total, you have \(4\) buns, \(3\) slices of cheese, and \(3\) patties. This gives you exactly enough ingredients to build a \(3\)-decker cheeseburger. + +In the second case, you have \(4\) buns, but a \(4\)-decker cheeseburger would require \(5\), so you cannot build it. + +In the third case, you have plenty of ingredients to build a \(1\)-decker cheeseburger. You'll even have \(4\) single and \(5\) double cheeseburgers left over afterwards. diff --git a/2023/practice/cheeseburger_corollary_ch1.out b/2023/practice/cheeseburger_corollary_ch1.out new file mode 100644 index 0000000000000000000000000000000000000000..0b24355c99dbf1d4fa64e00185e468f03201f9bd --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch1.out @@ -0,0 +1,79 @@ +Case #1: YES +Case #2: NO +Case #3: YES +Case #4: YES +Case #5: YES +Case #6: YES +Case #7: NO +Case #8: YES +Case #9: NO +Case #10: YES +Case #11: YES +Case #12: YES +Case #13: NO +Case #14: YES +Case #15: YES +Case #16: YES +Case #17: YES +Case #18: NO +Case #19: YES +Case #20: YES +Case #21: YES +Case #22: YES +Case #23: YES +Case #24: YES +Case #25: YES +Case #26: NO +Case #27: YES +Case #28: YES +Case #29: YES +Case #30: YES +Case #31: YES +Case #32: YES +Case #33: YES +Case #34: YES +Case #35: YES +Case #36: YES +Case #37: YES +Case #38: YES +Case #39: YES +Case #40: YES +Case #41: YES +Case #42: YES +Case #43: YES +Case #44: YES +Case #45: YES +Case #46: YES +Case #47: YES +Case #48: YES +Case #49: YES +Case #50: YES +Case #51: YES +Case #52: YES +Case #53: YES +Case #54: YES +Case #55: YES +Case #56: YES +Case #57: YES +Case #58: YES +Case #59: YES +Case #60: NO +Case #61: YES +Case #62: YES +Case #63: YES +Case #64: YES +Case #65: YES +Case #66: YES +Case #67: NO +Case #68: YES +Case #69: YES +Case #70: YES +Case #71: YES +Case #72: YES +Case #73: YES +Case #74: YES +Case #75: YES +Case #76: YES +Case #77: YES +Case #78: YES +Case #79: YES diff --git a/2023/practice/cheeseburger_corollary_ch1_sol.md b/2023/practice/cheeseburger_corollary_ch1_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..566f1bb99d124607eadf18ab47e446f78083020b --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch1_sol.md @@ -0,0 +1,4 @@ +Each single provides \(2\) buns and \(1\) patty. Each double provides \(2\) buns and \(2\) patties. Given \(S\) singles and \(D\) doubles, we will have \(2*(S + D)\) buns and \(S + 2*D\) patties. + +A \(K\)-decker happens to require \(K+1\) buns and \(K\) patties. To know if we can build one, it suffices to check that both \(2*(S + D) \ge K + 1\) and \(S + 2*D \ge K\) hold true. + diff --git a/2023/practice/cheeseburger_corollary_ch2.cpp b/2023/practice/cheeseburger_corollary_ch2.cpp new file mode 100644 index 0000000000000000000000000000000000000000..cf04c3e5296598a2a6ba89f6773540bf96956057 --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch2.cpp @@ -0,0 +1,38 @@ +#include +#include +using namespace std; + +using int64 = long long; + +int64 solve() { + int64 A, B, C; + cin >> A >> B >> C; + + // Watch out for this nasty bug: 2*x/y != 2*(x/y). + // Remember to floor before multiplying! + + // Option 1: Buy only singles. + int64 ans = C / A; + + // Option 2: Buy only doubles. + ans = max(ans, 2*(C / B) - 1); + + // Option 3: 1 single + maximum doubles. + if (C > A) { + ans = max(ans, 1 + 2*((C - A)/B)); + } + // Option 4: 2 singles + maximum doubles. + if (C > 2*A) { + ans = max(ans, 2 + 2*((C - 2*A)/B)); + } + return ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/practice/cheeseburger_corollary_ch2.in b/2023/practice/cheeseburger_corollary_ch2.in new file mode 100644 index 0000000000000000000000000000000000000000..123521087a056044d262ae05e65c45649675fc3d --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch2.in @@ -0,0 +1,81 @@ +80 +2 3 5 +2 3 2 +2 3 1 +5 1 100 +1 3 100 +1 1 1000000000000 +314 512 1024 +10 13 11234567890 +1000000000000 1 1000000000000 +1 1000000000000 1000000000000 +1000000000000 1 1 +1 1000000000000 1 +314 512 133769420 +6 9 497087563 +7 12 6439637337963783 +7 7 6977204123379581 +9 17 7379218941168776 +10 13 3337041873753082 +1 1 3771679948420674 +9 10 9259928947805118 +6 9 2903294259185990 +5 7 4495587747091262 +4 4 1760792368134037 +1 1 6044898352400565 +2 2 5384712727499747 +1 1 7851116204037232 +9 9 2395440648063235 +9 15 5802491041437611 +10 13 8839845532190096 +6 6 3465509398779757 +5 9 9152845331581750 +2 2 1081292147434158 +4 6 7754441745231444 +10 17 370045965243845 +4 6 2271224666368145 +5 8 550103143310362 +10 13 8689503031950123 +10 10 4042027009034951 +4 4 4198590241362000 +3 5 840843200459482 +2 3 9429605562755370 +4 7 6988873275022466 +5 9 2462869142646418 +2 2 9825445613196928 +7861 11257 5477539912208008 +13973 26419 3513626523439425 +9334 14595 5458585253152592 +8272 10247 5338458442301888 +48878 70096 1422160026469748 +25878 28689 9188723367817961 +63664 95087 9775891537969006 +74036 112850 549053549625598 +52245 85277 3528041057867931 +13988 25555 2154753727553324 +4930 6317 3001060110844245 +19749 37179 1140508669256069 +80006 121046 2203488461171151 +26597 37816 4245730305851899 +64922 110795 4504704366599950 +37177 52260 3627623246300954 +5089 7407 1785197717173513 +59617 76399 8216425361382011 +2407 4810 850600912749360 +49127 75508 3693288613316216 +74116 111994 1493371909130523 +99838 101928 3638110057611840 +60280 106158 382504020903546 +19716 29416 841088842498107 +6761462000443010 904477908128201 3043816924959190 +9966606336937688 9740005473217677 583445907010364 +3865717696012301 6136029900710052 3321279115901350 +4175966084693454 5040165207533386 1846283425419327 +650920191250544 3690753017565881 4021727189586015 +8025714536856354 9903000027803061 904922510325972 +9706554173387903 5097740681849255 2095245244912381 +6396320710057362 7814230344468181 3183274048075890 +725831654114702 4541733210835335 2742887072719959 +259149987802241 6689368820300403 3534721002568495 +3626621529746551 3995165959615617 3319701929960343 +9545048558841825 6899255672076999 9272818149758853 diff --git a/2023/practice/cheeseburger_corollary_ch2.md b/2023/practice/cheeseburger_corollary_ch2.md new file mode 100644 index 0000000000000000000000000000000000000000..f63092c7ec935a9216d04b86d6c883a469472202 --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch2.md @@ -0,0 +1,38 @@ +*This problem shares some similarities with A1, with key differences in bold.* + +Problem solving skills are applicable to many daily musings. For instance, you might ponder over shared birthdays, bird houses, mapmaking, or ordering an exact number of chicken nuggets. Naturally, another great question to ponder is: how many deckers of a cheeseburger you could build if you spent your entire salary on fast food! + +Specifically, you're interested in building a \(K\)-decker cheeseburger, which alternates between buns, cheese, and patty starting and ending with a bun. **Buying a single cheeseburger costs \(A\) dollars and a double cheeseburger costs \(B\) dollars**. Each provides you with two buns, though a single provides one patty and one cheese, while a double provides two patties and two cheese. + +{{PHOTO_ID:181863494933248|WIDTH:600}} + +You'd like to know **the biggest \(K\) for which you can build a \(K\)-decker cheeseburger by spending at most \(C\) dollars**. + + +# Constraints + +\(1 \leq T \leq 65\) +\(1 \leq A, B, C \leq 10^{16}\) + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case contains one line with three space-separated integers, \(A\), \(B\) and \(C\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by the largest possible \(K\) for which you can build a \(K\)-decker cheeseburger, or \(0\) if you cannot build even a \(1\)-decker cheeseburger. + + +# Sample Explanation + +In the first case, you can buy \(1\) single and \(1\) double cheeseburger. This gives you \(4\) buns, \(3\) slices of cheese, and \(3\) patties, exactly enough for a \(3\)-decker cheeseburger. + +In the second case, you can only afford to build a \(1\)-decker cheeseburger. That's coincidentally identical to a single cheeseburger. + +In the third case, you only have \(\$1\), while a single costs \(\$2\) and a double costs \(\$3\). No fast food for you today. + +In the fourth case, our best possible strategy is to spend all our money on \(100\) doubles, which we can use to build a \(199\)-decker cheeseburger. + +In the fifth case, our best possible strategy is to spend all our money on \(100\) singles, which we can use to build a \(100\)-decker cheeseburger. diff --git a/2023/practice/cheeseburger_corollary_ch2.out b/2023/practice/cheeseburger_corollary_ch2.out new file mode 100644 index 0000000000000000000000000000000000000000..7cfe6afdc1ef3face93b6ad6767fc70e274459d4 --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch2.out @@ -0,0 +1,80 @@ +Case #1: 3 +Case #2: 1 +Case #3: 0 +Case #4: 199 +Case #5: 100 +Case #6: 1999999999999 +Case #7: 3 +Case #8: 1728395059 +Case #9: 1999999999999 +Case #10: 1000000000000 +Case #11: 1 +Case #12: 1 +Case #13: 522536 +Case #14: 110463902 +Case #15: 1073272889660630 +Case #16: 1993486892394165 +Case #17: 868143404843385 +Case #18: 513391057500473 +Case #19: 7543359896841347 +Case #20: 1851985789561022 +Case #21: 645176502041330 +Case #22: 1284453642026074 +Case #23: 880396184067017 +Case #24: 12089796704801129 +Case #25: 5384712727499745 +Case #26: 15702232408074463 +Case #27: 532320144014051 +Case #28: 773665472191681 +Case #29: 1359976235721552 +Case #30: 1155169799593251 +Case #31: 2033965629240388 +Case #32: 1081292147434157 +Case #33: 2584813915077147 +Case #34: 43534819440452 +Case #35: 757074888789381 +Case #36: 137525785827590 +Case #37: 1336846620300017 +Case #38: 808405401806989 +Case #39: 2099295120680999 +Case #40: 336337280183792 +Case #41: 6286403708503579 +Case #42: 1996820935720704 +Case #43: 547304253921426 +Case #44: 9825445613196927 +Case #45: 973179339469 +Case #46: 265992393613 +Case #47: 748007571517 +Case #48: 1041955390318 +Case #49: 40577494477 +Case #50: 640574670975 +Case #51: 205619938329 +Case #52: 9730678769 +Case #53: 82743085658 +Case #54: 168636566429 +Case #55: 950153588995 +Case #56: 61352304755 +Case #57: 36407456027 +Case #58: 224546768872 +Case #59: 81316022683 +Case #60: 138829821901 +Case #61: 482029895280 +Case #62: 215092484491 +Case #63: 353680213201 +Case #64: 97825094381 +Case #65: 26668784204 +Case #66: 71385881357 +Case #67: 7206315508 +Case #68: 57185806533 +Case #69: 5 +Case #70: 0 +Case #71: 0 +Case #72: 0 +Case #73: 6 +Case #74: 0 +Case #75: 0 +Case #76: 0 +Case #77: 3 +Case #78: 13 +Case #79: 0 +Case #80: 1 diff --git a/2023/practice/cheeseburger_corollary_ch2_sol.md b/2023/practice/cheeseburger_corollary_ch2_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..812b87c4b7cb473f90d2f1cd49f59d2c771a85d0 --- /dev/null +++ b/2023/practice/cheeseburger_corollary_ch2_sol.md @@ -0,0 +1,10 @@ +Let's think about what ingredient is limiting us. If we've bought \(0\) singles, we're limited by buns. Otherwise we're limited by patties. So we should either: + +- First reasonable option: buy only doubles (we'll be limited by buns unfortunately, but maybe singles are so outrageously expensive that it's worth it), or +- Second reasonable option: buy \(1\) single first so that we're only limited by patties, then buy as many patties as possible. + +The second option is a bit trickier than it seems. Specifically, we might want to buy \(\lfloor C/A \rfloor\) singles (and no doubles) or \(1\) single (and as many doubles) **or \(2\) singles (and as many doubles)**. + +We might want to buy \(2\) singles because if we start with a single, then prefer doubles, we might run out of money for doubles, but be able to afford one more single. Intuitively it might sound silly to buy \(2\) singles when \(1\) double would be cheaper, but keep in mind that for the first \(2\) (and only the first \(2\)) singles you buy, this may actually be useful because you get an extra bun. If you had bought only doubles, you'd be limited by buns, not patties. + +In short, it is sufficient to consider \(S\) as one of \(\{0, 1, 2, \lfloor C/A \rfloor\}\), and then buying as many doubles as possible, and taking the best of these candidates. diff --git a/2023/practice/dim_sum_delivery.cpp b/2023/practice/dim_sum_delivery.cpp new file mode 100644 index 0000000000000000000000000000000000000000..038a59be3f64d5bafb2bfbff397a1a5be13944ca --- /dev/null +++ b/2023/practice/dim_sum_delivery.cpp @@ -0,0 +1,14 @@ +#include +using namespace std; + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": "; + int R, C, A, B; + cin >> R >> C >> A >> B; + cout << (R > C ? "YES" : "NO") << endl; + } + return 0; +} diff --git a/2023/practice/dim_sum_delivery.in b/2023/practice/dim_sum_delivery.in new file mode 100644 index 0000000000000000000000000000000000000000..d572900740a2fa19be6aee1475932c1f5099f7aa --- /dev/null +++ b/2023/practice/dim_sum_delivery.in @@ -0,0 +1,601 @@ +600 +2 2 1 1 +2 3 1 1 +2 3 1 2 +2 4 1 1 +2 4 1 2 +2 4 1 3 +2 100 1 1 +2 100 1 2 +2 100 1 3 +2 100 1 99 +2 999999999 1 1 +2 999999999 1 2 +2 999999999 1 3 +2 999999999 1 999999998 +2 1000000000 1 1 +2 1000000000 1 2 +2 1000000000 1 3 +2 1000000000 1 999999999 +3 2 1 1 +3 2 2 1 +3 3 1 1 +3 3 1 2 +3 3 2 1 +3 3 2 2 +3 4 1 1 +3 4 1 2 +3 4 1 3 +3 4 2 1 +3 4 2 2 +3 4 2 3 +3 100 1 1 +3 100 1 2 +3 100 1 3 +3 100 1 99 +3 100 2 1 +3 100 2 2 +3 100 2 3 +3 100 2 99 +3 999999999 1 1 +3 999999999 1 2 +3 999999999 1 3 +3 999999999 1 999999998 +3 999999999 2 1 +3 999999999 2 2 +3 999999999 2 3 +3 999999999 2 999999998 +3 1000000000 1 1 +3 1000000000 1 2 +3 1000000000 1 3 +3 1000000000 1 999999999 +3 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+598618548 1000000000 1 2 +100 351514573 1 3 +900200350 242692999 164767257 3 +1000000000 1000000000 990110168 2 +100 1000000000 3 875437474 +4 867643142 2 2 +999999999 3 618585901 2 +3 49194494 2 1 +3 440178503 1 1 +4 783105139 2 2 +826551905 100 1 49 +251333014 100 1 2 +70829133 100 3 26 +1000000000 634373833 687984129 3 +126406469 999999999 3 3 +999999999 1000000000 141127510 2 +999999999 3 768481208 2 +3 348674172 2 208198940 +3 560442205 2 3 +100 2 22 1 +4 100 3 56 +999999999 1000000000 38408940 1 +1000000000 100 872721048 64 +1000000000 352351844 795508322 1 +100 616551719 3 3 +1000000000 4 998306795 2 +3 1000000000 1 638408569 +270079655 4 13920983 1 +245635459 1000000000 2 2 +2 43534587 1 1 +372315310 428364244 1 1 +1000000000 445823784 2 2 +999999999 4 618409559 1 +349725997 3 3 2 +513324740 2 1 1 +809198703 3 2 1 +1000000000 4 120887411 1 +999999999 4 37069733 2 +57695794 1000000000 17394064 1 +100 39439397 2 35075686 +76553835 3 3 2 +2 999999999 1 235322552 +1000000000 743670139 3 3 +2 999999999 1 354427649 +2 611627491 1 2 +3 131597407 2 130390896 +275678673 999999999 1 1 +697757447 999999999 3 614333645 +189832940 1000000000 2 3 +999999999 396025382 2 2 +1000000000 100 3 62 +999999999 999999999 1 713021361 +142421367 2 48241495 1 +3 565428755 1 2 +4 999999999 1 344277279 +4 314723735 2 2 +4 100 2 31 +2 243229745 1 59406209 +2 501785784 1 334445275 +543390985 2 25843304 1 +163708539 999999999 3 3 +216458893 858261817 2 730048733 +999999999 130079639 121158886 3 +3 1000000000 1 859457977 +999999999 734992451 64730372 1 +249097053 100 3 2 +2 475952396 1 3 +990898699 2 1 1 +100 999999999 2 217038469 +702326696 999999999 3 1 +3 332461468 2 3 +3 999999999 2 930212147 +999999999 999999999 13776769 2 +903145896 4 200741956 3 +1000000000 354487415 2 1 +4 345870621 1 11170728 +1000000000 527180352 2 1 +1000000000 935367125 2 688653704 +3 525092475 2 3 +999999999 4 888595089 3 +1000000000 100 3 32 +1000000000 1000000000 3 165697225 +843094006 3 3 1 +4 21218471 2 2 +999999999 1000000000 3 218565021 +38475058 4 38289611 2 +999999999 999999999 3 97982365 +100 100 70 2 +820459816 1000000000 228363612 2 +241065083 1000000000 1 1 +1000000000 999999999 2 756841576 +1000000000 1000000000 244626410 3 +4 911940777 2 39960172 +100 492122489 3 2 +3 750409238 1 2 +999999999 919338507 247646166 1 +3 999999999 1 419422452 +1000000000 1000000000 432922762 516684653 +159636198 2 132750702 1 +999999999 306100163 2 1 +406955220 100 2 1 +564769898 999999999 3 3 +100 1000000000 3 881233701 +568263512 100 2 43 +2 251894516 1 3 +1000000000 293794518 859896680 2 +3 929445484 2 1 +999999999 3 976671636 2 +269564282 100 1 3 +204388631 3 2 2 +312920545 1000000000 309279031 2 +1000000000 1000000000 245797795 1 +997867345 170075009 374134420 2 +402854259 831040336 3 3 +100 461180180 3 3 +999999999 1000000000 534911742 345724526 +3 999999999 1 79992930 +1000000000 100 968047634 19 +1000000000 100 2 16 +268284003 3 3 1 +100 999999999 57 187975571 +4 999999999 2 471976874 +999999999 878891013 1 3 +4 249842301 1 1 +4 1000000000 2 812085677 +6565870 999999999 5153895 286487589 +100 305987626 2 48247012 +999999999 553819129 274277625 13275303 +690839093 999999999 2 3 +100 380837481 37 96807255 +804494411 2 1 1 +770491971 1000000000 3 1 +501599341 999999999 2 3 +100 999999999 83 1 +999999999 999999999 470395372 2 diff --git a/2023/practice/dim_sum_delivery.md b/2023/practice/dim_sum_delivery.md new file mode 100644 index 0000000000000000000000000000000000000000..804102cd0660ada3d17629843e6f534295e822ef --- /dev/null +++ b/2023/practice/dim_sum_delivery.md @@ -0,0 +1,36 @@ +*Nim Sum Dim Sum*, a bustling local dumpling restaurant, has two game theory-loving servers named, you guessed it, Alice and Bob. Its dining area can be represented as a two-dimensional grid of \(R\) rows (numbered \(1..R\) from top to bottom) by \(C\) columns (numbered \(1..C\) from left to right\). + +Currently, both of them are standing at coordinates \((1, 1)\) where there is a big cart of dim sum. Their job is to work together to push the cart to a customer at coordinates \((R, C)\). To make the job more interesting, they've turned it into a game. + +Alice and Bob will take turns pushing the cart. On Alice's turn, the cart must be moved between \(1\) and \(A\) units down. On Bob's turn, the cart must be moved between \(1\) and \(B\) units to the right. The cart may not be moved out of the grid. If the cart is already at row \(R\) on Alice's turn or column \(C\) on Bob's turn, then that person loses their turn. + +The "winner" is the person to ultimately move the cart to \((R, C)\) and thus get all the recognition from the customer. Alice pushes first. Does she have a guaranteed winning strategy? + + +# Constraints + +\(1 \leq T \leq 500\) +\(2 \leq R, C \leq 10^9\) +\(1 \leq A < R\) +\(1 \leq B < C\) + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case will contain one line with four space-separated integers, \(R\), \(C\), \(A\), and \(B\). + + +# Output Format + +For the \(i\)th test case, print `"Case #i: "` followed by `"YES"` if Alice has a guaranteed winning strategy, or `"NO"` otherwise. + + +# Sample Explanation + +The first case is depicted below, with Alice's moves in red and Bob's in blue. Alice moves down, and Bob moves right to win immediately. There is no other valid sequence of moves, so Alice has no guaranteed winning strategy. + +{{PHOTO_ID:842253013944047|WIDTH:500}} + +The second case is depicted below. One possible guaranteed winning strategy is if Alice moves \(3\) units down, then Bob can only move \(1\) unit, and finally Alice can win with \(1\) unit. + +{{PHOTO_ID:852013469652032|WIDTH:500}} diff --git a/2023/practice/dim_sum_delivery.out b/2023/practice/dim_sum_delivery.out new file mode 100644 index 0000000000000000000000000000000000000000..ec04da1e9843fa361ca4c5151c8b42c02c9767fb --- /dev/null +++ b/2023/practice/dim_sum_delivery.out @@ -0,0 +1,600 @@ +Case #1: NO +Case #2: NO +Case #3: NO +Case #4: NO +Case #5: NO +Case #6: NO +Case #7: NO +Case #8: NO +Case #9: NO +Case #10: NO +Case #11: NO +Case #12: NO +Case #13: NO +Case #14: NO +Case #15: NO +Case #16: NO +Case #17: NO +Case #18: NO +Case #19: YES +Case #20: YES +Case #21: NO +Case #22: NO +Case #23: NO +Case #24: NO +Case #25: NO +Case #26: NO +Case #27: NO +Case #28: NO +Case #29: NO +Case #30: NO +Case #31: NO +Case #32: NO +Case #33: NO +Case #34: NO +Case #35: NO +Case #36: NO +Case #37: NO +Case #38: NO +Case #39: NO +Case #40: NO +Case #41: NO +Case #42: NO +Case #43: NO +Case #44: NO +Case #45: NO +Case #46: NO +Case #47: NO +Case #48: NO +Case #49: NO +Case #50: NO +Case #51: NO +Case #52: NO +Case #53: NO +Case #54: NO +Case #55: YES +Case #56: YES +Case #57: YES +Case #58: YES +Case #59: YES +Case #60: YES +Case #61: YES +Case #62: YES +Case #63: YES +Case #64: NO +Case #65: NO +Case #66: NO +Case #67: NO +Case #68: NO +Case #69: NO +Case #70: NO +Case #71: NO +Case #72: NO +Case #73: NO +Case #74: NO +Case #75: NO +Case #76: NO +Case #77: NO +Case #78: NO +Case #79: NO +Case #80: NO +Case #81: NO +Case #82: NO +Case #83: NO +Case #84: NO +Case #85: NO +Case #86: NO +Case #87: NO +Case #88: NO +Case #89: NO +Case #90: NO +Case #91: NO +Case #92: NO +Case #93: NO +Case #94: NO +Case #95: NO +Case #96: NO +Case #97: NO +Case #98: NO +Case #99: NO +Case #100: NO +Case #101: NO +Case #102: NO +Case #103: NO +Case #104: NO +Case #105: NO +Case #106: NO +Case #107: NO +Case #108: NO +Case #109: YES +Case #110: YES +Case #111: YES +Case #112: YES +Case #113: YES +Case #114: YES +Case #115: YES +Case #116: YES +Case #117: YES +Case #118: YES +Case #119: YES +Case #120: YES +Case #121: YES +Case #122: YES +Case #123: YES +Case #124: YES +Case #125: YES +Case #126: YES +Case #127: YES +Case #128: YES +Case #129: YES +Case #130: YES +Case #131: YES +Case #132: YES +Case #133: NO +Case #134: NO +Case #135: NO +Case #136: NO +Case #137: NO +Case #138: NO +Case #139: NO +Case #140: NO +Case #141: NO +Case #142: NO +Case #143: NO +Case #144: NO +Case #145: NO +Case #146: NO +Case #147: NO +Case #148: NO +Case #149: NO +Case #150: NO +Case #151: NO +Case #152: NO +Case #153: NO +Case #154: NO +Case #155: NO +Case #156: NO +Case #157: NO +Case #158: NO +Case #159: NO +Case #160: NO +Case #161: NO +Case #162: NO +Case #163: NO +Case #164: NO +Case #165: NO +Case #166: NO +Case #167: NO +Case #168: NO +Case #169: NO +Case #170: NO +Case #171: NO +Case #172: NO +Case #173: NO +Case #174: NO +Case #175: NO +Case #176: NO +Case #177: NO +Case #178: NO +Case #179: NO +Case #180: NO +Case #181: YES +Case #182: YES +Case #183: YES +Case #184: YES +Case #185: YES +Case #186: YES +Case #187: YES +Case #188: YES +Case #189: YES +Case #190: YES +Case #191: YES +Case #192: YES +Case #193: YES +Case #194: YES +Case #195: YES +Case #196: YES +Case #197: YES +Case #198: YES +Case #199: YES +Case #200: YES +Case #201: YES +Case #202: YES +Case #203: YES +Case #204: YES +Case #205: YES +Case #206: YES +Case #207: YES +Case #208: YES +Case #209: YES +Case #210: YES +Case #211: YES +Case #212: YES +Case #213: YES +Case #214: YES +Case #215: YES +Case #216: YES +Case #217: YES +Case #218: YES +Case #219: YES +Case #220: YES +Case #221: NO +Case #222: NO +Case #223: NO +Case #224: NO +Case #225: NO +Case #226: NO +Case #227: NO +Case #228: NO +Case #229: NO +Case #230: NO +Case #231: NO +Case #232: NO +Case #233: NO +Case #234: NO +Case #235: NO +Case #236: NO +Case #237: NO +Case #238: NO +Case #239: NO +Case #240: NO +Case #241: NO +Case #242: NO +Case #243: NO +Case #244: NO +Case #245: NO +Case #246: NO +Case #247: NO +Case #248: NO +Case #249: NO +Case #250: NO +Case #251: NO +Case #252: NO +Case #253: YES +Case #254: YES +Case #255: YES +Case #256: YES +Case #257: YES +Case #258: YES +Case #259: YES +Case #260: YES +Case #261: YES +Case #262: YES +Case #263: YES +Case #264: YES +Case #265: YES +Case #266: YES +Case #267: YES +Case #268: YES +Case #269: YES +Case #270: YES +Case #271: YES +Case #272: YES +Case #273: YES +Case #274: YES +Case #275: YES +Case #276: YES +Case #277: YES +Case #278: YES +Case #279: YES +Case #280: YES +Case #281: YES +Case #282: YES +Case #283: YES +Case #284: YES +Case #285: YES +Case #286: YES +Case #287: YES +Case #288: YES +Case #289: YES +Case #290: YES +Case #291: YES +Case #292: YES +Case #293: YES +Case #294: YES +Case #295: YES +Case #296: YES +Case #297: YES +Case #298: YES +Case #299: YES +Case #300: YES +Case #301: YES +Case #302: YES +Case #303: YES +Case #304: YES +Case #305: YES +Case #306: YES +Case #307: YES +Case #308: YES +Case #309: NO +Case #310: NO +Case #311: NO +Case #312: NO +Case #313: NO +Case #314: NO +Case #315: NO +Case #316: NO +Case #317: NO +Case #318: NO +Case #319: NO +Case #320: NO +Case #321: NO +Case #322: NO +Case #323: NO +Case #324: NO +Case #325: YES +Case #326: NO +Case #327: YES +Case #328: NO +Case #329: YES +Case #330: YES +Case #331: YES +Case #332: NO +Case #333: YES +Case #334: YES +Case #335: NO +Case #336: NO +Case #337: YES +Case #338: YES +Case #339: NO +Case #340: NO +Case #341: YES +Case #342: YES +Case #343: NO +Case #344: YES +Case #345: NO +Case #346: NO +Case #347: NO +Case #348: NO +Case #349: YES +Case #350: NO +Case #351: YES +Case #352: YES +Case #353: NO +Case #354: YES +Case #355: YES +Case #356: YES +Case #357: NO +Case #358: NO +Case #359: YES +Case #360: NO +Case #361: YES +Case #362: NO +Case #363: NO +Case #364: NO +Case #365: YES +Case #366: NO +Case #367: YES +Case #368: NO +Case #369: NO +Case #370: NO +Case #371: NO +Case #372: YES +Case #373: NO +Case #374: YES +Case #375: NO +Case #376: YES +Case #377: NO +Case #378: NO +Case #379: YES +Case #380: YES +Case #381: YES +Case #382: YES +Case #383: YES +Case #384: YES +Case #385: NO +Case #386: NO +Case #387: YES +Case #388: YES +Case #389: YES +Case #390: YES +Case #391: NO +Case #392: YES +Case #393: NO +Case #394: NO +Case #395: YES +Case #396: NO +Case #397: YES +Case #398: YES +Case #399: NO +Case #400: YES +Case #401: YES +Case #402: YES +Case #403: YES +Case #404: NO +Case #405: NO +Case #406: YES +Case #407: YES +Case #408: NO +Case #409: NO +Case #410: NO +Case #411: NO +Case #412: YES +Case #413: YES +Case #414: YES +Case #415: NO +Case #416: YES +Case #417: YES +Case #418: NO +Case #419: YES +Case #420: NO +Case #421: NO +Case #422: NO +Case #423: NO +Case #424: NO +Case #425: NO +Case #426: NO +Case #427: NO +Case #428: NO +Case #429: NO +Case #430: NO +Case #431: YES +Case #432: NO +Case #433: NO +Case #434: NO +Case #435: YES +Case #436: NO +Case #437: YES +Case #438: NO +Case #439: YES +Case #440: NO +Case #441: YES +Case #442: YES +Case #443: YES +Case #444: NO +Case #445: NO +Case #446: NO +Case #447: NO +Case #448: NO +Case #449: YES +Case #450: YES +Case #451: YES +Case #452: NO +Case #453: NO +Case #454: NO +Case #455: NO +Case #456: NO +Case #457: YES +Case #458: YES +Case #459: NO +Case #460: YES +Case #461: YES +Case #462: YES +Case #463: NO +Case #464: NO +Case #465: NO +Case #466: NO +Case #467: NO +Case #468: YES +Case #469: NO +Case #470: NO +Case #471: NO +Case #472: YES +Case #473: NO +Case #474: NO +Case #475: NO +Case #476: YES +Case #477: YES +Case #478: YES +Case #479: YES +Case #480: NO +Case #481: NO +Case #482: YES +Case #483: NO +Case #484: NO +Case #485: YES +Case #486: NO +Case #487: NO +Case #488: YES +Case #489: YES +Case #490: NO +Case #491: YES +Case #492: NO +Case #493: YES +Case #494: NO +Case #495: NO +Case #496: NO +Case #497: YES +Case #498: YES +Case #499: YES +Case #500: YES +Case #501: YES +Case #502: YES +Case #503: YES +Case #504: NO +Case #505: NO +Case #506: YES +Case #507: NO +Case #508: YES +Case #509: NO +Case #510: NO +Case #511: NO +Case #512: NO +Case #513: NO +Case #514: NO +Case #515: YES +Case #516: YES +Case #517: NO +Case #518: YES +Case #519: NO +Case #520: NO +Case #521: NO +Case #522: NO +Case #523: NO +Case #524: NO +Case #525: YES +Case #526: NO +Case #527: NO +Case #528: YES +Case #529: NO +Case #530: YES +Case #531: YES +Case #532: NO +Case #533: YES +Case #534: NO +Case #535: NO +Case #536: NO +Case #537: NO +Case #538: NO +Case #539: YES +Case #540: YES +Case #541: NO +Case #542: YES +Case #543: YES +Case #544: NO +Case #545: YES +Case #546: YES +Case #547: NO +Case #548: YES +Case #549: NO +Case #550: NO +Case #551: YES +Case #552: NO +Case #553: NO +Case #554: NO +Case #555: NO +Case #556: YES +Case #557: NO +Case #558: NO +Case #559: NO +Case #560: NO +Case #561: YES +Case #562: NO +Case #563: NO +Case #564: YES +Case #565: YES +Case #566: YES +Case #567: NO +Case #568: NO +Case #569: YES +Case #570: NO +Case #571: YES +Case #572: NO +Case #573: YES +Case #574: YES +Case #575: YES +Case #576: NO +Case #577: NO +Case #578: YES +Case #579: NO +Case #580: NO +Case #581: NO +Case #582: NO +Case #583: YES +Case #584: YES +Case #585: YES +Case #586: NO +Case #587: NO +Case #588: YES +Case #589: NO +Case #590: NO +Case #591: NO +Case #592: NO +Case #593: YES +Case #594: NO +Case #595: NO +Case #596: YES +Case #597: NO +Case #598: NO +Case #599: NO +Case #600: NO diff --git a/2023/practice/dim_sum_delivery_sol.md b/2023/practice/dim_sum_delivery_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..2d226c21237e9c9ec657cce1685d7ed3bd548790 --- /dev/null +++ b/2023/practice/dim_sum_delivery_sol.md @@ -0,0 +1,26 @@ +If Alice reaches row \(R\) before Bob reaches row \(C\), then it's game over for Alice. Since each player now wants to get to the finish as slowly as possible, both have a simple dominating strategy of only moving \(1\) unit in their direction each turn, and \(R\) and \(C\) are the only things that matter. + +If \(R \le C\), Bob can always force Alice to reach row \(R\) first by moving \(1\) unit right at a time. Alice also only moves \(1\) unit at a time, because if she moves any faster, she'll just get stuck sooner. + +Example: Alice moves first: +``` +[ ][ ][ ][ ] +[x][ ][ ][ ] +[ ][ ][ ][ ] +``` + +Bob: +``` +[ ][ ][ ][ ] +[ ][x][ ][ ] +[ ][ ][ ][ ] +``` + +Alice moves, and gets stuck to watch Bob stroll to the finish line: +``` +[ ][ ][ ][ ] +[ ][ ][ ][ ] +[ ][x][ ][ ] +``` + +Conversely, if \(R > C\), then Alice can always force a win by moving \(1\) step at a time. Therefore we output "`YES`" if and only if \(R > C\), regardless of the values of \(A\) and \(B\). diff --git a/2023/practice/road_to_nutella.cpp b/2023/practice/road_to_nutella.cpp new file mode 100644 index 0000000000000000000000000000000000000000..62f677183c6aa4949d79d21ab30052d3db47294a --- /dev/null +++ b/2023/practice/road_to_nutella.cpp @@ -0,0 +1,241 @@ +#include +#include +#include +#include +#include +using namespace std; + +const int INF = (int)1e9, lgmax = 20; + +int N, M, Q; +vector> adj, block_adj; + +// Start of bridge-finding code. + +int timer, nblocks; +vector visit; +vector lowlink, tin, stack, block; +vector> blocks; +set> bridges; + +void dfs(int u, int p) { + visit[u] = true; + lowlink[u] = tin[u] = timer++; + stack.push_back(u); + int children = 0; + for (int v : adj[u]) { + if (v == p) { + continue; + } + if (visit[v]) { + lowlink[u] = min(lowlink[u], tin[v]); + } else { + dfs(v, u); + lowlink[u] = min(lowlink[u], lowlink[v]); + if (lowlink[v] > tin[u]) { + bridges.insert({u, v}); + } + children++; + } + } + if (lowlink[u] == tin[u]) { + vector tmp; + int v; + do { + v = stack.back(); + stack.pop_back(); + tmp.push_back(v); + } while (u != v); + blocks.push_back(tmp); + } +} + +void tarjan(int nodes) { + visit.assign(nodes, false); + lowlink.assign(nodes, 0); + tin.assign(nodes, 0); + bridges.clear(); + stack.clear(); + blocks.clear(); + timer = 0; + for (int i = 0; i < nodes; i++) { + if (!visit[i]) { + dfs(i, -1); + } + } + // Get node to block mapping. + block.assign(nodes, 0); + nblocks = blocks.size(); + for (int i = 0; i < nblocks; i++) { + for (int j : blocks[i]) { + block[j] = i; + } + } + // Get bridge-block forest adjacencies. + block_adj.assign(nblocks, {}); + for (int i = 0; i < nodes; i++) { + for (int j : adj[i]) { + if (block[i] != block[j]) { + block_adj[block[i]].push_back(block[j]); + } + } + } +} + +// Start of code to find odd cycles/distances to odd cycle blocks. + +bool no_odd_cycles; +vector color; +vector has_odd_cycle; +vector dist_to_odd; + +bool dfs_color(int u, int c = 1) { + // Color edges in search of an odd cycle. + color[u] = c; + int nextc = 3 - c; + for (int v : adj[u]) { + if (bridges.count({u, v}) || bridges.count({v, u})) { + continue; + } + if (color[v]) { + if (color[v] != nextc) { + return true; + } + } else if (dfs_color(v, nextc)) { + return true; + } + } + return false; +} + +void find_odd_cycles_and_distances() { + color.assign(N, 0); + has_odd_cycle.assign(nblocks, false); + dist_to_odd.assign(nblocks, INF); + queue q; + for (int i = 0; i < nblocks; i++) { + has_odd_cycle[i] = dfs_color(blocks[i].front()); + if (has_odd_cycle[i]) { + dist_to_odd[i] = 0; + q.push(i); + } + } + no_odd_cycles = q.empty(); + while (!q.empty()) { + int v = q.front(); + q.pop(); + for (int u : block_adj[v]) { + if (dist_to_odd[u] == INF) { + dist_to_odd[u] = dist_to_odd[v] + 1; + q.push(u); + } + } + } +} + +// Start of binary lifting code for path-min queries. + +vector depth; +vector> lift, liftval; + +void dfs_lift(int u, int p, int d) { + depth[u] = d; + lift[u][0] = p; + liftval[u][0] = dist_to_odd[u]; + for (int v : block_adj[u]) { + if (v != p) { + dfs_lift(v, u, d + 1); + } + } +} + +void init_binary_lifting() { + depth.assign(nblocks, -1); + lift.assign(nblocks, vector(lgmax, -1)); + liftval.assign(nblocks, vector(lgmax)); + dfs_lift(0, -1, 0); + for (int e = 1; e < lgmax; e++) { + for (int u = 0; u < nblocks; u++) { + if (lift[u][e - 1] != -1) { + lift[u][e] = lift[lift[u][e - 1]][e - 1]; + liftval[u][e] = min(liftval[u][e - 1], liftval[lift[u][e - 1]][e - 1]); + } + } + } +} + +int go_up(int u, int to_depth) { + if (depth[u] == to_depth) { + return u; + } + return go_up(lift[u][__builtin_ctz(depth[u] - to_depth)], to_depth); +} + +int get_lca(int u, int v, int maxlift) { + if (depth[u] != depth[v]) { + if (depth[u] > depth[v]) { + return get_lca(go_up(u, depth[v]), v, lgmax - 1); + } + return get_lca(u, go_up(v, depth[u]), lgmax - 1); + } + if (u == v) { + return u; + } + if (lift[u][0] == lift[v][0]) { + return lift[u][0]; + } + while (lift[u][maxlift] == lift[v][maxlift]) { + --maxlift; + } + return get_lca(lift[u][maxlift], lift[v][maxlift], maxlift); +} + +int path_min_go_up(int u, int to_depth) { + if (depth[u] == to_depth) { + return dist_to_odd[u]; + } + int e = __builtin_ctz(depth[u] - to_depth); + return min(path_min_go_up(lift[u][e], to_depth), liftval[u][e]); +} + +int path_min(int u, int v) { + int lca = get_lca(u, v, lgmax - 1); + return min(path_min_go_up(u, depth[lca]), path_min_go_up(v, depth[lca])); +} + +// Main Solution. + +long long solve() { + cin >> N >> M; + adj.assign(N, {}); + for (int i = 0, u, v; i < M; i++) { + cin >> u >> v; + --u, --v; + adj[u].push_back(v); + adj[v].push_back(u); + } + // Find bridges and bridge block tree. + tarjan(N); + // Find odd cycle blocks, and min-distance to odd cycles from each block. + find_odd_cycles_and_distances(); + // Initialize binary lifting for path mins. + init_binary_lifting(); + // Process queries. + long long ans = 0; + cin >> Q; + for (int i = 0, a, b; i < Q; i++) { + cin >> a >> b; + --a, --b; + ans += no_odd_cycles ? -1 : path_min(block[a], block[b]); + } + return ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/practice/road_to_nutella.in b/2023/practice/road_to_nutella.in new file mode 100644 index 0000000000000000000000000000000000000000..446bba9277de8c6e344d8aed9fb8a8af89d771d4 --- /dev/null +++ b/2023/practice/road_to_nutella.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:37239cb51da58b8fd94fd96d1164b12cc2d464cd0810e856ec7ec0d1054f7e88 +size 21165992 diff --git a/2023/practice/road_to_nutella.md b/2023/practice/road_to_nutella.md new file mode 100644 index 0000000000000000000000000000000000000000..90b7870d6e3b589ef27a525266d82ca5a4620ee4 --- /dev/null +++ b/2023/practice/road_to_nutella.md @@ -0,0 +1,43 @@ +You and your friends have drawn a really big connected graph in sidewalk chalk with \(N\) nodes (numbered from \(1..N\)) and \(M\) edges. \(Q\) times, there will be a race from node \(a_i\) to node \(b_i\)​, with a chance to win a coveted hazelnut chocolate snack. By the unbreakable rules of hop-scotch, everyone must travel along a path from node \(a_i\) to node \(b_i\) using edges in the graph, alternating which foot touches the ground at each node, starting each race with their left foot on \(a_i\). + +Your friends will make a mad dash for the chocolate along the **shortest path** from \(a_i\) to \(b_i\)​. You on the other hand are looking for a more interesting challenge, and are allowed to take *any* path, potentially including any nodes (even \(b_i\)) multiple times. You want to end at node \(b_i\)​, but with the following conditions: + - You must finish on a different foot from everyone who took the shortest path. +- To make things interesting, you'd like to minimize the number of edges you travel through more than once. + +{{PHOTO_ID:903178538089777|WIDTH:600}} + +*An illustration of the first sample. Your friends take a shortest path (blue), and you can take the path in red. The path in red travels through \(1\) edge multiple times: the edge connecting nodes \(6\) and \(8\).* + +For each query, is it possible to fulfill your two conditions? If so, add the minimum number of edges you have to travel through multiple times to your answer. If not, add \(-1\) to your answer. + +# Constraints + +\(1 \leq T \leq 140\) +\(1 \leq N, M, Q \leq 3*10^5\) + +The sum of \(N\), \(M\), and \(Q\) over all cases are each no more than \(3*10^6\). +There will be at most one edge connecting the same pair of nodes directly. +The graph is connected. +No edge will connect a node with itself. +For all queries, \(a_i \neq b_i\). That is, the start and end of each race will be on different nodes. + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each test case, there is first a line containing two space-separated integers, \(N\) and \(M\). \(M\) lines follow, each containing two integers describing the endpoints of an edge. Then, there is a line containing a single integer \(Q\). \(Q\) lines follow, the \(i\)th of which contains two space-separated integers \(a_i\) and \(b_i\). + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by the sum of answers to all queries in that case. The answer to a query is the minimum number of edges you'd need to travel through more than once to reach node \(b_i\) on a the other foot than your friends, or \(-1\) if it isn't possible to do so. + +# Sample Explanation + +The graph for the first case is shown above. The first query asks us to go from node \(7\) to node \(1\). The shortest paths (one shown in blue) will start on the left and end on the right foot, but we can end our our left by taking the red path. We'll cross the edge from \(6\) to \(8\) more than once, so the answer to this query is \(1\). The answers to the queries are \([1, 0, 2, 0, 2]\) respectively. + +In the second case, the query answers are \([-1, -1]\) respectively. + +In the third case, the query answers are \([0, 0]\) respectively. + +In the fourth case, the query answers are \([-1, -1]\) respectively. + + + diff --git a/2023/practice/road_to_nutella.out b/2023/practice/road_to_nutella.out new file mode 100644 index 0000000000000000000000000000000000000000..6e7c2296fa4ae67ee09c196e5b099f21cdac3f51 --- /dev/null +++ b/2023/practice/road_to_nutella.out @@ -0,0 +1,160 @@ +Case #1: 5 +Case #2: -2 +Case #3: -2 +Case #4: 0 +Case #5: 270676 +Case #6: 121294 +Case #7: 3051989226 +Case #8: 265273926 +Case #9: 1785 +Case #10: 12671 +Case #11: 0 +Case #12: 0 +Case #13: 1 +Case #14: 1 +Case #15: 0 +Case #16: 0 +Case #17: 0 +Case #18: 0 +Case #19: -27 +Case #20: 0 +Case #21: 0 +Case #22: 0 +Case #23: 0 +Case #24: 0 +Case #25: 0 +Case #26: 0 +Case #27: 0 +Case #28: 0 +Case #29: 0 +Case #30: 0 +Case #31: 0 +Case #32: 0 +Case #33: 0 +Case #34: 0 +Case #35: 2 +Case #36: 0 +Case #37: 4 +Case #38: 0 +Case #39: 0 +Case #40: -15 +Case #41: 2 +Case #42: 0 +Case #43: 0 +Case #44: 0 +Case #45: 0 +Case #46: 0 +Case #47: 0 +Case #48: 0 +Case #49: 2 +Case #50: 2 +Case #51: 0 +Case #52: 0 +Case #53: 0 +Case #54: 0 +Case #55: 0 +Case #56: 0 +Case #57: 0 +Case #58: 0 +Case #59: 0 +Case #60: 1 +Case #61: 11 +Case #62: 10 +Case #63: -25 +Case #64: 3 +Case #65: -18 +Case #66: 4 +Case #67: 0 +Case #68: 0 +Case #69: 0 +Case #70: 0 +Case #71: -32 +Case #72: 0 +Case #73: 0 +Case #74: -8 +Case #75: 13 +Case #76: 2 +Case #77: 4 +Case #78: 13 +Case #79: 2 +Case #80: 21 +Case #81: 2 +Case #82: -27 +Case #83: -7 +Case #84: 5 +Case #85: -37 +Case #86: 0 +Case #87: -34 +Case #88: 0 +Case #89: -36 +Case #90: 0 +Case #91: 0 +Case #92: 0 +Case #93: -28 +Case #94: 1 +Case #95: -19 +Case #96: -38 +Case #97: 0 +Case #98: -13 +Case #99: 0 +Case #100: 1 +Case #101: -23 +Case #102: -41 +Case #103: 0 +Case #104: 0 +Case #105: -20 +Case #106: 2 +Case #107: 0 +Case #108: 0 +Case #109: 29 +Case #110: 8 +Case #111: 4 +Case #112: 0 +Case #113: -21 +Case #114: 0 +Case #115: 0 +Case #116: 4 +Case #117: 0 +Case #118: 45 +Case #119: 0 +Case #120: 0 +Case #121: 1 +Case #122: 33 +Case #123: 0 +Case #124: 0 +Case #125: 9 +Case #126: 1 +Case #127: 0 +Case #128: 0 +Case #129: 1 +Case #130: 3 +Case #131: 1 +Case #132: 0 +Case #133: 0 +Case #134: 7 +Case #135: 0 +Case #136: 0 +Case #137: 15 +Case #138: 0 +Case #139: -7 +Case #140: 1 +Case #141: 0 +Case #142: 19 +Case #143: 2 +Case #144: 0 +Case #145: 6 +Case #146: 76 +Case #147: 3 +Case #148: 22 +Case #149: 0 +Case #150: 2 +Case #151: 4 +Case #152: -29 +Case #153: 3 +Case #154: 0 +Case #155: 17 +Case #156: -10 +Case #157: -32 +Case #158: 6 +Case #159: 0 +Case #160: 25 diff --git a/2023/practice/road_to_nutella_sol.md b/2023/practice/road_to_nutella_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..9e5b96697d26377244424955297dc184665e17a9 --- /dev/null +++ b/2023/practice/road_to_nutella_sol.md @@ -0,0 +1,15 @@ +It's easier to think about this problem if instead of thinking about which foot we're standing on, we consider the path length. You'll end on your left foot (the same one you started with) if the path contains an even number of edges, and on your right foot if it contains an odd number of edges. So really we care only about path lengths modulo \(2\). + +If you're familiar with graph theory, you'll know it's difficult to efficiently answer shortest path queries between nodes in a general graph. This gives a hint that we probably can't and won't need to find our friends' path to solve this problem, reducing the space of possible approaches a fair bit. + +Consider a tree. Notice that given two nodes, all possible paths (simple or non-simple) between them will have the same parity. Every edge will be traveled an odd number of times if it's on the simple path between those two nodes, or an even number of times if it's not on that simple path. So if the problem were given on a tree, then we'd have a solution: the answer is always impossible. In fact, it's even impossible on bipartite graphs, as if the graph is two-colorable, then each color represents the only foot that could ever touch that node. + +When *is* it possible to get a path with different parity? Consider a graph consisting of only a cycle of odd length. Between any pair of nodes, there will have one even length path, and one odd length path. One will be shorter, the other will be longer. Furthermore, notice that the longer path will never repeat any edge, *meaning that here we know the answer is \(0\) even without knowing the distance between the nodes!* + +In general, we'll want to consider the bridge-block tree (a.k.a. bridge-tree) of the given graph. That is, find all the [bridges](https://en.wikipedia.org/wiki/Bridge\_(graph_theory)), and condense each maximal component without bridges into a single node (the block, or \(2\)-edge-connected component). This can be done in \(\mathcal{O}(N+M)\) using Tarjan's bridge-finding algorithm. We'll be left with a tree where the only edges are bridges. These bridges are the only possible edges we might need to walk multiple times. (Why? By the definition of a bridge, everything else is part of a \(2\)-edge-connected component, meaning no matter where we want to go, there's one path there and a completely disjoint path back) + +Our path from \(a_i\) to \(b_i\) therefore will always be: take some (possibly empty) part of the direct path from \(a_i\) to \(b_i\), then take some path to a block with an odd cycle, then take a path back to where you were, then finish the path from \(a_i\) to \(b_i\). Of all odd cycle blocks we could visit, we'll want to pick one which is the fewest bridges away from the direct path between \(a_i\) to \(b_i\). That number of bridges is what we'll add to our answer. + +Famously, it can be shown that a graph is bipartite if and only if it contains no odd cycles. So, after finding each block, we'll first mark if it has an odd cycle by trying to \(2\)-color it using DFS. Then, we can find the distance from each block to the nearest odd cycle block using a single multi-source BFS (initializing the queue with all the odd cycle blocks and expanding outwards). + +Finally, given two query nodes \(a_i\) and \(b_i\), we'll look up their blocks, then do a path-min query, getting the minimum "distance to odd cycle block" across every block on the direct path. Path-min queries on a tree can be handled in \(\mathcal{O}(\log N\)\) using binary-lifting, heavy-light decomposition, or a link-cut tree. The overall running time will be \(\mathcal{O}(N + M + Q \log N)\). diff --git a/2023/practice/two_apples_a_day.cpp b/2023/practice/two_apples_a_day.cpp new file mode 100644 index 0000000000000000000000000000000000000000..26a9bd8469504ced5669595c1a2a943be41ac2ac --- /dev/null +++ b/2023/practice/two_apples_a_day.cpp @@ -0,0 +1,54 @@ +#include +#include +#include +using namespace std; + +const int INF = (int)1e9 + 5; + +int N, sz; +vector a; + +int try_sum(int sum) { + int skipped = 0; + int l = 0, r = sz - 1; + int res = sum / 2; + while (l <= r) { + if (a[l] + a[r] == sum) { + l++; + r--; + continue; + } + skipped++; + if (a[l] + a[r] < sum) { + res = sum - a[l++]; + } else { + res = sum - a[r--]; + } + } + return (skipped <= 1 && res > 0) ? res : INF; +} + +int solve() { + cin >> N; + sz = 2*N - 1; + a.resize(sz); + for (int i = 0; i < sz; i++) { + cin >> a[i]; + } + sort(a.begin(), a.end()); + int ans = min({ + try_sum(a[1] + a[sz - 1]), // remove first + try_sum(a[0] + a[sz - 1]), // remove middle + try_sum(a[0] + a[sz - 2]) // remove last + }); + return ans == INF ? -1 : ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/practice/two_apples_a_day.in b/2023/practice/two_apples_a_day.in new file mode 100644 index 0000000000000000000000000000000000000000..0c67f5a069ad30569175befb962e3d09c7ab2a32 --- /dev/null +++ b/2023/practice/two_apples_a_day.in @@ -0,0 +1,181 @@ +90 +3 +6 3 1 2 5 +2 +7 7 7 +1 +1 +3 +1 9 1 1 4 +4 +1 9 1 1 4 9 9 +4 +1 9 10 1 4 6 9 +3 +1000000000 2 10 4 999999994 +11 +4 28 36 62 22 2 30 37 20 42 42 22 38 27 22 44 62 2 34 60 26 +15 +2 1 1 2 2 2 1 1 1 2 1 2 1 1 2 2 1 2 2 1 1 1 2 2 2 1 2 1 1 +9 +5 7 4 9 4 10 4 8 3 7 8 3 7 3 10 5 6 +1 +2 +1 +1000000000 +1 +9 +2 +3 3 1 +3 +3 2 3 2 4 +4 +2 6 7 3 1 3 5 +5 +1 5 4 3 5 2 8 4 7 +2 +4 4 1 +3 +2 1 3 4 3 +4 +1 2 1 1 2 2 2 +5 +5 4 3 4 1 2 3 3 5 +6 +3 2 2 3 2 3 2 4 2 1 3 +7 +1 2 3 1 2 2 2 2 2 3 1 3 2 +8 +1 4 4 2 2 2 4 4 4 5 5 2 4 1 3 +9 +4 3 4 6 6 4 6 3 5 3 1 2 5 1 3 3 2 +10 +1 2 2 3 3 2 2 2 1 3 2 1 2 2 3 1 3 1 3 +11 +1 2 2 1 1 1 1 1 2 1 2 2 2 2 1 1 2 1 2 1 2 +2 +2 2 3 +3 +5 1 3 4 5 +4 +4 5 2 4 3 4 3 +5 +4 4 3 1 3 2 3 4 4 +6 +3 2 3 2 4 5 3 4 4 4 5 +7 +3 1 5 1 3 3 3 5 5 4 3 2 3 +8 +3 5 4 2 2 5 3 2 1 5 1 4 1 4 3 +9 +5 1 1 3 3 4 1 1 2 4 3 5 5 2 3 3 3 +10 +4 1 3 2 5 3 4 5 2 3 4 2 1 4 5 1 3 5 4 +11 +2 2 2 3 5 2 5 2 1 1 5 3 2 3 1 3 2 5 2 2 5 +5 +54 8 20 40 45 15 52 60 6 +15 +91 64 45 12 47 7 17 46 29 68 14 62 44 9 87 74 4 61 27 45 23 46 84 30 79 86 77 5 82 +25 +19 6 13 14 24 7 21 22 27 16 13 12 29 21 18 15 36 18 30 17 14 19 6 10 11 14 23 14 29 23 17 22 12 28 8 21 30 18 15 17 21 5 21 5 14 20 16 20 25 +35 +22 15 14 4 35 15 16 20 7 10 22 17 12 23 22 9 4 3 16 14 9 21 16 24 14 13 8 10 16 6 10 17 11 6 18 4 3 21 11 9 23 15 12 19 9 13 10 12 25 7 23 16 1 20 8 25 10 5 2 18 3 21 17 17 5 11 19 5 1 +45 +13 8 10 14 2 12 3 7 17 4 17 15 17 7 6 8 11 1 4 11 4 12 8 8 9 5 14 18 9 10 7 6 12 17 6 10 20 10 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211563846 414124782 341374054 232602106 293574805 411325792 296419152 335376822 204044128 372882924 428738022 515159560 421570644 246460698 360923926 284566684 184015834 385053152 445593170 328609327 439506926 300745911 215665932 319633662 202103428 407604366 375811926 524029694 349701404 480443850 414669020 253793500 529294754 382888474 434925720 203957078 227942494 327098382 215932690 196878860 271964392 505194938 496866423 234061890 300882074 212038610 200012886 381373118 421228512 505307850 343166497 496732725 519900257 389514702 385026008 283047263 411374320 481663188 485288898 511988808 267346330 226116604 189654764 534477972 529046962 255316498 492149414 525678396 242830388 diff --git a/2023/practice/two_apples_a_day.md b/2023/practice/two_apples_a_day.md new file mode 100644 index 0000000000000000000000000000000000000000..f97efc748c7069db439086f60e5933e6a0011985 --- /dev/null +++ b/2023/practice/two_apples_a_day.md @@ -0,0 +1,34 @@ +“An apple a day keeps the doctor away” is Steve’s motto. His other motto, “You can never have too much of a good thing,” holds true for both apples and mottos. Steve would like to eat two apples per day for the next \(N\) days, but with strict adherence to his third motto “Consistency is key.” Specifically, he’d like the sum of the two apple weights he eats over the next \(N\) days to be the same for each day. + +Steve has already purchased \(2*N-1\) apples, the \(i\)th of which weighs \(A_i\) ounces. He'd like to buy one more apple that's as light as possible to fulfill his goal. Steve can buy an apple of any positive integer weight in ounces from the store. Is it possible for him to reach his goal, and if so, what weight apple should he buy? + +{{PHOTO_ID:1563872647765708|WIDTH:600}} + + +*The above image depicts the solution to the first sample. Each day, Steve will eat two apples totalling \(7\) oz. Steve must buy a \(4\) oz apple to make this happen.* + +# Constraints +\(1 \leq T \leq 70\) +\(1 \leq N \leq 3*10^5\) +The sum of \(N\) over all cases is at most \(600{,}000\) +\(1 \leq A_i \leq 10^9\) + +# Input Format +Input begins with an integer \(T\), the number of test cases. Each test case starts with a single integer \(N\). The next line contains \(2*N-1\) space-separated integers \(A_1, ..., A_{2*N - 1}\). + +# Output Format +For the \(i\)th test case, print "`Case #i:` " followed a single integer, the smallest possible apple weight in ounces that Steve can buy so that he can eat two apples for the next \(N\) days and have the sum of apple weights be the same every day, or \(-1\) if doing so is impossible. + +# Sample Explanation + +In the first case, if Steve buys a \(4\) oz apple, he can group his apples as shown above. For this input, there's no way to succeed by buying any apple below \(4\) oz. + +In the second case, Steve can buy a \(7\) oz apple, and eat two apples totaling \(14\) oz each day. + +In the third case, any apple weight will suffice, so Steve will buy the lightest one possible. + +In the fourth case, no matter what weight apple Steve attempts to buy, it is impossible for him to achieve his goal. + +Please note, as demonstrated in the seventh case, that it's possible for the answer to exceed \(10^9\). + + diff --git a/2023/practice/two_apples_a_day.out b/2023/practice/two_apples_a_day.out new file mode 100644 index 0000000000000000000000000000000000000000..038fb88e874b0479c318438207f9a3944c16ea35 --- /dev/null +++ b/2023/practice/two_apples_a_day.out @@ -0,0 +1,80 @@ +Case #1: 4 +Case #2: 7 +Case #3: 1 +Case #4: -1 +Case #5: 6 +Case #6: -1 +Case #7: 1000000002 +Case #8: 42 +Case #9: 2 +Case #10: -1 +Case #11: 1 +Case #12: 1 +Case #13: 1 +Case #14: 1 +Case #15: 1 +Case #16: 5 +Case #17: 6 +Case #18: 1 +Case #19: 2 +Case #20: 1 +Case #21: -1 +Case #22: 3 +Case #23: 2 +Case #24: -1 +Case #25: -1 +Case #26: 1 +Case #27: 2 +Case #28: 1 +Case #29: -1 +Case #30: 3 +Case #31: -1 +Case #32: 3 +Case #33: 1 +Case #34: 3 +Case #35: 5 +Case #36: -1 +Case #37: -1 +Case #38: -1 +Case #39: -1 +Case #40: -1 +Case #41: -1 +Case #42: -1 +Case #43: 1 +Case #44: 1 +Case #45: 1 +Case #46: 1 +Case #47: 1 +Case #48: 1 +Case #49: 1 +Case #50: 1 +Case #51: 1 +Case #52: 3 +Case #53: 1 +Case #54: 30 +Case #55: 7 +Case #56: 129 +Case #57: 38 +Case #58: 46 +Case #59: 235 +Case #60: 76 +Case #61: 198 +Case #62: 367 +Case #63: 40895814 +Case #64: -1 +Case #65: -1 +Case #66: 1 +Case #67: -1 +Case #68: -1 +Case #69: -1 +Case #70: 6 +Case #71: 1 +Case #72: 2 +Case #73: -1 +Case #74: -1 +Case #75: 1 +Case #76: 1 +Case #77: -1 +Case #78: 1 +Case #79: 5 +Case #80: 13 diff --git a/2023/practice/two_apples_a_day_sol.md b/2023/practice/two_apples_a_day_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..81353ae80d79c6859523a695447ce1036a412cd9 --- /dev/null +++ b/2023/practice/two_apples_a_day_sol.md @@ -0,0 +1,13 @@ +If there's only one element, we don't care what it is, a \(1\) will match it. + +Otherwise, let's solve this backwards: instead of attempting to add a number in, we'll consider starting with a working solution and removing some number from it. Let's denote the daily sum of apple weights as \(K\). We can imagine having a sorted array of \(2*N\) elements where the first and last elements sum to \(K\), the second and second last sum \(K\), etc. + +Now, consider removing one element from the \(2*N\) sorted numbers. There are three cases: + +1. We removed the first element. Then, \(K\) is the new first element plus the second last element. +2. We removed the last element. Then, \(K\) is the new last element plus the second element. +3. We removed some element in the middle. Then, \(K\) is the sum of the first and last elements. + +With \(3\) candidates for \(K\), we'll pick the smallest that works. To check if a \(K\) works, we can use two pointers \(l\) and \(r\) moving from the outside inwards on the sorted input array. If the left and right values sum less than \(K\), we record a "skip" (noting the element that would've been removed) and increment \(l\). If it sums more, we decrement \(r\), else we advance both closer. In the end, \(K\) works if there is at most a single skip (and that removed element is positive). + +Running two pointers to check one candidate \(K\) takes \(\mathcal{O}(N)\) time, so the overall running time on \(3\) candidates is still \(\mathcal{O}(N)\). diff --git a/2023/round1/163905706775126.jpg b/2023/round1/163905706775126.jpg new file mode 100644 index 0000000000000000000000000000000000000000..fa54208309f70b0461e8806eae564adecf8b3879 --- /dev/null +++ b/2023/round1/163905706775126.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f179fdf460e2a68880492e6c41c1e6f943dbadbd8467f3f8ac57f1815c73658f +size 23594 diff --git a/2023/round1/287687020772716.jpg b/2023/round1/287687020772716.jpg new file mode 100644 index 0000000000000000000000000000000000000000..80ec505277aa823363cfb82ed2ee09b8dc26e96c --- /dev/null +++ b/2023/round1/287687020772716.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eea4a3ef3480381af88bdee8d8b7552cd479a7692f53cfc005b7544add916f8d +size 46209 diff --git a/2023/round1/838516607729325.jpg b/2023/round1/838516607729325.jpg new file mode 100644 index 0000000000000000000000000000000000000000..d315e0eb64c5e029bf8449d9003f07760ae32467 --- /dev/null +++ b/2023/round1/838516607729325.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cd4142652979bcc61deae14678a5ef9a94b7ad5fbe011fa460edebfbc72367c2 +size 41076 diff --git a/2023/round1/back_in_black_ch1.cpp b/2023/round1/back_in_black_ch1.cpp new file mode 100644 index 0000000000000000000000000000000000000000..a7425cb0cd2533cbe8c98119d284a01be2aad87c --- /dev/null +++ b/2023/round1/back_in_black_ch1.cpp @@ -0,0 +1,48 @@ +#include +#include +using namespace std; + +int solve() { + int N, Q; + cin >> N; + vector white(N + 1); + for (int i = 1; i <= N; i++) { + char c; + cin >> c; + white[i] = c - '0'; + } + vector press(N + 1); + cin >> Q; + for (int i = 0, b; i < Q; i++) { + cin >> b; + press[b] = press[b] ^ 1; + } + // Apply presses. + for (int i = 1; i <= N; i++) { + if (press[i]) { + for (int j = i; j <= N; j += i) { + white[j] = white[j] ^ 1; + } + } + } + // Convert to black greedily. + int num_presses = 0; + for (int i = 1; i <= N; i++) { + if (white[i]) { + num_presses++; + for (int j = i; j <= N; j += i) { + white[j] = white[j] ^ 1; + } + } + } + return num_presses; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/round1/back_in_black_ch1.in b/2023/round1/back_in_black_ch1.in new file mode 100644 index 0000000000000000000000000000000000000000..0e75f2a1d0dfdea1cf171f227065d1983261465f --- /dev/null +++ b/2023/round1/back_in_black_ch1.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bbe37cc01f2aa034446fae053c8322726d8c7a57d2fe49e197414d1e1da86265 +size 39373117 diff --git a/2023/round1/back_in_black_ch1.md b/2023/round1/back_in_black_ch1.md new file mode 100644 index 0000000000000000000000000000000000000000..7c518c592551a99d993832fcaf4fc438cd1fc169 --- /dev/null +++ b/2023/round1/back_in_black_ch1.md @@ -0,0 +1,33 @@ +*This problem shares some similarities with C2, with key differences in bold.* + +A friend who works at Metal Platforms Inc just lent you a curious puzzle, offering you tickets to a metal concert if you can solve it. + +The puzzle consists of \(N\) buttons in a row, numbered from \(1\) to \(N\). The initial state of button \(i\) is white if \(S_i = 1\), or black if \(S_i = 0\). Pressing a button \(k\) toggles the state of itself as well as every \(k\)th button. Your friend challenges you to return the puzzle to him with all buttons back in black. + +Life is hard enough without siblings pushing your buttons. Unfortunately, your brother has taken the puzzle and will push \(Q\) buttons sequentially, the \(i\)th being button \(B_i\). + +After your brother **has pushed all \(Q\) buttons**, you'd like to know the minimum number of button presses required to turn all the buttons black. + +**Constraints** + +\(1 \le T \le 70\) +\(1 \le N \le 4{,}000{,}000\) +\(1 \le Q \le 4{,}000{,}000\) + +The sum of \(N\) and \(Q\) over all cases will be at most \(9{,}000{,}000\). + +**Input Format** + +Input begins with a single integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\). Then, there is a line containing a bitstring \(S\) of length \(N\). Then, there is a line containing a single integer \(Q\). \(Q\) lines follow, the \(i\)th of which contains a single integer \(B_i\). + +**Output Format** + +For the \(i\)th test case, output a single line containing `"Case #i: "` followed by a single integer, the number of button presses needed to turn all buttons black **after all \(Q\) button presses.** + +**Sample Explanation** + +The first sample case is depicted below. After your brother presses the first button, the state of the puzzle is \(0101\). The best strategy is to press the second button, turning all lights off. + +{{PHOTO_ID:287687020772716|WIDTH:400}} + +In the second case, the puzzle starts as \(0001\), and after each button press becomes \(0100\), \(0110\), \(0011\), and \(0010\) respectively. Pressing only the third button will return the puzzle to all zeros. diff --git a/2023/round1/back_in_black_ch1.out b/2023/round1/back_in_black_ch1.out new file mode 100644 index 0000000000000000000000000000000000000000..aca307f8fbb8683aa183c6f35f44d09a979a94d7 --- /dev/null +++ b/2023/round1/back_in_black_ch1.out @@ -0,0 +1,96 @@ +Case #1: 1 +Case #2: 1 +Case #3: 4 +Case #4: 4 +Case #5: 0 +Case #6: 86747 +Case #7: 38042 +Case #8: 0 +Case #9: 1 +Case #10: 2 +Case #11: 3 +Case #12: 3 +Case #13: 1 +Case #14: 516885 +Case #15: 500557 +Case #16: 2025363 +Case #17: 46 +Case #18: 15 +Case #19: 18 +Case #20: 31 +Case #21: 42 +Case #22: 27 +Case #23: 2 +Case #24: 31 +Case #25: 13 +Case #26: 17 +Case #27: 29 +Case #28: 37 +Case #29: 56 +Case #30: 14 +Case #31: 9 +Case #32: 35 +Case #33: 41 +Case #34: 16 +Case #35: 19 +Case #36: 23 +Case #37: 11 +Case #38: 5 +Case #39: 15 +Case #40: 10 +Case #41: 47 +Case #42: 25 +Case #43: 10 +Case #44: 24 +Case #45: 12 +Case #46: 35 +Case #47: 17 +Case #48: 21 +Case #49: 36 +Case #50: 14 +Case #51: 40 +Case #52: 16 +Case #53: 25 +Case #54: 7 +Case #55: 25 +Case #56: 29 +Case #57: 12 +Case #58: 24 +Case #59: 27 +Case #60: 42 +Case #61: 8 +Case #62: 23 +Case #63: 9 +Case #64: 27 +Case #65: 24 +Case #66: 18 +Case #67: 4 +Case #68: 15 +Case #69: 13 +Case #70: 39 +Case #71: 19 +Case #72: 27 +Case #73: 37 +Case #74: 46 +Case #75: 8 +Case #76: 40 +Case #77: 37 +Case #78: 14 +Case #79: 11 +Case #80: 11 +Case #81: 36 +Case #82: 20 +Case #83: 21 +Case #84: 30 +Case #85: 39 +Case #86: 15 +Case #87: 31 +Case #88: 29 +Case #89: 4 +Case #90: 6 +Case #91: 10 +Case #92: 25 +Case #93: 21 +Case #94: 22 +Case #95: 20 +Case #96: 30 diff --git a/2023/round1/back_in_black_ch1_sol.md b/2023/round1/back_in_black_ch1_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..bdfb565d6eee631e129601179e7dd7a523237173 --- /dev/null +++ b/2023/round1/back_in_black_ch1_sol.md @@ -0,0 +1,32 @@ +First, note that pressing any button twice is equivalent to doing nothing. We can read in all \(Q\) button presses into a boolean array \(P_{1..N}\), where \(P_i\) stores for button \(i\) the total number of presses modulo \(2\). + +Next, we'll need to apply this array of presses to our original array \(S_{1..N}\). Naively, it would look something like: +``` +for (int i = 1; i <= N; i++) { + if (P[i]) { + for (int j = i; j <= N; j += i) { + S[j] ^= 1; + } + } +} +``` + +For \(i = 1\), the inner loop runs for \(N\) steps. For \(i = 2\), the inner loop runs for \(N / 2\) steps. So forth, until a total of \(N + \frac{N}{2} + \frac{N}{3} + ... + \frac{N}{N}\) steps. Factoring out \(N\), we get \(N*(1 + \frac{1}{2} + ... + \frac{1}{N})\). The second factor is the [harmonic series](https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)), which converges to roughly \(\log_2 (N)\). Thus, such a nested loop would only take \(\mathcal{O}(N \log N)\) time. + +After we've applied the presses to \(S\), we now need to convert every \(1\) to \(0\) with the minimum number of button presses. Consider the following greedy algorithm: + +``` +int num_presses = 0; +for (int i = 1; i <= N; i++) { + if (S[i]) { + num_presses++; + for (int j = i; j <= N; j += i) { + S[j] ^= 1; + } + } +} +``` + +We scan from left to right, and each time we see a \(1\), we'll simply press the button and apply the changes to the rest of the array. In fact, this is not a greedy method, but the only method we can convert all the buttons to \(0\). A button can only be changed by pressing itself, or some button before it. If it were not optimal to clear the leftmost \(1\) by pressing it directly, then we would have to press another \(1\) before it, which contradicts the former \(1\) being leftmost. + +The time complexity of this approach algorithm is analyzed similarly as the original application of the \(Q\) button presses, so the overall running time is \(\mathcal{O}(N \log N)\). diff --git a/2023/round1/back_in_black_ch2.cpp b/2023/round1/back_in_black_ch2.cpp new file mode 100644 index 0000000000000000000000000000000000000000..44e877a6a027e1d75515b513c75bf046073c9255 --- /dev/null +++ b/2023/round1/back_in_black_ch2.cpp @@ -0,0 +1,47 @@ +#include +#include +using namespace std; + +long long solve() { + int N, Q; + cin >> N; + vector white(N + 1); + for (int i = 1; i <= N; i++) { + char c; + cin >> c; + white[i] = c - '0'; + } + vector press(N + 1); + int num_presses = 0; + for (int i = 1; i <= N; i++) { + if (white[i]) { + num_presses++; + press[i] = 1; + for (int j = i; j <= N; j += i) { + white[j] = white[j] ^ 1; + } + } + } + long long ans = 0; + cin >> Q; + for (int i = 0, b; i < Q; i++) { + cin >> b; + if (press[b]) { + num_presses--; + } else { + num_presses++; + } + press[b] = press[b] ^ 1; + ans += num_presses; + } + return ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/round1/back_in_black_ch2.in b/2023/round1/back_in_black_ch2.in new file mode 100644 index 0000000000000000000000000000000000000000..f83d64dfee27770a210cc9230fdb1fb5be3f21ba --- /dev/null +++ b/2023/round1/back_in_black_ch2.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9546598c0287c5f491430af5f77725af7f01577cb76d6e79e3a68b2dc5157503 +size 39382254 diff --git a/2023/round1/back_in_black_ch2.md b/2023/round1/back_in_black_ch2.md new file mode 100644 index 0000000000000000000000000000000000000000..9bd259f37314bbad56303c8c29e8108abe180c58 --- /dev/null +++ b/2023/round1/back_in_black_ch2.md @@ -0,0 +1,31 @@ +*This problem shares some similarities with C1, with key differences in bold.* + +A friend who works at Metal Platforms Inc just lent you a curious puzzle, offering you tickets to a metal concert if you can solve it. + +The puzzle consists of \(N\) buttons in a row, numbered from \(1\) to \(N\). The initial state of button \(i\) is white if \(S_i = 1\), or black if \(S_i = 0\). Pressing a button \(k\) toggles the state of itself as well as every \(k\)th button. Your friend challenges you to return the puzzle to him with all buttons back in black. + +Life is hard enough without siblings pushing your buttons. Unfortunately, your brother has taken the puzzle and will push \(Q\) buttons sequentially, the \(i\)th being button \(B_i\). + +After your brother **pushes each button**, you'd like to add to your answer the minimum number of button presses required to turn all the buttons black. + +**Constraints** + +\(1 \le T \le 75\) +\(1 \le N \le 4{,}000{,}000\) +\(1 \le Q \le 4{,}000{,}000\) + +The sum of \(N\) and \(Q\) over all cases will be at most \(9{,}000{,}000\). + +**Input Format** + +Input begins with a single integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\). Then, there is a line containing a bitstring \(S\) of length \(N\). Then, there is a line containing a single integer \(Q\). \(Q\) lines follow, the \(i\)th of which contains a single integer \(B_i\). + +**Output Format** + +For the \(i\)th test case, output a single line containing `"Case #i: "` followed by a single integer, the sum of the number of button presses needed to turn all buttons black **after each of the \(Q\) presses.** + +**Sample Explanation** + +In the first case, after your brother presses the first button, the state of the puzzle is \(0101\). The best strategy is to press the second button, turning all lights off. + +In the second case, the puzzle starts as \(0001\), and after each button press becomes \(0100\), \(0110\), \(0011\), and \(0010\) respectively. The answers after each query are \(2\), \(3\), \(2\), and \(1\) respectively, so we output their sum of \(8\). diff --git a/2023/round1/back_in_black_ch2.out b/2023/round1/back_in_black_ch2.out new file mode 100644 index 0000000000000000000000000000000000000000..b541f233e5cf3c9032a706b349521a4ee58eb00e --- /dev/null +++ b/2023/round1/back_in_black_ch2.out @@ -0,0 +1,100 @@ +Case #1: 1 +Case #2: 8 +Case #3: 36 +Case #4: 0 +Case #5: 0 +Case #6: 11333979994 +Case #7: 4584086000 +Case #8: 50 +Case #9: 50 +Case #10: 0 +Case #11: 0 +Case #12: 0 +Case #13: 0 +Case #14: 23 +Case #15: 27 +Case #16: 19 +Case #17: 23 +Case #18: 480637563786 +Case #19: 500470223042 +Case #20: 7861756929460 +Case #21: 77 +Case #22: 573 +Case #23: 2074 +Case #24: 1764 +Case #25: 307 +Case #26: 1881 +Case #27: 4117 +Case #28: 2453 +Case #29: 1594 +Case #30: 682 +Case #31: 3217 +Case #32: 1709 +Case #33: 1665 +Case #34: 2171 +Case #35: 10 +Case #36: 2393 +Case #37: 2738 +Case #38: 4652 +Case #39: 1294 +Case #40: 1924 +Case #41: 339 +Case #42: 2154 +Case #43: 68 +Case #44: 1766 +Case #45: 3759 +Case #46: 1043 +Case #47: 128 +Case #48: 1945 +Case #49: 118 +Case #50: 1668 +Case #51: 379 +Case #52: 1151 +Case #53: 2449 +Case #54: 2648 +Case #55: 1421 +Case #56: 44 +Case #57: 3706 +Case #58: 333 +Case #59: 142 +Case #60: 3088 +Case #61: 1262 +Case #62: 91 +Case #63: 645 +Case #64: 559 +Case #65: 2363 +Case #66: 2043 +Case #67: 650 +Case #68: 836 +Case #69: 683 +Case #70: 67 +Case #71: 1750 +Case #72: 56 +Case #73: 573 +Case #74: 785 +Case #75: 768 +Case #76: 139 +Case #77: 2329 +Case #78: 120 +Case #79: 13 +Case #80: 3880 +Case #81: 2917 +Case #82: 1101 +Case #83: 937 +Case #84: 136 +Case #85: 656 +Case #86: 1632 +Case #87: 1943 +Case #88: 2395 +Case #89: 3518 +Case #90: 724 +Case #91: 1462 +Case #92: 115 +Case #93: 141 +Case #94: 1237 +Case #95: 1347 +Case #96: 6 +Case #97: 313 +Case #98: 377 +Case #99: 140 +Case #100: 353 diff --git a/2023/round1/back_in_black_ch2_sol.md b/2023/round1/back_in_black_ch2_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..388d3597ae5bac38b9fdb8b5c42df95c07554edb --- /dev/null +++ b/2023/round1/back_in_black_ch2_sol.md @@ -0,0 +1,7 @@ +In this part, it won't be efficient enough to directly simulate zeroing out the array in \(\mathcal{O}(N \log N)\) after each of the \(Q\) queries. Instead, we can track the number of presses needed as we go. + +Before your brother presses anything, we can first compute the array \(P_{1..N}\) indicating whether a press is needed to zero out the initial array \(S_{1..N}\), using the same greedy algorithm from chapter 1. We can then keep a counter of the minimum number of presses needed. This part takes \(\mathcal{O}(N \log N)\) time. + +From this point, if your brother presses some button \(i\) while previously \(P_i = 1\), then he has done us a favor, and we would no longer need to press that button to zero out the array. We can set \(P_i = 0\) and decrement our counter. Otherwise, if he pressed \(i\) while \(P_i = 0\) then we would now need to set \(P_i = 1\) and increment our counter. + +The key is that the counter always holds the optimal number of presses to zero out the array, so we increment the overall answer by the counter after each of the \(Q\) queries. Each query takes \(\mathcal{O}(1)\) to process, so overall running time is \(\mathcal{O}(N \log N + Q)\). diff --git a/2023/round1/bohemian_rap-sody.cpp b/2023/round1/bohemian_rap-sody.cpp new file mode 100644 index 0000000000000000000000000000000000000000..affec975d2b141d21895f8e0790d39864247df10 --- /dev/null +++ b/2023/round1/bohemian_rap-sody.cpp @@ -0,0 +1,164 @@ +#include +#include +#include +#include +#include +#include +using namespace std; + +using int64 = long long; + +int64 N, Q, sqrtsz; +vector freq, ff; + +void inc(int64 x) { + freq[x]++; + ff[freq[x]]++; +} + +void dec(int64 x) { + ff[freq[x]]--; + freq[x]--; +} + +int64 get_ans() { + int64 ans = ff[1]; + for (int i = 2; i <= ans; i++) { + int64 need = ans - (i - 1); + if (ff[i] < need) { + ans -= need - ff[i]; + } + } + return ans; +} + +int64 solve() { + cin >> N; + + vector vs(N); + for (int i = 0; i < N; i++) { + cin >> vs[i]; + } + int64 sumc = 0; + int maxt = 0; + for (int i = 0; i < N; i++) { + reverse(vs[i].begin(), vs[i].end()); + sumc += vs[i].size(); + maxt = max(maxt, (int)vs[i].size()); + } + + ff = vector(N + 10); + freq = vector(sumc + 10); + + int idt = 0; + vector> prox(sumc + 10, vector(26)); + + for (auto &s : vs) { + int est = 0; + for (auto &c : s) { + int ic = c - 'a'; + if (!prox[est][ic]) { + idt++; + prox[est][ic] = idt; + } + est = prox[est][ic]; + } + } + + vector vest(N); + set active; + for (int i = 0; i < N; i++) { + active.insert(i); + } + vector>> vq(maxt + 1); + + cin >> Q; + vector ans(Q); + for (int i = 0; i < Q; i++) { + int64 l, r, k; + cin >> l >> r >> k; + l--, r--; + if (k <= maxt) { + vq[k].emplace_back(l, r, i); + } + } + + // Reset ff only by the number of active! (in reality you only need sqrt). + // No need to reset freq, because they are all different states! + for (int k = 1; k <= maxt; k++) { + for (int i = 0; i <= (int)active.size(); i++) { + ff[i] = 0; + } + + for (auto it = active.begin(); it != active.end();) { + int i = *it; + if (k > (int)vs[i].size()) { + it = active.erase(it); + } else { + int ic = vs[i][k - 1] - 'a'; + vest[i] = prox[vest[i]][ic]; + it++; + } + } + vector vid(active.begin(), active.end()); + + vector> vqq; + vqq.reserve(vq[k].size()); + for (auto &[lc, rc, ia] : vq[k]) { + lc = lower_bound(vid.begin(), vid.end(), lc) - vid.begin(); + rc = upper_bound(vid.begin(), vid.end(), rc) - vid.begin(); + rc--; + if (lc <= rc) { + vqq.emplace_back(lc, rc, ia); + } + } + vq[k] = vqq; + sqrtsz = sqrt(vid.size()) + 1; + sort(vq[k].begin(), vq[k].end(), + [](tuple &a, tuple &b) { + int64 la, ra, ia; + tie(la, ra, ia) = a; + int64 lb, rb, ib; + tie(lb, rb, ib) = b; + if (la / sqrtsz != lb / sqrtsz) { + return la < lb; + } + return ra < rb; + }); + + int l = 0, r = -1; + for (auto &[lc, rc, ia] : vq[k]) { + while (r < rc) { + r++; + inc(vest[vid[r]]); + } + while (l > lc) { + l--; + inc(vest[vid[l]]); + } + while (l < lc) { + dec(vest[vid[l]]); + l++; + } + while (r > rc) { + dec(vest[vid[r]]); + r--; + } + ans[ia] = get_ans(); + } + } + int64 res = 0; + for (int i = 0; i < Q; i++) { + res += ans[i]; + } + return res; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/round1/bohemian_rap-sody.in b/2023/round1/bohemian_rap-sody.in new file mode 100644 index 0000000000000000000000000000000000000000..fb3ba9dbde9ffe9a2563b71a8252518bac5dca8a --- /dev/null +++ b/2023/round1/bohemian_rap-sody.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e78e15e34dc945d0c8be6bbf463934816b6190ec03ff39bf9c9b3b61dfc5a047 +size 24649825 diff --git a/2023/round1/bohemian_rap-sody.md b/2023/round1/bohemian_rap-sody.md new file mode 100644 index 0000000000000000000000000000000000000000..d9beba334124f7ac71f60fd91830212cd15d8639 --- /dev/null +++ b/2023/round1/bohemian_rap-sody.md @@ -0,0 +1,49 @@ +*Is this a real verse? Is this just fantasy? +Caught in a mixtape, no escape from this melody. +Open your ears, listen to the beats and see. +I'm just a rapper, I need no sympathy, +Because it's easy come, easy go, little high, little low, +Any way the bars flow doesn't really matter to me, to me.* + +As an up-and-coming rapper, you're trying to distinguish yourself by writing a song with the most bohemian rhyme scheme! + +Your vocabulary consists of \(N\) words, \(W_1\), \(…\), \(W_N\) made up of only lowercase letters. We consider two words to *\(K\)-rhyme* with each other if they have the same suffix of length \(K\). For example, “mixtape” 3-rhymes with “escape”. If two words \(K\)-rhyme for \(K > 1\), they also \((K-1)\)-rhyme. + +You are faced with \(Q\) bohemian rap-queries, the \(i\)th of which specifies \((A_i, B_i, K_i)\). You’d like to assign each word \(W_{A_i}\), \(…\), \(W_{B_i}\) to at most one group such that: + +* the number of words in each group is unique and nonzero, +* groups only contain words with length at least \(K_i\), +* within each group, all pairs of words \(K_i\)-rhyme with each other, and +* no two words from different groups \(K_i\)-rhyme. + +The answer to each query is the maximum possible number of nonempty groups. For each test case, output the sum of the answers over all queries. + +**Constraints** + +\(1 \le T \le 20\) +\(1 \le N \le 600{,}000\) +\(\sum |W_i| \le 3{,}000{,}000\) +\(1 \le Q \le 400{,}000\) +\(1 \le A_i \le B_i \le N\) +\(1 \le K_i \le 1{,}000{,}000\) +All words consist of only letters \(\{\texttt{`a'}, ..., \texttt{`z'}\}\) +All words within a test case are distinct. + +The sum of \(N\) across all test cases is at most \(2{,}000{,}000\). +The sum of \(Q\) across all test cases is at most \(1{,}000{,}000\). +The sum of word lengths across all test cases is at most \(10{,}000{,}000\). + +**Input Format** + +Input begins with an integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\). \(N\) lines follow, the \(i\)th of which contains the string \(W_i\). Then, there is a line containing a single integer \(Q\). \(Q\) lines follow, the \(i\)th of which contains \(3\) space-separated integers \(A_i\), \(B_i\), and \(K_i\). + +**Output Format** + +For the \(i\)th test case, print "`Case #i:` " followed by a single integer, the sum of answers over all queries. + +**Sample Explanation** + +For the first query in the first test case, we are allowed to choose from all \(W_1, ..., W_8\) words to form \(3\) groups where words of the same group \(1\)-rhyme with each other. One possible choice of the groups is: \(\{\)"low"\(\}\), \(\{\)"sympathy", "fantasy", "melody"\(\}\), \(\{\)"mixtape", "escape"\(\}\) with "come" and "high" going unused. Therefore the answer to the first query is \(3\). The answers to the remaining queries are \(1\), \(2\) and \(2\), with the overall sum of answers for this test case equal to \(8\). + +In the second test case, the answers to the \(3\) queries are \([2, 1, 0]\) with the sum being \(3\). + diff --git a/2023/round1/bohemian_rap-sody.out b/2023/round1/bohemian_rap-sody.out new file mode 100644 index 0000000000000000000000000000000000000000..1f06c2fa0e11ed965b54e1df0cb71bac4b9281af --- /dev/null +++ b/2023/round1/bohemian_rap-sody.out @@ -0,0 +1,25 @@ +Case #1: 8 +Case #2: 3 +Case #3: 15 +Case #4: 45519460 +Case #5: 3 +Case #6: 4 +Case #7: 21017 +Case #8: 5 +Case #9: 7 +Case #10: 70212 +Case #11: 11 +Case #12: 16 +Case #13: 200596 +Case #14: 232071628 +Case #15: 68068408 +Case #16: 31889927 +Case #17: 3 +Case #18: 4 +Case #19: 6741 +Case #20: 5 +Case #21: 7 +Case #22: 24492 +Case #23: 11 +Case #24: 16 +Case #25: 70705 diff --git a/2023/round1/bohemian_rap-sody_sol.md b/2023/round1/bohemian_rap-sody_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..10975ab677ec60105bccf143dd280fa6e58e2911 --- /dev/null +++ b/2023/round1/bohemian_rap-sody_sol.md @@ -0,0 +1,7 @@ +The problem is equivalent to finding the maximum \(x\) where you can make \(x\) groups with distinct sizes \(1, ..., x\), and all words that belong to a certain group pairwise \(K\)-rhyme with each other. + +The first step is to simplify things by reversing all strings so that we're dealing with prefixes of length \(K\) instead of suffixes. Then, order queries by increasing \(K\). For equal \(K\), we sort queries based on [Mo’s algorithm](https://cp-algorithms.com/data_structures/sqrt_decomposition.html#mos-algorithm). + +Note that there are \(\sum |W_i|\) possible prefixes across all the words. We can uniquely identify each prefix by a rolling hashing on each word, or by inserting each word into a trie and using the trie node ID or memory address. These are the values for which we are maintaining the frequency within Mo’s algorithm. + +At each query being handled by Mo's algorithm, we should be careful to only process active words for the current size \(K\). The overall time complexity is \(\mathcal{O}((\sum |W_i|)*\sqrt N)\). In practice, our model solution solves the full test set in under \(10\) seconds. diff --git a/2023/round1/here_comes_santa_claus.cpp b/2023/round1/here_comes_santa_claus.cpp new file mode 100644 index 0000000000000000000000000000000000000000..6f4cf25451fb1be5d5dec0708b69c9c0c0b7cf98 --- /dev/null +++ b/2023/round1/here_comes_santa_claus.cpp @@ -0,0 +1,33 @@ +#include +#include +#include +#include +using namespace std; + +double solve() { + int N; + cin >> N; + vector X(N); + for (int i = 0; i < N; i++) { + cin >> X[i]; + } + sort(X.begin(), X.end()); + double ans = X[N-1] + X[N-2] - (X[0] + X[1]); + if (N == 5) { + ans = max( + X[N-1] + X[N-3] - (X[0] + X[1]), + X[N-1] + X[N-2] - (X[0] + X[2]) + ); + } + return ans / 2; +} + +int main() { + int T; + cin >> T; + cout.precision(1); + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << fixed << solve() << endl; + } + return 0; +} diff --git a/2023/round1/here_comes_santa_claus.in b/2023/round1/here_comes_santa_claus.in new 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911912176 321816734 +5 +870 143 900 767 465 +5 +825091323 400774504 930562087 409992963 509192680 +5 +958 1084 1484 823 1331 +5 +858523084 471766505 314253610 915803897 285700013 +5 +96386 6814 55877 8245 26485 +5 +525690203 434302497 484331282 920373956 2305853 +5 +8119 18497 55754 3415 50827 +5 +27171115 922482835 238516391 835616355 927841962 +5 +2 2 1 2 4 +5 +689407424 788767265 971675203 442110187 292594024 diff --git a/2023/round1/here_comes_santa_claus.md b/2023/round1/here_comes_santa_claus.md new file mode 100644 index 0000000000000000000000000000000000000000..3f043fc7d6fcf635b309957bf87b8a61265581f4 --- /dev/null +++ b/2023/round1/here_comes_santa_claus.md @@ -0,0 +1,36 @@ +*…right down Santa Claus Lane!* + +Santa Claus Lane is home to \(N\) elves, the \(i\)th of whom lives \(X_i\) meters from the start of the lane. As Santa's little helper, you're tasked to assign elves to work on toys for good little girls and boys. Specifically, you must assign each elf to work on some toy so that at least \(2\) toys get worked on, and no elf works on a toy alone. + +All elves working on the same toy will meet at the point which minimizes the farthest distance that any of them would need to walk. Formally, if the elves assigned to a given toy live at \(X_1\), \(X_2\), \(…\), then they will meet at point \(P\) such that \(\max(|X_1 - P|\), \(|X_2 - P|\), \(…\)\()\) is as small as possible. + +For instance, the first sample case is depicted below: + +{{PHOTO_ID:838516607729325|WIDTH:700}} + +Santa is supervising, and you reckon he could use some exercise. Among all valid assignments of elves to toys, what's the farthest Santa would need to walk to visit all meeting points? Santa may start and end anywhere, but will try to walk as little as possible after seeing your assignments. + +# Constraints + +\(1 \leq T \leq 20\) +\(4 \leq N \leq 10^5 \) +\(1 \leq X_i \leq 10^9 \) + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case begins with one line containing the integer \(N\), followed by a second line containing the the \(N\) integers \(X_1 ... X_N\). + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by a single real number, the farthest distance Santa would need to walk in meters. Your answer will be considered correct if it differs from the jury's answer by at most \(10^{-6}\) meters or at most \(0.0001\)%, whichever is larger. + +# Sample Explanation + +In the first sample case, elves living at \(1 \,\text{m}\) and \(3 \,\text{m}\) will work on a toy. Elves living at \(7 \,\text{m}\), \(12 \,\text{m}\), and \(13 \,\text{m}\) will work on another toy, and elves at \(17 \,\text{m}\) and \(23 \,\text{m}\) will work on the third toy. The toys will be made at \(2 \,\text{m}\), \(10 \,\text{m}\) and \(20 \,\text{m}\) respectively. Santa would need to walk at least \(18 \,\text{m}\) in total to visit every meeting point. + +The second sample case is depicted below. No elf is allowed to work alone and we must make two toys. One optimal way of doing this is to have the leftmost \(3\) elves work on one toy at \(2 \,\text{m}\), and the rest work on the other toy at \(4.5 \,\text{m}\). This would maximize the distance Santa would have to walk among all valid elf assignments. + +{{PHOTO_ID:163905706775126|WIDTH:400}} + +In the third case, the two toys will be made at \(55 \,\text{m}\) and \(5500 \,\text{m}\). + diff --git a/2023/round1/here_comes_santa_claus.out b/2023/round1/here_comes_santa_claus.out new file mode 100644 index 0000000000000000000000000000000000000000..b3f2be408ce02ddf809b1f32591567ebea88d5e2 --- /dev/null +++ b/2023/round1/here_comes_santa_claus.out @@ -0,0 +1,24 @@ +Case #1: 18 +Case #2: 2.5 +Case #3: 5445 +Case #4: 3.5 +Case #5: 6.5 +Case #6: 58.5 +Case #7: 483.5 +Case #8: 512070178.5 +Case #9: 591879804.5 +Case #10: 786748034.5 +Case #11: 997872980 +Case #12: 999983020 +Case #13: 999978784.5 +Case #14: 999982689 +Case #15: 529.5 +Case #16: 422843113 +Case #17: 454 +Case #18: 508430231.5 +Case #19: 59482 +Case #20: 484048444 +Case #21: 42334.5 +Case #22: 748885405.5 +Case #23: 1.5 +Case #24: 463189208 diff --git a/2023/round1/here_comes_santa_claus_sol.md b/2023/round1/here_comes_santa_claus_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..121f93cbff43f3b43206ed27352f845cf4c451ea --- /dev/null +++ b/2023/round1/here_comes_santa_claus_sol.md @@ -0,0 +1,8 @@ +For a given assignment, Santa will always walk from the leftmost toy to the rightmost toy. It's in our interest to get these two meeting points as far as possible. This is equivalent to independently minimizing the leftmost meeting point, while maximing the rightmost meeting point. + +For all elves assigned to a toy, the meeting point will always be the average of the leftmost and rightmost elf. To minimize the leftmost meeting point, it's intuitive that grouping only the \(2\) leftmost elves will get the average as small as possible. Likewise, we group the \(2\) rightmost elves. + +We can sort the elves by ascending \(x\)-coordinates \(X_1' < ... < X_N'\), find the left meeting point as \((X_2' - X_1')/2\) and the right meeting point as \((X_{N}' - X_{N-1}')/2\). The answer (i.e. the distance of Santa's walk) will be the difference between those two meeting points. + +The only special case occurs if there are \(N = 5\) elves. If we did the above, we would be left with a single elf in the middle who isn't allowed to work on a toy alone. We must instead include the middle elf in one of the other two groups. When \(N \ge 6\), there are at least \(2\) middle elves in our main strategy, so they won't pose an issue. + diff --git a/2023/round1/sum_41_ch1.cpp b/2023/round1/sum_41_ch1.cpp new file mode 100644 index 0000000000000000000000000000000000000000..8c544dbb7845e24cbe526996052f3e03ce6660e6 --- /dev/null +++ b/2023/round1/sum_41_ch1.cpp @@ -0,0 +1,46 @@ +#include +#include +using namespace std; + +void solve() { + int P; + cin >> P; + int curr_sum = 0; + vector factors; + for (int i = 2; i*i <= P; ) { + if (P % i == 0) { + factors.push_back(i); + curr_sum += i; + P /= i; + } else { + i++; + } + } + if (P > 1) { + curr_sum += P; + factors.push_back(P); + } + if (curr_sum > 41) { + cout << " -1"; + return; + } + while (curr_sum < 41) { + factors.push_back(1); + curr_sum++; + } + cout << " " << factors.size(); + for (int f : factors) { + cout << " " << f; + } +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ":"; + solve(); + cout << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round1/sum_41_ch1.in b/2023/round1/sum_41_ch1.in new file mode 100644 index 0000000000000000000000000000000000000000..790763c86cb9d5e9856f970938ff91215bc183db --- /dev/null +++ b/2023/round1/sum_41_ch1.in @@ -0,0 +1,977 @@ +976 +20592 +2400 +900360029 +6840 +16128 +15876 +54432 +110250 +31200 +8960 +5280 +150528 +43200 +56700 +4608 +725573 +673921 +30 +194241 +36864 +633 +125 +5824 +19008 +279936 +170 +21383 +5657 +900 +17550 +22176 +116 +787320 +68040 +14336 +420 +2403381 +16224 +813 +46464 +46 +18 +25600 +4160 +40 +8826131 +1530 +8087519 +15120 +72900 +56448 +94770 +19656 +180000 +672 +17920 +70400 +39600 +165888 +128 +80 +21 +7560 +28672 +29160 +3 +116640 +4536 +1817 +34 +33108 +225792 +94 +364500 +16800 +13056 +1764 +20 +5250 +1249 +4050 +12000 +16000 +131980 +50000 +98 +3375 +1000000000 +938017885 +5120 +11664 +69 +8424 +74088 +153600 +216 +7455307 +559 +35 +196608 +782 +97200 +172032 +388800 +29400 +216000 +28800 +8660 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0000000000000000000000000000000000000000..931d84f3133ba87f3ae3f9163e956e3d42a65630 --- /dev/null +++ b/2023/round1/sum_41_ch1.md @@ -0,0 +1,21 @@ +*This problem shares some similarities with B2, with key differences in bold.* + +Given a positive integer \(P\), please find an array of at most \(100\) positive integers which have a sum of \(41\) and a product of \(P\), or output \(-1\) if no such array exists. + +If multiple such arrays exist, **you may output any one of them**. + +# Constraints +\(1 \leq T \leq 965\) +\(1 \leq P \leq 10^9\) + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, there is one line containing a single integer \(P\). + +# Output Format + +For the \(i\)th test case, if there is no such array, print "`Case #i: -1`". Otherwise, print "`Case #i:` " followed by the integer \(N\), the size of your array, followed by the array itself as \(N\) more space-separated positive integers. + +# Sample Explanation +In the first sample, we must find an array with product \(2023\), and sum \(41\). One possible answer is \([7, 17, 17]\). + diff --git a/2023/round1/sum_41_ch1.out b/2023/round1/sum_41_ch1.out new file mode 100644 index 0000000000000000000000000000000000000000..b56669066e21079bf225c12064501e4f3bf46069 --- /dev/null +++ b/2023/round1/sum_41_ch1.out @@ -0,0 +1,976 @@ +Case #1: 5 4 4 9 11 13 +Case #2: 4 2 6 8 25 +Case #3: -1 +Case #4: 4 6 6 10 19 +Case #5: 6 1 4 4 6 12 14 +Case #6: 6 3 3 3 4 7 21 +Case #7: 6 3 6 6 6 6 14 +Case #8: 8 1 3 3 5 5 7 7 10 +Case #9: 6 2 4 5 5 12 13 +Case #10: 5 1 8 8 10 14 +Case #11: 5 2 3 4 10 22 +Case #12: 8 1 4 4 4 6 7 7 8 +Case #13: 6 4 4 4 5 9 15 +Case #14: 7 3 3 3 3 5 10 14 +Case #15: 6 1 2 4 4 6 24 +Case #16: -1 +Case #17: -1 +Case #18: 12 1 1 1 1 1 1 1 1 1 1 1 30 +Case #19: -1 +Case #20: 7 1 4 4 4 4 12 12 +Case #21: -1 +Case #22: 13 1 1 1 1 1 1 1 1 1 1 1 5 25 +Case #23: 6 1 1 4 8 13 14 +Case #24: 6 1 4 4 9 11 12 +Case #25: 9 3 3 3 3 3 4 4 6 12 +Case #26: 4 1 1 5 34 +Case #27: -1 +Case #28: -1 +Case #29: 6 1 1 1 3 15 20 +Case #30: 6 1 3 5 9 10 13 +Case #31: 5 4 6 6 11 14 +Case #32: 10 1 1 1 1 1 1 1 1 4 29 +Case #33: 11 3 3 3 3 3 3 3 3 3 4 10 +Case #34: 7 3 3 3 4 6 7 15 +Case #35: 6 1 2 8 8 8 14 +Case #36: 5 1 1 2 7 30 +Case #37: -1 +Case #38: 6 1 4 4 6 13 13 +Case #39: -1 +Case #40: 7 1 4 4 4 6 11 11 +Case #41: 18 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 23 +Case #42: 24 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 18 +Case #43: 6 4 4 4 4 5 20 +Case #44: 4 4 4 13 20 +Case #45: 2 1 40 +Case #46: -1 +Case #47: 6 1 1 1 6 15 17 +Case #48: -1 +Case #49: 5 2 7 8 9 15 +Case #50: 7 3 3 3 3 9 10 10 +Case #51: 6 4 4 6 6 7 14 +Case #52: 8 3 3 3 3 3 3 10 13 +Case #53: 6 2 3 3 6 13 14 +Case #54: 7 4 5 5 5 6 6 10 +Case #55: 5 1 1 2 16 21 +Case #56: 6 1 4 4 8 10 14 +Case #57: 8 1 2 4 4 4 5 10 11 +Case #58: 6 3 3 4 10 10 11 +Case #59: 8 2 3 4 4 4 6 6 12 +Case #60: 7 1 1 1 1 1 4 32 +Case #61: 19 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 20 +Case #62: 21 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 21 +Case #63: 6 1 1 6 9 10 14 +Case #64: 7 1 2 4 4 8 8 14 +Case #65: 7 3 3 3 3 3 6 20 +Case #66: 39 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 +Case #67: 8 3 3 3 3 3 4 10 12 +Case #68: 5 1 4 6 9 21 +Case #69: -1 +Case #70: 8 1 1 1 1 1 1 1 34 +Case #71: -1 +Case #72: 8 2 4 4 4 4 7 7 9 +Case #73: -1 +Case #74: 9 3 3 3 3 3 5 5 6 10 +Case #75: 5 3 4 10 10 14 +Case #76: 5 4 4 4 12 17 +Case #77: 4 3 3 7 28 +Case #78: 22 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 20 +Case #79: 6 1 1 5 5 14 15 +Case #80: -1 +Case #81: 6 1 3 3 3 6 25 +Case #82: 6 1 4 5 5 6 20 +Case #83: 7 1 2 4 4 5 5 20 +Case #84: -1 +Case #85: 8 1 1 4 5 5 5 10 10 +Case #86: 22 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 14 +Case #87: 7 1 1 1 3 5 15 15 +Case #88: -1 +Case #89: -1 +Case #90: 5 1 4 8 8 20 +Case #91: 7 2 3 3 3 3 3 24 +Case #92: 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 23 +Case #93: 5 1 6 9 12 13 +Case #94: 7 3 3 3 4 7 7 14 +Case #95: 8 2 3 4 4 4 4 10 10 +Case #96: 8 1 1 1 1 1 1 8 27 +Case #97: -1 +Case #98: -1 +Case #99: 7 1 1 1 1 1 1 35 +Case #100: 9 1 4 4 4 4 4 4 4 12 +Case #101: -1 +Case #102: 7 3 3 3 6 6 10 10 +Case #103: 8 3 4 4 4 4 4 4 14 +Case #104: 9 3 3 3 3 4 4 5 6 10 +Case #105: 6 2 3 5 7 10 14 +Case #106: 8 2 3 4 5 5 6 6 10 +Case #107: 7 2 3 3 4 4 5 20 +Case #108: -1 +Case #109: -1 +Case #110: 9 2 4 4 4 4 4 4 6 9 +Case #111: 6 4 4 4 8 9 12 +Case #112: 5 1 4 6 7 23 +Case #113: 9 1 1 1 1 1 1 2 15 18 +Case #114: 9 1 1 1 1 1 3 9 10 14 +Case #115: 7 3 3 3 4 4 4 20 +Case #116: 7 3 4 4 4 6 6 14 +Case #117: 15 1 1 1 1 1 1 1 1 1 1 1 1 1 4 24 +Case #118: 8 2 3 3 3 4 4 10 12 +Case #119: 10 1 1 1 1 1 1 1 1 2 31 +Case #120: 6 3 3 3 10 11 11 +Case #121: 8 1 2 4 4 4 8 8 10 +Case #122: 9 1 4 4 4 5 5 5 6 7 +Case #123: 6 1 1 3 9 9 18 +Case #124: -1 +Case #125: 13 1 1 1 1 1 1 1 1 1 1 1 1 29 +Case #126: 7 1 1 1 1 1 17 19 +Case #127: 7 2 4 4 4 7 8 12 +Case #128: -1 +Case #129: 7 2 3 4 5 6 7 14 +Case #130: -1 +Case #131: 7 2 4 4 4 6 7 14 +Case #132: 4 1 5 15 20 +Case #133: -1 +Case #134: 7 2 3 4 4 6 10 12 +Case #135: 5 4 4 7 9 17 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The problem reduces to finding any factorization \(P = f_1 * ... * f_k\) such that every \(f_i \ge 2\) and \(\sum f_i \le 41\). We can simply pad out the list with \(1\)s until we reach \(41\). + +Let's consider the prime factorization of \(P = p_1^{k_1} * p_2^{k_2} ... p_M^{k_M}\). We claim that if an answer exists, it can always be attained by taking the prime factorization and padding out the rest with \(1\)s. + +To see why this is true, note that for any two integers \(a, b > 1\) it's always true that \(a*b \ge a+b\). Since composite factors can only be attained by merging prime factors, the overall sum can only stay the same or increase if we use composite factors. We might as well just use the prime factorization if it sums less than or equal to \(41\) (otherwise the answer does not exist), and then pad with \(1\)s. + +The prime factorization can be generated in \(\mathcal{O}(\sqrt P)\), and padding up to \(41\) takes constant time, so the overall running time is \(\mathcal{O}(\sqrt P)\). diff --git a/2023/round1/sum_41_ch2.cpp b/2023/round1/sum_41_ch2.cpp new file mode 100644 index 0000000000000000000000000000000000000000..b8f53f88fab0c506bb6add8c45af188bd73b3854 --- /dev/null +++ b/2023/round1/sum_41_ch2.cpp @@ -0,0 +1,81 @@ +#include +#include +#include +using namespace std; + +int curr_sum, best_len; +vector divisors, curr, best_ans; + +int maxsteps = 0; +int steps = 0; + +void rec(int P) { + steps++; + if (P == 1) { + int len = curr.size() + (41 - curr_sum); + if (len < best_len) { + best_len = len; + best_ans = curr; + } + return; + } + for (int d : divisors) { + if (P % d == 0 && curr_sum + d <= 41) { + curr.push_back(d); + curr_sum += d; + rec(P / d); + curr_sum -= d; + curr.pop_back(); + } + } +} + +void solve() { + int P; + cin >> P; + if (P == 1) { + best_ans.assign(41, 1); + } else { + divisors.clear(); + // Precompute all divisors. + for (int i = 2; i <= 41; i++) { + if (P % i == 0) { + divisors.push_back(i); + } + } + curr.clear(); + steps = 0; + curr_sum = 0; + best_ans.clear(); + best_len = (int)1e9; + rec(P); + if (steps > maxsteps) { + maxsteps = steps; + cerr << P << endl; + } + } + if (!best_ans.empty()) { + int best_sum = accumulate(best_ans.begin(), best_ans.end(), 0); + for (; best_sum < 41; best_sum++) { + best_ans.push_back(1); + } + cout << " " << best_ans.size(); + for (int f : best_ans) { + cout << " " << f; + } + } else { + cout << " -1"; + } +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ":"; + solve(); + cout << endl; + } + cerr << maxsteps; + return 0; +} \ No newline at end of file diff --git a/2023/round1/sum_41_ch2.in 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b/2023/round1/sum_41_ch2.md @@ -0,0 +1,21 @@ +*This problem shares some similarities with B1, with key differences in bold.* + +Given a positive integer \(P\), please find an array of at most \(100\) positive integers which have a sum of \(41\) and a product of \(P\), or output \(-1\) if no such array exists. + +If multiple such arrays exist, **print one with the fewest number of elements**. If there are multiple with the fewest number of elements, you may print any one of them. + +# Constraints +\(1 \leq T \leq 960\) +\(1 \leq P \leq 10^9\) + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, there is one line containing a single integer \(P\). + +# Output Format + +For the \(i\)th test case, if there is no such array, print "`Case #i: -1`". Otherwise, print "`Case #i:` " followed by the integer \(N\), the size of your array, followed by the array itself as \(N\) more space-separated positive integers. + +# Sample Explanation +In the first sample, we must find an array with product \(2023\), and sum \(41\). One possible answer is \([7, 17, 17]\). + diff --git a/2023/round1/sum_41_ch2.out b/2023/round1/sum_41_ch2.out new file mode 100644 index 0000000000000000000000000000000000000000..d4fd08ff8ef8c792acc8776982f9b1216855fd64 --- /dev/null +++ b/2023/round1/sum_41_ch2.out @@ -0,0 +1,976 @@ +Case #1: 5 3 6 8 11 13 +Case #2: 3 10 15 16 +Case #3: -1 +Case #4: 4 5 8 9 19 +Case #5: 5 2 7 8 12 12 +Case #6: 5 2 7 9 9 14 +Case #7: 6 2 6 7 8 9 9 +Case #8: 8 1 2 5 5 5 7 7 9 +Case #9: 6 2 3 5 8 10 13 +Case #10: 4 7 8 10 16 +Case #11: 4 4 6 11 20 +Case #12: 7 3 4 4 7 7 8 8 +Case #13: 6 2 4 6 9 10 10 +Case #14: 6 3 5 5 7 9 12 +Case #15: 4 3 6 16 16 +Case #16: -1 +Case #17: -1 +Case #18: 12 1 1 1 1 1 1 1 1 1 1 1 30 +Case #19: -1 +Case #20: 5 8 8 8 8 9 +Case #21: -1 +Case #22: 13 1 1 1 1 1 1 1 1 1 1 1 5 25 +Case #23: 5 1 4 7 13 16 +Case #24: 6 2 3 3 6 11 16 +Case #25: 8 2 3 6 6 6 6 6 6 +Case #26: 4 1 1 5 34 +Case #27: -1 +Case #28: -1 +Case #29: 3 5 6 30 +Case #30: 6 2 3 3 5 13 15 +Case #31: 5 3 7 8 11 12 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At each step, we consider every possible candidate of \(f_i \in [2, 41]\). Since \(f_i\) has to be a divisor of \(P\) within \([2,41]\), we can just precompute these. We start the backtracking with \(P\), and at each step, divide by each choice of \(f_i\) until either \(P\) becomes \(1\), or the sum of factors used so far exceeds \(41\). + +Will this backtracking be efficient enough? First, note that the deepest we can go is \(k=20\) by choosing each \(f_i=2\) (since minimizing each factor maximizes the array length). This mean \(P \le 2^k \approx 10^6\). It's somewhat difficult to prove an upper bound on this recurrence, but intuitively, the larger the factor we pick at each step, the fewer remaining branching factors there will be. Emprically, the model solution passes the full test set in under \(10\) seconds on a modest system, taking no more than \(20\) million recursive steps on any test case. + +Another approach is to enumerate all [partitions](https://en.wikipedia.org/wiki/Partition\_(number\_theory)) of \(41\), of which there are [under 45000](https://www.wolframalpha.com/input?i=number+of+partitions+of+41). There are many known [algorithms](https://code.activestate.com/recipes/218332/) to do so, taking linear or constant steps per partition. Even exponential ones will likely work in our submission window. After generating each partition of size \(k\), we can check whether the product equals \(P\) in \(\mathcal{O}(k)\), for an overall time complexity of \(\mathcal{O}(45000*k)\). diff --git a/2023/round1/today_is_gonna_be_a_great_day.cpp b/2023/round1/today_is_gonna_be_a_great_day.cpp new file mode 100644 index 0000000000000000000000000000000000000000..cce28f52531471c0464b4c3c7902ea2f21332e44 --- /dev/null +++ b/2023/round1/today_is_gonna_be_a_great_day.cpp @@ -0,0 +1,154 @@ +#include +#include +using namespace std; + +const long long MOD = (int)1e9 + 7; + +struct Node { + long long maxval, maxflip; + int idx_of_maxval, idx_of_maxflip; + + Node() {} + Node(int i, long long v): + maxval(v), idx_of_maxval(i), + maxflip(((long long)v * (MOD - 1)) % MOD), idx_of_maxflip(i) {} +}; + +template +class segment_tree { + static T join_values(const T &a, const T &b) { + Node res; + if (a.maxval > b.maxval || + (a.maxval == b.maxval && a.idx_of_maxval < b.idx_of_maxval)) { + res.maxval = a.maxval; + res.idx_of_maxval = a.idx_of_maxval; + } else { + res.maxval = b.maxval; + res.idx_of_maxval = b.idx_of_maxval; + } + if (a.maxflip > b.maxflip || + (a.maxflip == b.maxflip && a.idx_of_maxflip < b.idx_of_maxflip)) { + res.maxflip = a.maxflip; + res.idx_of_maxflip = a.idx_of_maxflip; + } else { + res.maxflip = b.maxflip; + res.idx_of_maxflip = b.idx_of_maxflip; + } + return res; + } + + static void join_value_with_delta(T &v, const D &d, int len) { + if (d) { // Flip all values in v.lo .. v.hi. + swap(v.maxval, v.maxflip); + swap(v.idx_of_maxval, v.idx_of_maxflip); + } + } + + static D join_deltas(const D &d1, const D &d2) { + return d1 ^ d2; + } + + int len; + std::vector value; + std::vector delta; + std::vector pending; + + template + void build(int i, int lo, int hi, It arr) { + if (lo == hi) { + value[i] = Node(lo, *(arr + lo)); + return; + } + int mid = lo + (hi - lo)/2; + build(i*2 + 1, lo, mid, arr); + build(i*2 + 2, mid + 1, hi, arr); + value[i] = join_values(value[i*2 + 1], value[i*2 + 2]); + } + + void push_delta(int i, int lo, int hi) { + if (pending[i]) { + join_value_with_delta(value[i], delta[i], hi - lo + 1); + if (lo != hi) { + int l = 2*i + 1, r = 2*i + 2; + delta[l] = pending[l] ? join_deltas(delta[l], delta[i]) : delta[i]; + delta[r] = pending[r] ? join_deltas(delta[r], delta[i]) : delta[i]; + pending[l] = pending[r] = true; + } + pending[i] = false; + } + } + + T query(int i, int lo, int hi, int tgt_lo, int tgt_hi) { + push_delta(i, lo, hi); + if (lo == tgt_lo && hi == tgt_hi) { + return value[i]; + } + int mid = lo + (hi - lo)/2; + if (tgt_lo <= mid && mid < tgt_hi) { + return join_values( + query(i*2 + 1, lo, mid, tgt_lo, std::max(tgt_hi, mid)), + query(i*2 + 2, mid + 1, hi, std::max(tgt_lo, mid + 1), tgt_hi)); + } + if (tgt_lo <= mid) { + return query(i*2 + 1, lo, mid, tgt_lo, std::min(tgt_hi, mid)); + } + return query(i*2 + 2, mid + 1, hi, std::max(tgt_lo, mid + 1), tgt_hi); + } + + void update(int i, int lo, int hi, int tgt_lo, int tgt_hi, const D &d) { + push_delta(i, lo, hi); + if (hi < tgt_lo || lo > tgt_hi) { + return; + } + if (tgt_lo <= lo && hi <= tgt_hi) { + delta[i] = d; + pending[i] = true; + push_delta(i, lo, hi); + return; + } + update(2*i + 1, lo, (lo + hi)/2, tgt_lo, tgt_hi, d); + update(2*i + 2, (lo + hi)/2 + 1, hi, tgt_lo, tgt_hi, d); + value[i] = join_values(value[2*i + 1], value[2*i + 2]); + } + + public: + template + segment_tree(It lo, It hi) + : len(hi - lo), value(4*len), delta(4*len), pending(4*len, false) { + build(0, 0, len - 1, lo); + } + + int size() const { return len; } + T query(int lo, int hi) { return query(0, 0, len - 1, lo, hi); } + void update(int i, const D &d) { update(0, 0, len - 1, i, i, d); } + void update(int lo, int hi, const D &d) { update(0, 0, len - 1, lo, hi, d); } +}; + +long long solve() { + int N, Q; + cin >> N; + vector A(N); + for (int i = 0; i < N; i++) { + cin >> A[i]; + } + segment_tree S(A.begin(), A.end()); + long long ans = 0; + cin >> Q; + for (int i = 0, l, r; i < Q; i++) { + cin >> l >> r; + l--; + r--; + S.update(l, r, 1); + ans += S.query(0, N - 1).idx_of_maxval+1; + } + return ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round1/today_is_gonna_be_a_great_day.in b/2023/round1/today_is_gonna_be_a_great_day.in new file mode 100644 index 0000000000000000000000000000000000000000..665df05182b7116c227821f2b47ea5c0ed197c15 --- /dev/null +++ b/2023/round1/today_is_gonna_be_a_great_day.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2e35792f53f2bb5b364dd6af9dcce9d2d20dbf97eed08ff60fd5fb526f110b0e +size 39924894 diff --git a/2023/round1/today_is_gonna_be_a_great_day.md b/2023/round1/today_is_gonna_be_a_great_day.md new file mode 100644 index 0000000000000000000000000000000000000000..8f3cfe1d1b1c082b87df5910117f7a8594932988 --- /dev/null +++ b/2023/round1/today_is_gonna_be_a_great_day.md @@ -0,0 +1,31 @@ +*This could possibly be the best day ever! +And the forecast says that tomorrow will likely be a billion and 6 times better. +So make every minute count, jump up, jump in, and seize the day, +And let's make sure that in every single possible way… today is gonna be a great day!* + +There's \(N\) days of summer vacation, and the \(i\)th day initially has greatness \(A_i\). However, the days just keep getting better! In particular, there will be \(Q\) changes, the \(i\)th of which changes the greatness of every day from \(L_i\) to \(R_i\) to be \(1\,000\,000\,006\) times what they were before! + +After each query, you'd like to know: which day's greatness modulo \(1\,000\,000\,007\) is the largest? If multiple days tie, we're interested in the earliest of them. For example, if there are \(N = 4\) days with greatnesses \([4, 1, 4, 2]\), then days \(1\) and \(3\) have the largest remainder, so our answer would be \(1\). To keep the output size small, please print only the sum of answers to all queries. + +**Constraints** + +\(1 \le T \le 45\) +\(1 \le N \le 1{,}000{,}004\) +\(1 \le A_i \le 1{,}000{,}000{,}006\) +\(1 \le Q \le 500{,}000\) +\(1 \le L_i \le R_i \le N\) + +The sum of \(N\) across all test cases is at most \(3{,}000{,}000\). + +**Input Format** + +Input begins with a single integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\). Then, there is a line containing \(N\) space-separated integers \(A_1, ..., A_N\). Then, there is a line containing a single integer \(Q\). \(Q\) lines follow, the \(i\)th of which contains two space-separated integers \(L_i\) and \(R_i\). + +**Output Format** + +For the \(i\)th test case, print a single line containing "`Case #i: `" followed by a single integer, the sum of the greatest day(s) after each of the \(Q\) queries. + +**Sample Explanation** + +In the first test case, there are \(N=3\) days with initial greatnesses of \([1, 20, 30]\). The first query modifies all \(3\) days, transforming their greatnesses \([1\,000\,000\,006, 999\,999\,987, 999\,999\,977]\), where the maximum is on day \(1\). After the second query, day \(3\) will have the largest greatness of \(999\,999\,977\). The overall answer is the sum of the greatest days, which is \(1 + 3 = 4\). + diff --git a/2023/round1/today_is_gonna_be_a_great_day.out b/2023/round1/today_is_gonna_be_a_great_day.out new file mode 100644 index 0000000000000000000000000000000000000000..cf896fc6e529832d835b600bc20bce580b854c6c --- /dev/null +++ b/2023/round1/today_is_gonna_be_a_great_day.out @@ -0,0 +1,50 @@ +Case #1: 4 +Case #2: 1 +Case #3: 507 +Case #4: 1020 +Case #5: 1413 +Case #6: 741698 +Case #7: 513437 +Case #8: 338543 +Case #9: 480658 +Case #10: 516826 +Case #11: 387047 +Case #12: 187178 +Case #13: 490358 +Case #14: 145604187953 +Case #15: 92597003624 +Case #16: 114981219828 +Case #17: 50841603 +Case #18: 33310284 +Case #19: 30831649 +Case #20: 33726129 +Case #21: 34921785 +Case #22: 57832922 +Case #23: 51920086 +Case #24: 60559471 +Case #25: 44533777 +Case #26: 68117533 +Case #27: 55851457 +Case #28: 69023488 +Case #29: 42376940 +Case #30: 31206969 +Case #31: 48913678 +Case #32: 44119261 +Case #33: 65007476 +Case #34: 71752852 +Case #35: 41012180 +Case #36: 50003893 +Case #37: 87090264 +Case #38: 40792538 +Case #39: 51913811 +Case #40: 65522986 +Case #41: 41143021 +Case #42: 55469276 +Case #43: 51383440 +Case #44: 57522676 +Case #45: 38820765 +Case #46: 37450548 +Case #47: 70376064 +Case #48: 59786707 +Case #49: 36279473 +Case #50: 63806322 diff --git a/2023/round1/today_is_gonna_be_a_great_day_sol.md b/2023/round1/today_is_gonna_be_a_great_day_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..3996e952142873780647ce1144eb175f7642e54d --- /dev/null +++ b/2023/round1/today_is_gonna_be_a_great_day_sol.md @@ -0,0 +1,14 @@ +The key insight is that for the operation of multiplying a number \(x\) by \(1{,}000{,}000{,}006\) modulo \(1{,}000{,}000{,}007\), doing it twice will just get us \(x\) back again. Therefore, the \(i\)th day will only ever take on two greatness values: \(A_i\) and \((A_i * 1{,}000{,}000{,}006) \text{ mod } 1{,}000{,}000{,}007\). We can consider an update of \(L_i..R_i\) to be "flipping" every value in the range to their complementary value. + +Let's build a [segment tree](https://en.wikipedia.org/wiki/Segment_tree) on the array. Each node is responsible for some subarray \(A_{l..r}\) and will maintain the following information: + +* the maximum greatnesses value in that range (i.e. the current greatnesses) +* the day (index) of said maximum value +* the maximum of the flipped greatnesses in that range (i.e. what the greatnesses will become after an update) +* the day (index) of said flipped maximum + +When merging two child nodes to a parent, we set the max initial value in the parent node to be the max of the children's initial values, breaking ties by lowest day index. Likewise for the flipped max and index. + +Using the well-known lazy propagation technique, we can update ranges in time \(\mathcal{O}(\log N)\). For each node affected by an update, we can simply swap the first two fields with the last two. After each update, we can query the entire array (just access the root node) for the day of the current maximum greatness. + +Across \(Q\) queries, we get an overall running time of \(\mathcal{O}(Q \log N)\). diff --git a/2023/round2/1327337367897977.jpg b/2023/round2/1327337367897977.jpg new file mode 100644 index 0000000000000000000000000000000000000000..541e5bdfcdeb84aa4fe4a557802b82ef9899eb1b --- /dev/null +++ b/2023/round2/1327337367897977.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e7b43c3504242921a063f7c3269c6be78b77c6f8029ce1ee87b6ba6e09d0298f +size 281665 diff --git a/2023/round2/142163805651048.jpg b/2023/round2/142163805651048.jpg new file mode 100644 index 0000000000000000000000000000000000000000..728e3883d01dd6823b31fec3c614763bd500341c --- /dev/null +++ b/2023/round2/142163805651048.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b972fe29ec5a56e73cc3260cccbe2cee42b7c00f75094456039744ef21685e22 +size 218913 diff --git a/2023/round2/1491241961665655.jpg b/2023/round2/1491241961665655.jpg new file mode 100644 index 0000000000000000000000000000000000000000..728e3883d01dd6823b31fec3c614763bd500341c --- /dev/null +++ b/2023/round2/1491241961665655.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b972fe29ec5a56e73cc3260cccbe2cee42b7c00f75094456039744ef21685e22 +size 218913 diff --git a/2023/round2/311797128262237.jpg b/2023/round2/311797128262237.jpg new file mode 100644 index 0000000000000000000000000000000000000000..1f044d27956fbaa6938e020f700630572270ff18 --- /dev/null +++ b/2023/round2/311797128262237.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7c394c92c1f0adf45c65abbf285526d5bd9f7bfb439c8a988f4fe42a74d22159 +size 148641 diff --git a/2023/round2/359057163229199.jpg b/2023/round2/359057163229199.jpg new file mode 100644 index 0000000000000000000000000000000000000000..c5fa9b08a01829d05aa55974a4a10dd729b45682 --- /dev/null +++ b/2023/round2/359057163229199.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:54ba28881baad615f671c7fe5718ddd9af5b801cf7b88921f8031fe1fb63bcf1 +size 39941 diff --git a/2023/round2/632037365762295.jpg b/2023/round2/632037365762295.jpg new file mode 100644 index 0000000000000000000000000000000000000000..541e5bdfcdeb84aa4fe4a557802b82ef9899eb1b --- /dev/null +++ b/2023/round2/632037365762295.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e7b43c3504242921a063f7c3269c6be78b77c6f8029ce1ee87b6ba6e09d0298f +size 281665 diff --git a/2023/round2/636025968710743.jpg b/2023/round2/636025968710743.jpg new file mode 100644 index 0000000000000000000000000000000000000000..6d366c275682a6912b337751b9cdf6b14f8786fa --- /dev/null +++ b/2023/round2/636025968710743.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e0ad51f814ceb3e7d1412d9f3ab96ba31bf5089c965942c24706018a98d8e47 +size 57160 diff --git a/2023/round2/804957498302389.jpg b/2023/round2/804957498302389.jpg new file mode 100644 index 0000000000000000000000000000000000000000..cc0774b755224ea55d8fd7ce45d1c9835ec0325c --- /dev/null +++ b/2023/round2/804957498302389.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d2adc80b8dd128a5cb6cbeec376240816017b272ea39800111c4c762dc465444 +size 458569 diff --git a/2023/round2/850220283519385.jpg b/2023/round2/850220283519385.jpg new file mode 100644 index 0000000000000000000000000000000000000000..1f044d27956fbaa6938e020f700630572270ff18 --- /dev/null +++ b/2023/round2/850220283519385.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7c394c92c1f0adf45c65abbf285526d5bd9f7bfb439c8a988f4fe42a74d22159 +size 148641 diff --git a/2023/round2/975143063563862.jpg b/2023/round2/975143063563862.jpg new file mode 100644 index 0000000000000000000000000000000000000000..cc0774b755224ea55d8fd7ce45d1c9835ec0325c --- /dev/null +++ b/2023/round2/975143063563862.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d2adc80b8dd128a5cb6cbeec376240816017b272ea39800111c4c762dc465444 +size 458569 diff --git a/2023/round2/meta_game.cpp b/2023/round2/meta_game.cpp new file mode 100644 index 0000000000000000000000000000000000000000..773c0c6cd15c82cf11b371000ae7efdf72bd7391 --- /dev/null +++ b/2023/round2/meta_game.cpp @@ -0,0 +1,100 @@ +#include +#include +#include +using namespace std; + +int N; +deque A, B; + +bool is_metalike_now() { + for (int i = 0; i < N; i++) { + if (i+1 < (N+1)/2 && !(A[i] < B[i])) { + return false; + } + if (i+1 > (N+1)/2 && !(A[i] > B[i])) { + return false; + } + if (A[i] != B[N - i - 1]) { + return false; + } + } + return true; +} + +// Simulate 2*N seconds. +// Given a vector M of O(1) candidate midpoints (index of the middle elem if N +// is odd, or right after the middle if N even), return the first time that: +// - some m in M becomes the midpoint floor(N/2) (both even and odd N), and +// - is_meta_like() holds true. +int check(vector M) { + for (int t = 0; t <= 2*N; t++) { + for (int m : M) { + if (m == N/2 && is_metalike_now()) { + return t; + } + } + A.push_back(B[0]); + B.push_back(A[0]); + A.pop_front(); + B.pop_front(); + for (int i = 0; i < (int)M.size(); i++) { + if (--M[i] < 0) { + M[i] += N; + } + } + } + return -1; +} + + +int solve() { + cin >> N; + A.resize(N); + B.resize(N); + for (int i = 0; i < N; i++) { + cin >> A[i]; + } + for (int i = 0; i < N; i++) { + cin >> B[i]; + } + // Handle any equal pairs. + vector eq; + for (int i = 0; i < N; i++) { + if (A[i] == B[i]) { + if (N % 2 == 0) { + return -1; + } + eq.push_back(i); + } + } + if (eq.size() > 1) { + return -1; + } + if (eq.size() == 1) { + return check({eq[0]}); + } + // From here on, guaranteed that A[i] != B[i]. + // Check the number of times that A[i] < B[i] flips. + vector flipped; + for (int i = 1; i < N; i++) { + if ((A[i - 1] < B[i - 1]) != (A[i] < B[i])) { + flipped.push_back(i); + } + } + if (flipped.size() > 1) { + return -1; + } + // Might as well consider elements 0 and N-1 to be midpoints. + flipped.push_back(0); + flipped.push_back(N-1); + return check(flipped); +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round2/meta_game.in b/2023/round2/meta_game.in new file mode 100644 index 0000000000000000000000000000000000000000..ea0a1108236c9ff60ae5ad301e253fffc7cf654c --- /dev/null +++ b/2023/round2/meta_game.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dc8ff4227c65c647b1324b136b769f375b19493fca9f9b3a76dbf8a89b859bf7 +size 56090268 diff --git a/2023/round2/meta_game.md b/2023/round2/meta_game.md new file mode 100644 index 0000000000000000000000000000000000000000..8277be3cce9a16fa7a3fe03955a550361073903d --- /dev/null +++ b/2023/round2/meta_game.md @@ -0,0 +1,51 @@ +Hacker Cup contest strategy often involves a metagame, where choosing which problems to work on might just be an important decision. On a Quest to become more Pro, you encounter an oracle promising to teach you the contest meta if you play her own Meta-game. + +The oracle presents a peg board with \(2N\) moving dots. The initial \(y\)-positions of the dots are given as two arrays \(A_{1..N}\) and \(B_{1..N}\). Each second, simultaneously, \(A_1\) will move to the end of \(B\), while \(B_1\) will move to the end of \(A\) (with all elements shifting left accordingly). + +You can connect the dots to form a *Meta-like logo* if all of the following are true: +* For the first half of both arrays, each dot in \(A\) is below the corresponding dot in \(B\). +* For the last half of both arrays, each dot in \(A\) is above the corresponding dot in \(B\). +* \(A\) equals the reverse of \(B\). + +Formally: +* \(A_i < B_i\) for every \(i < (N+1)/2\) +* \(A_i > B_i\) for every \(i > (N+1)/2\) +* \(A_i = B_{N-i+1}\) for every \(i = 1..N\) + +Note that if \(N\) is odd, the arrays' middle elements are not subject to the first two constraints. + +The following is a visualization of a Meta-like logo (corresponding to the first sample case), with dots in \(A\) shown in red and dots in \(B\) shown in blue. + +{{PHOTO_ID:359057163229199|WIDTH:500}} + +You must answer the oracle: how many seconds must pass before the first time a Meta-like logo appears? If one never appears, output \(-1\). + +# Constraints +\(1 \leq T \leq 135\) +\(2 \leq N \leq 2{,}000{,}000\) +\(0 \leq A_i, B_i \leq 1{,}000{,}000{,}000\) + +The sum of \(N\) across all test cases is at most \(9{,}000{,}000\). + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\). Then, there is a line containing integers \(A_1, ..., A_N\). Then, there is a line containing integers \(B_1, ..., B_N\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by a single integer, the number of seconds that must pass before a Meta-like logo appears, or \(-1\) if that will never happen. + + +# Sample Explanation + +The first test case is shown above. \(A\) and \(B\) already form a Meta-like logo, so the answer is 0. + +The second case is not initially a Meta-like logo, for several reasons. One reason is that it is not symmetric. Specifically, the \([3, 3, 2, 3, 5 ,6]\) is not the reverse of \([4, 4, 6, 5, 3 ,2]\). After \(1\) second though, this case turns into the case above and is Meta-like. + +The third and fourth cases will never turn into a Meta-like logo, no matter how many seconds we wait. + +In the fifth case, after 6 seconds we see the first Meta-like logo. In this case \(A = [1, 1, 2, 2]\) and \(B = [2, 2, 1, 1]\). + + diff --git a/2023/round2/meta_game.out b/2023/round2/meta_game.out new file mode 100644 index 0000000000000000000000000000000000000000..efce65df5d650d9baf5aae3b17cbf9524c4b1924 --- /dev/null +++ b/2023/round2/meta_game.out @@ -0,0 +1,135 @@ +Case #1: 0 +Case #2: 1 +Case #3: -1 +Case #4: -1 +Case #5: 6 +Case #6: -1 +Case #7: 7 +Case #8: 2 +Case #9: 0 +Case #10: 0 +Case #11: 0 +Case #12: -1 +Case #13: -1 +Case #14: 3207591 +Case #15: 596658 +Case #16: 0 +Case #17: 2 +Case #18: 0 +Case #19: 15 +Case #20: 1 +Case #21: 1 +Case #22: 1 +Case #23: 5 +Case #24: 14 +Case #25: 0 +Case #26: 8 +Case #27: 2 +Case #28: 5 +Case #29: 2 +Case #30: 5 +Case #31: 0 +Case #32: 7 +Case #33: 10 +Case #34: 13 +Case #35: 0 +Case #36: 15 +Case #37: 1 +Case #38: 8 +Case #39: 1 +Case #40: 2 +Case #41: 0 +Case #42: 18 +Case #43: 1 +Case #44: 1 +Case #45: 3 +Case #46: 0 +Case #47: 2 +Case #48: 2 +Case #49: 1 +Case #50: 0 +Case #51: 6 +Case #52: 4 +Case #53: 6 +Case #54: 3 +Case #55: 10 +Case #56: 15 +Case #57: 13 +Case #58: 2 +Case #59: 12 +Case #60: 12 +Case #61: 4 +Case #62: 7 +Case #63: 12 +Case #64: 17 +Case #65: 1 +Case #66: -1 +Case #67: -1 +Case #68: -1 +Case #69: -1 +Case #70: -1 +Case #71: -1 +Case #72: -1 +Case #73: -1 +Case #74: -1 +Case #75: -1 +Case #76: -1 +Case #77: -1 +Case #78: -1 +Case #79: 0 +Case #80: -1 +Case #81: -1 +Case #82: -1 +Case #83: -1 +Case #84: -1 +Case #85: -1 +Case #86: -1 +Case #87: -1 +Case #88: -1 +Case #89: -1 +Case #90: -1 +Case #91: -1 +Case #92: -1 +Case #93: -1 +Case #94: -1 +Case #95: -1 +Case #96: 2 +Case #97: -1 +Case #98: -1 +Case #99: -1 +Case #100: -1 +Case #101: -1 +Case #102: -1 +Case #103: -1 +Case #104: -1 +Case #105: -1 +Case #106: -1 +Case #107: -1 +Case #108: 0 +Case #109: -1 +Case #110: -1 +Case #111: 3 +Case #112: -1 +Case #113: -1 +Case #114: -1 +Case #115: -1 +Case #116: -1 +Case #117: -1 +Case #118: 52 +Case #119: 114 +Case #120: 116 +Case #121: 19 +Case #122: 216 +Case #123: -1 +Case #124: 196 +Case #125: -1 +Case #126: -1 +Case #127: 140 +Case #128: -1 +Case #129: 146 +Case #130: 250 +Case #131: 97 +Case #132: 128 +Case #133: 8 +Case #134: -1 +Case #135: 80 diff --git a/2023/round2/meta_game_sol.md b/2023/round2/meta_game_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..62cc37275d3a6779cf045d7536193d83660fd979 --- /dev/null +++ b/2023/round2/meta_game_sol.md @@ -0,0 +1,27 @@ +Note that over time, a pair of dots that are next to each other will always stay next to each other. + +If \(N\) is even and \(A_i = B_i\) for any \(i\), we know that an answer is not possible by definition. If \(N\) is odd, then there can be at most \(1\) place where \(A_i = B_i\) (the midpoint of the metalike-logo), so we can find it and get the answer in \(\mathcal{O}(N)\), or return \(-1\) if there's more than one \(i\) where \(A_i = B_i\). + +Otherwise, every \(A_i \ne B_i\), and we can first just think about each \(i\) as the lower vs higher dot. For a meta-like logo, the relationship always look like this: +``` +(even N) (odd N) +3 2 3 5 6 4 -> LLLHHH 3 2 3 x 5 6 4 -> LLLxHHH +4 6 5 3 2 3 -> HHHLLL 4 6 5 x 3 2 3 -> HHHxLLL +``` +Let's first consider even \(N\) by simulating the first pattern in reverse: +``` +(t) (t-1) (t-2) (t-N/2) (t-N) +LLLHHH LLLLHH LLLLLH LLLLLL HLLLLL HHLLLL HHHLLL +HHHLLL HHHHLL HHHHHL HHHHHH LHHHHH LLHHHH LLLHHH + +(t-N-1) (t-2N) +HHHHLL HHHHHL HHHHHH LHHHHH LHHHHH LLHHHH LLLHHH +LLLLHH LLLLLH LLLLLL HLLLLL HLLLLL HHLLLL HHHLLL +``` +Thus the pattern repeats after \(2N\) steps. If an answer exists, then at any given time: +* for even \(N\), there's at most \(1\) place when \(A_i < B_i\) inverts to \(A_{i+1} > B_{i+1}\) (or the reverse). +* for odd \(N\), the middle element eithers conform to its left or right side, so this still holds. + +Therefore, the midpoint of the metalike logo either be this inversion point, or the first/last element. Either way, there are \(\mathcal{O}(1)\) possible candidates for the midpoints. After we've found these candidates, we can simulate \(2N\) steps on the input arrays using a double-ended queue (updating the positions of the midpoint candidates at each step). If at any time a midpoint candidate actually shifts to the midpoint, we can do a check to see if \(A\) equals the reverse of \(B\). + +There are \(\mathcal{O}(1)\) midpoint candidates, and each can only reach the midpoint twice in our \(2N\) simulated time steps. Thus, the reverse condition is checked at most \(\mathcal{O}(1)\) times. The overall running time is \(\mathcal{O}(N)\). diff --git a/2023/round2/ready_go_part_1.cpp b/2023/round2/ready_go_part_1.cpp new file mode 100644 index 0000000000000000000000000000000000000000..fe930a13665e9ee9592e66c533d715377eacbf3c --- /dev/null +++ b/2023/round2/ready_go_part_1.cpp @@ -0,0 +1,54 @@ +#include +#include +#include +#include +using namespace std; + +vector> A; +vector> visit; +set> empty_adj; + +void rec(int r, int c) { + visit[r][c] = true; + for (auto [r2, c2] : {pair{r - 1, c}, {r, c - 1}, {r + 1, c}, {r, c + 1}}) { + if (A[r2][c2] == '.') { + empty_adj.insert({r2, c2}); + } + if (A[r2][c2] == 'W' && !visit[r2][c2]) { + rec(r2, c2); + } + } +} + +string solve() { + int R, C; + cin >> R >> C; + A.assign(R + 2, vector(C + 2, 'B')); + for (int i = 1; i <= R; i++) { + for (int j = 1; j <= C; j++) { + cin >> A[i][j]; + } + } + visit.assign(R + 2, vector(C + 2, false)); + for (int i = 1; i <= R; i++) { + for (int j = 1; j <= C; j++) { + if (A[i][j] == 'W' && !visit[i][j]) { + empty_adj.clear(); + rec(i, j); + if (empty_adj.size() == 1) { + return "YES"; + } + } + } + } + return "NO"; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round2/ready_go_part_1.in b/2023/round2/ready_go_part_1.in new file mode 100644 index 0000000000000000000000000000000000000000..b105d4239e8497c9aea8662dad7a23121178f7bc --- /dev/null +++ b/2023/round2/ready_go_part_1.in @@ -0,0 +1,2599 @@ +239 +4 4 +.B.. +BW.. +.B.. +.... +4 5 +WWWB. +WB... +WB.WB +B.W.. +4 5 +WWWB. +WB... +WB.WB +..W.. +3 5 +..W.. +.WBWW +.B..W +1 2 +W. +1 2 +B. +3 3 +WWW +W.W +WWW +3 3 +.W. +W.W +.W. +3 3 +.WW +WWW +WWW +3 3 +.WW +WWW +WW. +19 19 +BWBBBB..BWWWWWB..W. +BWWWWWB.BWBBBB..... +.BBWBB..BWBBBBB.... +..BWBBBBBWWWWWWB... +..BWWWWWWWBBBBB..B. +..BBBWBBBB......... +....BWBBBBBBBBBBBBB +....BWWWWWWWWWWWWWW +..B.BWBBBBBBBWBBBBB +.BWBBWB.....BWB.... 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+WWWWWWWWWWWWWW.WW +WWWW..WWWW.WWW.WW +WWWWW.W.WWBWW.WWW +WW.WW.WW.WWWWWWWW +.W...W..WWW.WWW.. +WWWWWWWWWWW.B.W.W +WWWWWWWWWWWWWWW.W +..W..WW.WWWWWWWWW +WWW.WWWWWWWWWWWWW +..W.W.WW.W.WWWWWW +WWWWWWWWWWW.W.W.W diff --git a/2023/round2/ready_go_part_1.md b/2023/round2/ready_go_part_1.md new file mode 100644 index 0000000000000000000000000000000000000000..3fd3cc03e5942e1c3b073018313a7f225dc6c645 --- /dev/null +++ b/2023/round2/ready_go_part_1.md @@ -0,0 +1,58 @@ +_This problem shares some similarities with A2, with key differences in bold._ + +_Atari 2600? More like Atari 2600 BCE!_ + +The classic board game _Go_ is a two-player game played on an \(R \times C\) board. One player places white stones while the other places black stones. On a player's turn, they may place a stone in any empty space. A curiosity of Go is that stones are placed on the intersections of grid lines rather than between the lines, so an in-progress \(5 \times 5\) game looks like this: + +{{PHOTO_ID:1491241961665655|WIDTH:180}} + +An orthogonally contiguous set of stones of the same color is called a *group*. A group of stones is captured (and removed from the board) once no stones in the group has an adjacent empty space. + +You're playing as Black and it's your turn. Given a valid board (i.e. no groups have \(0\) adjacent empty spaces), **is it possible to capture any white stones on this turn** with a single black stone? + +Here are some examples of captures. If a black stone is placed at the point marked with a triangle, a single white stone will be captured: + +{{PHOTO_ID:632037365762295|WIDTH:300}} + +Here, Black can capture a group of \(3\) white stones. Note that this move is valid even though the new black stone has no adjacent empty spaces at the moment it's placed: + +{{PHOTO_ID:850220283519385|WIDTH:240}} + +Black can even capture multiple groups at once. Here, Black captures a group of \(2\) stones and a group of \(3\) stones: + +{{PHOTO_ID:804957498302389|WIDTH:400}} + +The Go board is represented as a character array \(A\) where \(A_{i, j}\) is one of: +* `B` for a black stone +* `W` for a white stone +* `.` for an empty space + + +# Constraints + +\(1 \leq T \leq 160\) +\(1 \leq R, C \leq \textbf{19}\) +\(A_{i, j} \in \{\)'`.`', '`B`', '`W`'\(\}\) + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case begins with a line containing two integers \(R\) and \(C\). Then, \(R\) lines follow, the \(i\)th of which contains \(C\) characters \(A_{i, 1}\) through \(A_{i,C}\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by "`YES`" if you can capture any white stones, or "`NO`" if you cannot. + + +# Sample Explanation + +The first sample case is the first capture example shown above. Black can capture the lone white stone. + +In the second case, Black can capture a group of \(5\) white stones. + +The third case is the same as the second case, except that the group of \(5\) white stones has two empty adjacent spaces, so it cannot be captured in one term. The other white stones can also not be captured on this turn. + +In the fourth case, White could capture a black stone if it was their turn, but Black cannot capture any white stones. + +The fifth case is the second capture example shown above. Black can capture \(3\) white stones by playing in the upper-left corner. diff --git a/2023/round2/ready_go_part_1.out b/2023/round2/ready_go_part_1.out new file mode 100644 index 0000000000000000000000000000000000000000..dd2815d13a154df17fc94384be5d9c4615a73a3f --- /dev/null +++ b/2023/round2/ready_go_part_1.out @@ -0,0 +1,239 @@ +Case #1: YES +Case #2: YES +Case #3: NO +Case #4: NO +Case #5: YES +Case #6: NO +Case #7: YES +Case #8: NO +Case #9: YES +Case #10: NO +Case #11: YES +Case #12: NO +Case #13: NO +Case #14: YES +Case #15: NO +Case #16: NO +Case #17: YES +Case #18: YES +Case #19: NO +Case #20: NO +Case #21: NO +Case #22: YES +Case #23: YES +Case #24: NO +Case #25: YES +Case #26: YES +Case #27: NO +Case #28: YES +Case #29: YES +Case #30: NO +Case #31: YES +Case #32: YES +Case #33: NO +Case #34: NO +Case #35: YES +Case #36: NO +Case #37: YES +Case #38: YES +Case #39: YES +Case #40: NO +Case #41: YES +Case #42: NO +Case #43: NO +Case #44: YES +Case #45: YES +Case #46: YES +Case #47: YES +Case #48: YES +Case #49: YES +Case #50: NO +Case #51: YES +Case #52: YES +Case #53: YES +Case #54: YES +Case #55: NO +Case #56: NO +Case #57: NO +Case #58: YES +Case #59: YES +Case #60: YES +Case #61: YES +Case #62: NO +Case #63: YES +Case #64: YES +Case #65: NO +Case #66: YES +Case #67: YES +Case #68: NO +Case #69: NO +Case #70: YES +Case #71: YES +Case #72: YES +Case #73: YES +Case #74: YES +Case #75: YES +Case #76: NO +Case #77: YES +Case #78: YES +Case #79: YES +Case #80: NO +Case #81: YES +Case #82: YES +Case #83: YES +Case #84: NO +Case #85: YES +Case #86: NO +Case #87: NO +Case #88: NO +Case #89: NO +Case #90: YES +Case #91: NO +Case #92: YES +Case #93: NO +Case #94: YES +Case #95: YES +Case #96: NO +Case #97: YES +Case #98: YES +Case #99: NO +Case #100: NO +Case #101: YES +Case #102: YES +Case #103: NO +Case #104: NO +Case #105: NO +Case #106: NO +Case #107: NO +Case #108: NO +Case #109: NO +Case #110: NO +Case #111: YES +Case #112: NO +Case #113: NO +Case #114: YES +Case #115: NO +Case #116: YES +Case #117: YES +Case #118: YES +Case #119: NO +Case #120: NO +Case #121: YES +Case #122: YES +Case #123: NO +Case #124: NO +Case #125: NO +Case #126: YES +Case #127: NO +Case #128: NO +Case #129: NO +Case #130: NO +Case #131: NO +Case #132: YES +Case #133: YES +Case #134: NO +Case #135: NO +Case #136: YES +Case #137: NO +Case #138: YES +Case #139: YES +Case #140: NO +Case #141: YES +Case #142: YES +Case #143: YES +Case #144: NO +Case #145: YES +Case #146: NO +Case #147: NO +Case #148: YES +Case #149: NO +Case #150: NO +Case #151: NO +Case #152: NO +Case #153: YES +Case #154: YES +Case #155: NO +Case #156: YES +Case #157: NO +Case #158: NO +Case #159: NO +Case #160: NO +Case #161: YES +Case #162: NO +Case #163: NO +Case #164: NO +Case #165: YES +Case #166: NO +Case #167: NO +Case #168: NO +Case #169: YES +Case #170: NO +Case #171: YES +Case #172: NO +Case #173: NO +Case #174: NO +Case #175: YES +Case #176: NO +Case #177: NO +Case #178: NO +Case #179: NO +Case #180: NO +Case #181: YES +Case #182: NO +Case #183: NO +Case #184: NO +Case #185: NO +Case #186: NO +Case #187: NO +Case #188: NO +Case #189: YES +Case #190: YES +Case #191: YES +Case #192: YES +Case #193: NO +Case #194: NO +Case #195: NO +Case #196: NO +Case #197: NO +Case #198: NO +Case #199: YES +Case #200: YES +Case #201: NO +Case #202: NO +Case #203: NO +Case #204: NO +Case #205: NO +Case #206: YES +Case #207: YES +Case #208: NO +Case #209: YES +Case #210: NO +Case #211: YES +Case #212: NO +Case #213: NO +Case #214: NO +Case #215: NO +Case #216: NO +Case #217: YES +Case #218: YES +Case #219: YES +Case #220: NO +Case #221: NO +Case #222: YES +Case #223: NO +Case #224: NO +Case #225: YES +Case #226: YES +Case #227: NO +Case #228: YES +Case #229: NO +Case #230: NO +Case #231: NO +Case #232: NO +Case #233: NO +Case #234: NO +Case #235: NO +Case #236: NO +Case #237: NO +Case #238: YES +Case #239: NO diff --git a/2023/round2/ready_go_part_1_sol.md b/2023/round2/ready_go_part_1_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..f6371b6b497a5710f0c50563dd3d79d9ed47fed8 --- /dev/null +++ b/2023/round2/ready_go_part_1_sol.md @@ -0,0 +1,5 @@ +A group of white stones can be captured if and only if there's exactly \(1\) empty space adjacent to the group, since playing a black stone in that space would leave the group with \(0\) adjacent empty spaces. + +Loop through each space on the board, and when you encounter a white stone, use any search algorithm (a straightforward breadth-first search will suffice) to find all of the white stones that are part of the same group. Once you know which stones are in the group, you can check all \(4\) orthogonal neighbors of each white stone to see how many empty spaces are adjacent. Take care not to count the same empty space multiple times if it's adjacent to multiple white stones in the group (a set is a useful structure for this). + +If you find any white group with exactly \(1\) adjacent empty space, the answer is "`YES`", otherwise the answer is "`NO`". This solution has a time complexity of \(\mathcal{O}(R * C\)). diff --git a/2023/round2/ready_go_part_2.cpp b/2023/round2/ready_go_part_2.cpp new file mode 100644 index 0000000000000000000000000000000000000000..e6a2de33fa1152e7f64947537a50166ba3f3d638 --- /dev/null +++ b/2023/round2/ready_go_part_2.cpp @@ -0,0 +1,60 @@ +#include +#include +#include +#include +#include +#include +using namespace std; + +vector> A; +vector> visit; +set> empty_adj; +map, int> captured; + +int rec(int r, int c) { + visit[r][c] = true; + int cnt = 1; + for (auto [r2, c2] : {pair{r - 1, c}, {r, c - 1}, {r + 1, c}, {r, c + 1}}) { + if (A[r2][c2] == '.') { + empty_adj.insert({r2, c2}); + } else if (A[r2][c2] == 'W' && !visit[r2][c2]) { + cnt += rec(r2, c2); + } + } + return cnt; +} + +int solve() { + int R, C; + cin >> R >> C; + A.assign(R + 2, vector(C + 2, 'B')); + for (int i = 1; i <= R; i++) { + for (int j = 1; j <= C; j++) { + cin >> A[i][j]; + } + } + visit.assign(R + 2, vector(C + 2, false)); + captured.clear(); + int ans = 0; + for (int i = 1; i <= R; i++) { + for (int j = 1; j <= C; j++) { + if (A[i][j] == 'W' && !visit[i][j]) { + empty_adj.clear(); + int cnt = rec(i, j); + if (empty_adj.size() == 1) { + ans = max(ans, captured[*empty_adj.begin()] += cnt); + } + } + } + } + return ans; +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round2/ready_go_part_2.in b/2023/round2/ready_go_part_2.in new file mode 100644 index 0000000000000000000000000000000000000000..371856d9ecc55f2bde38303191ab427c8c089a3e --- /dev/null +++ b/2023/round2/ready_go_part_2.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:834e5c7044d7c9017869eaa693b19344a826008c5a1e6018a64b47f616bddd26 +size 11973169 diff --git a/2023/round2/ready_go_part_2.md b/2023/round2/ready_go_part_2.md new file mode 100644 index 0000000000000000000000000000000000000000..d6d14aa6b73cea22b7925ef44a5dcbf5c6d4fa38 --- /dev/null +++ b/2023/round2/ready_go_part_2.md @@ -0,0 +1,57 @@ +_This problem shares some similarities with A1, with key differences in bold._ + +_Atari 2600? More like Atari 2600 BCE!_ + +The classic board game _Go_ is a two-player game played on an \(R \times C\) board. One player places white stones while the other places black stones. On a player's turn, they may place a stone in any empty space. A curiosity of Go is that stones are placed on the intersections of grid lines rather than between the lines, so an in-progress \(5 \times 5\) game looks like this: + +{{PHOTO_ID:142163805651048|WIDTH:180}} + +An orthogonally contiguous set of stones of the same color is called a *group*. A group of stones is captured (and removed from the board) once no stones in the group has an adjacent empty space. + +You're playing as Black and it's your turn. Given a valid board (i.e. no groups have \(0\) adjacent empty spaces), **what’s the maximum number of white stones you can capture** with a single black stone? + +Here are some examples of captures. If a black stone is placed at the point marked with a triangle, a single white stone will be captured: + +{{PHOTO_ID:1327337367897977|WIDTH:300}} + +Here, Black can capture a group of \(3\) white stones. Note that this move is valid even though the new black stone has no adjacent empty spaces at the moment it's placed: + +{{PHOTO_ID:311797128262237|WIDTH:240}} + +Black can even capture multiple groups at once. Here, Black captures a group of \(2\) stones and a group of \(3\) stones: + +{{PHOTO_ID:975143063563862|WIDTH:400}} + +The Go board is represented as a character array \(A\) where \(A_{i, j}\) is one of: +* `B` for a black stone +* `W` for a white stone +* `.` for an empty space + + +# Constraints + +\(1 \leq T \leq 150\) +\(1 \leq R, C \leq \textbf{3{,}000}\) +\(A_{i, j} \in \{\)'`.`', '`B`', '`W`'\(\}\) + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case begins with a line containing two integers \(R\) and \(C\). Then, \(R\) lines follow, the \(i\)th of which contains \(C\) characters \(A_{i, 1}\) through \(A_{i,C}\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by a single integer, the maximum number of white stones you can capture on your turn. + + +# Sample Explanation + +In the first case, Black can capture 3 white stones by playing in the bottom-right corner. + +In the second case, there are no white stones that can be captured. + +In the third case, Black can capture both white groups at once, for a total of 6 + 3 = 9 white stones. + +In the fourth case, there are 6 different white stones that can be captured, but Black can capture at most 4 of them at once (by playing in the center of the board). + diff --git a/2023/round2/ready_go_part_2.out b/2023/round2/ready_go_part_2.out new file mode 100644 index 0000000000000000000000000000000000000000..09138e144d0ac4a325e4b2f17e1e8738e4801380 --- /dev/null +++ b/2023/round2/ready_go_part_2.out @@ -0,0 +1,239 @@ +Case #1: 1 +Case #2: 5 +Case #3: 0 +Case #4: 0 +Case #5: 1 +Case #6: 0 +Case #7: 8 +Case #8: 0 +Case #9: 8 +Case #10: 0 +Case #11: 90 +Case #12: 0 +Case #13: 0 +Case #14: 1 +Case #15: 0 +Case #16: 0 +Case #17: 1 +Case #18: 1 +Case #19: 0 +Case #20: 0 +Case #21: 0 +Case #22: 1 +Case #23: 1 +Case #24: 0 +Case #25: 1 +Case #26: 1 +Case #27: 0 +Case #28: 1 +Case #29: 1 +Case #30: 0 +Case #31: 1 +Case #32: 1 +Case #33: 0 +Case #34: 0 +Case #35: 2 +Case #36: 0 +Case #37: 4 +Case #38: 4 +Case #39: 61 +Case #40: 3600185 +Case #41: 959113 +Case #42: 0 +Case #43: 1842 +Case #44: 0 +Case #45: 0 +Case #46: 11 +Case #47: 0 +Case #48: 165 +Case #49: 0 +Case #50: 20 +Case #51: 0 +Case #52: 86 +Case #53: 58 +Case #54: 74 +Case #55: 0 +Case #56: 7 +Case #57: 0 +Case #58: 52 +Case #59: 0 +Case #60: 90 +Case #61: 48 +Case #62: 583 +Case #63: 0 +Case #64: 21 +Case #65: 0 +Case #66: 10 +Case #67: 6 +Case #68: 148 +Case #69: 71 +Case #70: 44 +Case #71: 14 +Case #72: 221 +Case #73: 0 +Case #74: 2 +Case #75: 1 +Case #76: 195 +Case #77: 0 +Case #78: 16 +Case #79: 0 +Case #80: 17 +Case #81: 7 +Case #82: 1 +Case #83: 21 +Case #84: 20 +Case #85: 28 +Case #86: 94 +Case #87: 15 +Case #88: 61 +Case #89: 0 +Case #90: 67 +Case #91: 25 +Case #92: 204 +Case #93: 34 +Case #94: 61 +Case #95: 52 +Case #96: 29 +Case #97: 0 +Case #98: 16 +Case #99: 85 +Case #100: 40 +Case #101: 97 +Case #102: 59 +Case #103: 0 +Case #104: 31 +Case #105: 69 +Case #106: 20 +Case #107: 115 +Case #108: 0 +Case #109: 14 +Case #110: 4 +Case #111: 36 +Case #112: 47 +Case #113: 105 +Case #114: 24 +Case #115: 39 +Case #116: 272 +Case #117: 149 +Case #118: 383 +Case #119: 156 +Case #120: 135 +Case #121: 0 +Case #122: 4 +Case #123: 14 +Case #124: 29 +Case #125: 1 +Case #126: 6 +Case #127: 10 +Case #128: 66 +Case #129: 0 +Case #130: 113 +Case #131: 37 +Case #132: 251 +Case #133: 115 +Case #134: 64 +Case #135: 0 +Case #136: 226 +Case #137: 12 +Case #138: 2 +Case #139: 0 +Case #140: 15 +Case #141: 87 +Case #142: 139 +Case #143: 0 +Case #144: 44 +Case #145: 9 +Case #146: 149 +Case #147: 0 +Case #148: 11 +Case #149: 56 +Case #150: 246 +Case #151: 42 +Case #152: 64 +Case #153: 131 +Case #154: 114 +Case #155: 1 +Case #156: 384 +Case #157: 0 +Case #158: 78 +Case #159: 81 +Case #160: 149 +Case #161: 29 +Case #162: 288 +Case #163: 71 +Case #164: 57 +Case #165: 58 +Case #166: 66 +Case #167: 39 +Case #168: 89 +Case #169: 49 +Case #170: 42 +Case #171: 85 +Case #172: 430 +Case #173: 10 +Case #174: 94 +Case #175: 58 +Case #176: 235 +Case #177: 1 +Case #178: 25 +Case #179: 20 +Case #180: 253 +Case #181: 52 +Case #182: 1 +Case #183: 1 +Case #184: 76 +Case #185: 66 +Case #186: 20 +Case #187: 30 +Case #188: 58 +Case #189: 43 +Case #190: 15 +Case #191: 69 +Case #192: 85 +Case #193: 50 +Case #194: 42 +Case #195: 0 +Case #196: 43 +Case #197: 49 +Case #198: 54 +Case #199: 16 +Case #200: 119 +Case #201: 1 +Case #202: 1 +Case #203: 0 +Case #204: 0 +Case #205: 36 +Case #206: 22 +Case #207: 28 +Case #208: 37 +Case #209: 50 +Case #210: 30 +Case #211: 35 +Case #212: 54 +Case #213: 0 +Case #214: 5 +Case #215: 65 +Case #216: 10 +Case #217: 1 +Case #218: 3 +Case #219: 1 +Case #220: 80 +Case #221: 34 +Case #222: 51 +Case #223: 1 +Case #224: 36 +Case #225: 9 +Case #226: 12 +Case #227: 40 +Case #228: 57 +Case #229: 13 +Case #230: 20 +Case #231: 33 +Case #232: 6 +Case #233: 20 +Case #234: 195 +Case #235: 34 +Case #236: 70 +Case #237: 185 +Case #238: 10 +Case #239: 29 diff --git a/2023/round2/ready_go_part_2_sol.md b/2023/round2/ready_go_part_2_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..7e48d41fdd2539c3649432483a873143daec934b --- /dev/null +++ b/2023/round2/ready_go_part_2_sol.md @@ -0,0 +1,7 @@ +We can augment our solution to part 1 with a bit more information to solve part 2. Additionally, now that the bounds are larger, depth-first search solutions will need to be more careful about stack size. Breadth-first solutions won't have this problem. + +We'll build up a table, \(\text{captured}(i, j)\), the number of white stones you can capture if you play a black stone at \((i, j)\). Initially every entry in this table is \(0\). + +When you run your search to find all of the white stones in a given group, keep track of the total number of white stones in the group. Call this \(K\). If and only if this group turns out to have exactly \(1\) adjacent empty space at \((i, j)\), then increment \(\text{captured}(i, j)\) by \(K\), indicating that we'll capture these \(K\) stones if we play at \((i, j)\). If multiple white groups share the same single adjacent empty space, we'll be summing the sizes of all of those groups. + +At the end, your answer is the maximum value in \(\text{captured}\). This solution still has a time complexity of \(\mathcal{O}(R * C\)). diff --git a/2023/round2/tower_rush.cpp b/2023/round2/tower_rush.cpp new file mode 100644 index 0000000000000000000000000000000000000000..ee423b34391bf6c459a966b88253043778bdd2e0 --- /dev/null +++ b/2023/round2/tower_rush.cpp @@ -0,0 +1,189 @@ +#include +#include +#include +using namespace std; + +const int MAX = 1000010; +const int MOD = 1000000007; +using LL = long long; + +class mint { + LL val; + + mint& normalize(LL x) { + val = x % MOD; + if (val < 0) { + val += MOD; + } + return *this; + } + + static LL inverse(LL a) { + LL u = 0, v = 1, m = MOD; + while (a != 0) { + LL t = m / a; + m -= t * a; + swap(a, m); + u -= t * v; + swap(u, v); + } + assert(m == 1); + return u; + } + + public: + mint(LL x = 0) { normalize(x); } + + LL operator()() const { return val; } + + template + explicit operator U() const { return static_cast(val); } + + mint& operator=(LL x) { return normalize(x); } + mint& operator=(const mint& x) { val = x.val; return *this; } + + mint& operator+=(const mint& x) { return normalize(val + x.val); } + mint& operator-=(const mint& x) { return normalize(val - x.val); } + mint& operator*=(const mint& x) { return normalize(val * x.val); } + mint& operator/=(const mint& x) { return *this *= mint(inverse(x.val)); } + + mint& operator+=(LL x) { return *this += mint(x); } + mint& operator-=(LL x) { return *this -= mint(x); } + mint& operator*=(LL x) { return *this *= mint(x); } + mint& operator/=(LL x) { return *this /= mint(x); } + + mint& operator++() { return *this += 1; } + mint& operator--() { return *this -= 1; } + mint operator++(int) { mint z(*this); ++*this; return z; } + mint operator--(int) { mint z(*this); --*this; return z; } + + friend mint operator+(mint x, const mint& y) { return x += y; } + friend mint operator*(mint x, const mint& y) { return x *= y; } + friend mint operator-(mint x, const mint& y) { return x -= y; } + friend mint operator/(mint x, const mint& y) { return x /= y; } + + friend mint operator+(mint x, LL y) { return x += y; } + friend mint operator*(mint x, LL y) { return x *= y; } + friend mint operator-(mint x, LL y) { return x -= y; } + friend mint operator/(mint x, LL y) { return x /= y; } + + friend mint operator+(LL x, mint y) { return y += x; } + friend mint operator*(LL x, mint y) { return y *= x; } + friend mint operator-(LL x, const mint& y) { mint z(x); return z -= y; } + friend mint operator/(LL x, const mint& y) { mint z(x); return z /= y; } + + bool operator <(const mint& x) const { return val < x.val; } + bool operator==(const mint& x) const { return val == x.val; } + bool operator >(const mint& x) const { return val > x.val; } + bool operator!=(const mint& x) const { return val != x.val; } + bool operator<=(const mint& x) const { return val <= x.val; } + bool operator>=(const mint& x) const { return val >= x.val; } + + bool operator <(LL x) const { return val < x; } + bool operator==(LL x) const { return val == x; } + bool operator >(LL x) const { return val > x; } + bool operator!=(LL x) const { return val != x; } + bool operator<=(LL x) const { return val <= x; } + bool operator>=(LL x) const { return val >= x; } +}; + +class factorial : vector { + void lazy_eval(int n) { + for (LL p = size(); n >= p; ++p) { + push_back(back() * p); + } + } + + public: + factorial() { push_back(1); } + + mint choose(LL n, LL k) { + if (n < 0 || k < 0 || n < k) { + return 0; + } + if (k == 0 || k == n) { + return 1; + } + if (n >= MOD && n <= k + MOD - 1) { + return 0; + } + lazy_eval(n); + return at(n) / (at(k) * at(n - k)); + } + + mint operator[](LL n) { + lazy_eval(n); + return at(n); + } +}; + +factorial F; +vector> divisors(MAX); +vector mu(MAX); + +void build_mu() { + vector B(MAX); + vector primes; + mu[1] = 1LL; + for (LL i = 2; i < MAX; i++) { + if (!B[i]) { + primes.push_back(i); + mu[i] = -1; + } + for (auto& p : primes) { + LL k = (LL)i * p; + if (k > MAX) { + break; + } + B[k] = 1; + if (i % p != 0) { + mu[k] = -mu[i]; + } else { + mu[k] = 0; + break; + } + } + } +} + +LL solve() { + int N, K, D; + cin >> N >> K >> D; + vector H(N), freq(MAX); + for (int i = 0; i < N; i++) { + cin >> H[i]; + for (LL d : divisors[H[i]]) { + freq[d]++; + } + } + mint ans = 0; + for (LL d : divisors[D]) { + for (LL j = 0; j < MAX; j++) { + LL i = (LL)d * j; + if (i > MAX) { + break; + } + ans += F.choose(freq[i], K) * mu[j]; + } + } + return (LL)(ans * F[K]); +} + +int main() { + ios_base::sync_with_stdio(false); + cin.tie(nullptr); + + for (int i = 1; i < MAX; i++) { + for (int j = i; j < MAX; j += i) { + divisors[j].push_back(i); + } + } + build_mu(); + + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round2/tower_rush.in b/2023/round2/tower_rush.in new file mode 100644 index 0000000000000000000000000000000000000000..3573478438b333f2e2554e08c5c35dd68bad1313 --- /dev/null +++ b/2023/round2/tower_rush.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f6822413effe1f27177a35be9cafffd6608f78b8fae45da30ee1d7b8af84c182 +size 48259197 diff --git a/2023/round2/tower_rush.md b/2023/round2/tower_rush.md new file mode 100644 index 0000000000000000000000000000000000000000..fbbe442f54f3e27d6d6347b25902726f2105bdc8 --- /dev/null +++ b/2023/round2/tower_rush.md @@ -0,0 +1,43 @@ +Alice and Bob are playing *Tower Rush,* a two-phase game involving \(N\) types of blocks, numbered from \(1\) to \(N\). Blocks of type \(i\) have height \(H_i\), and no two types of blocks have the same height. + +Phase 1 of the game consists of \(K \ge 2\) alternating turns, with Alice going first. On each player's turn, they will choose a block type that has not yet been chosen. Note that if \(K\) is odd, this phase would end with Alice having chosen one more type of block than Bob. From this point on, players have access to infinitely many blocks of each of the types they chose. + +Phase 2 consists of an indefinite number of alternating turns, with Bob going first. Each player starts with a tower of height \(0\). On each player's turn, they may pick a block of any type \(i\) available to them, and extend their tower height by \(H_i\). They may also choose to skip their turn, leaving their tower unchanged. + +In a break from tradition, Alice and Bob want to work *together* to see if it's possible for Alice to build a tower that's exactly \(D\) units taller than Bob's. In how many different ways can phase 1 be played out such that it will be possible for Alice to get her tower to be exactly \(D\) units taller than Bob's in phase 2? Two ways are considered different if there exists an \(i\) such that different block types are chosen on turn \(i\) between the two ways. + +As this value may be large, output it modulo \(1{,}000{,}000{,}007\). + + +# Constraints + +\(1 \leq T \leq 24\) +\(2 \leq N \leq 1{,}000{,}000\) +\(2 \leq K \leq \min(N, 20)\) +\(1 \leq D \leq 1{,}000{,}000\) +\(1 \leq H_i \leq 1{,}000{,}000\) + +The sum of \(N\) across all test cases is at most \(7{,}500{,}000\). + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, the first line contains the integers \(N\), \(K\), and \(D\). Then, there is a line containing the \(N\) integers \(H_1\), ..., \(H_N\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by a single integer, the number of unique ways to accomplish Bob's goal, modulo \(1{,}000{,}000{,}007\). + + +# Sample Explanation + +In the first sample case, there are \(54\) valid choices of blocks in phase 1. One such example is \(\{H_2 = 4,\, H_3 = 5\}\) for Alice and \(\{H_4 = 7\}\) for Bob. Four turns in phase 2 can go as follows: +* Bob adds \(H_4 = 7\). His tower now has height \(7\). +* Alice adds \(H_3 = 5\). Her tower now has height \(5\). +* Bob chooses not to add a block. His tower still has height \(7\). +* Alice adds \(H_3 = 5\). Her tower now has height \(10\). + +Alice’s tower is now exactly \(3\) units taller than Bob’s, so this is a valid choice. + +An example of an invalid choice in the first test case would be \(\{H_1 = 2,\, H_2=4\}\) for Alice and \(\{H_5 = 8\}\) for Bob, since no matter how they build their towers, it is impossible for the difference in the height of their towers to ever be \(3\). diff --git a/2023/round2/tower_rush.out b/2023/round2/tower_rush.out new file mode 100644 index 0000000000000000000000000000000000000000..08e5ef28c2d4e2a4f3ad449fa35f6a4b8c8f0341 --- /dev/null +++ b/2023/round2/tower_rush.out @@ -0,0 +1,25 @@ +Case #1: 54 +Case #2: 0 +Case #3: 62 +Case #4: 29520 +Case #5: 5800 +Case #6: 859449537 +Case #7: 431202326 +Case #8: 650698650 +Case #9: 698331763 +Case #10: 629804697 +Case #11: 776102753 +Case #12: 256267520 +Case #13: 920528907 +Case #14: 830255246 +Case #15: 297647105 +Case #16: 376167472 +Case #17: 902540749 +Case #18: 425681072 +Case #19: 425681072 +Case #20: 37480848 +Case #21: 65212767 +Case #22: 910399849 +Case #23: 65569096 +Case #24: 919134035 +Case #25: 296106217 diff --git a/2023/round2/tower_rush_sol.md b/2023/round2/tower_rush_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..44323f7e39fcf8482af330070530a67e4dabbfb1 --- /dev/null +++ b/2023/round2/tower_rush_sol.md @@ -0,0 +1,9 @@ +First note that, for a set of blocks, it does not matter how they are distributed between the players. That is, if two blocks get swapped this will not change the outcome (possible or not to have difference of d). This comes from the fact that for a given distribution of blocks, we can always form two towers of equal height (with height being the LCM of all elements in the set). + +With that, we can effectively either remove or add a block with height \(H_i\) to a tower, which corresponds to either adding or subtracting \(H_i\) to the current difference between the towers. It is a well known result that given integers \((a,b)\), with the equation \(a*x + b*y = k\), where \(x\) and \(y\) can assume any integer values, \(k\) can assume any value that is a multiple of \(\text{gcd}(a,b)\). This results can is also valid for multiple variables (see Bézout's identity). For a given set of blocks, we can make it so that the difference assumes any value that is a multiple of the gcd of the chosen blocks. + +Now the problem simplifies to, for each divisor \(d\) of \(D\), consider how many subsets of size \(K\) are there such that the gcd is \(d\). This can be solved in \(\mathcal{O}(MAXV/i)\). It is easy to convert this problem to, how many subsets of size \(k\) are there such that GCD of values is \(1\). This final problem can be solved with Inclusion-Exclusion principle: for each integer \(x\), it is easy to count how many subsets of size \(k\) are there such that the gcd is a multiple of \(x\) (let's call this \(f(x)\)), this is simply \(\binom{\text{freq}[x]}{k}\) where \(\text{freq}[x]\) is how many integers are divisible by \(x\). The solution is + +\[\sum_{x=1}^{MAXV} [f(x)*\mu(x)]\] + +where \(\mu(x)\) is the Mobious function (that is, for square free integers, \(\mu(x)\) is \(1\) if \(x\) has an even number of prime factors, or \(-1\) if it has an odd number of prime factors, which is related to inclusion exclusion). This works in \(\mathcal{O}(N \log N)\) for any value of \(K\), not necessarily \(K \le 20\). diff --git a/2023/round2/wiki_race.cpp b/2023/round2/wiki_race.cpp new file mode 100644 index 0000000000000000000000000000000000000000..1704b3d1f501126e7a175ba216b4d35e4bca58d9 --- /dev/null +++ b/2023/round2/wiki_race.cpp @@ -0,0 +1,88 @@ +#include +#include +#include +#include +using namespace std; + +int N, leaves; +vector> ch; +vector> S(N); + +// Collapse nodes with 1 ch, u->v->(w1,w2,...) into u->(w1, w2, ...). +void dfs1(int u) { + if (ch[u].empty()) { + leaves++; + return; + } + while (ch[u].size() == 1) { + int v = ch[u][0]; + S[u].insert(S[v].begin(), S[v].end()); + ch[u] = ch[v]; + } + for (int v : ch[u]) { + dfs1(v); + } +} + +int dfs2(int u, const string &s) { + if (ch[u].empty()) { + return S[u].count(s); + } + int has = 0; + for (int v : ch[u]) { + int res = dfs2(v, s); + if (res == -1) { + return -1; + } + has += res; + } + if (has == (int)ch[u].size()) { + return 1; + } + if (has + 1 == (int)ch[u].size()) { + return S[u].count(s); + } + return -1; +} + +int solve() { + cin >> N; + ch.assign(N, {}); + S.assign(N, {}); + for (int i = 1, p; i < N; i++) { + cin >> p; + ch[p - 1].push_back(i); + } + unordered_map freq; + for (int i = 0, m; i < N; i++) { + cin >> m; + string s; + for (int j = 0; j < m; j++) { + cin >> s; + S[i].insert(s); + freq[s]++; + } + } + // Collapse any single child chains in the tree, and count leaves. + leaves = 0; + dfs1(0); + // Try all words with freq[k] >= L (there can only be O(W/L). + int ans = 0; + for (const auto &[s, f] : freq) { + if (f >= leaves && dfs2(0, s) == 1) { + ans++; + } + } + return ans; +} + +int main() { + ios_base::sync_with_stdio(false); + cin.tie(nullptr); + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round2/wiki_race.in b/2023/round2/wiki_race.in new file mode 100644 index 0000000000000000000000000000000000000000..2744a5be8374414ce2cf13362092834c46321459 --- /dev/null +++ b/2023/round2/wiki_race.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e287cfd7e39ab2fe215a709acb49dcc376028384e84132dfbf4e3a82efab6659 +size 60270150 diff --git a/2023/round2/wiki_race.md b/2023/round2/wiki_race.md new file mode 100644 index 0000000000000000000000000000000000000000..0950c55c8d6b5950c8010b11bf0690e2f32d30ad --- /dev/null +++ b/2023/round2/wiki_race.md @@ -0,0 +1,49 @@ +You and some friends are playing [Wiki Race](https://en.wikipedia.org/wiki/Wikipedia:Wiki\_Game) on an online encyclopedia of \(N\) pages, numbered from \(1\) to \(N\). Page \(i\) covers \(M_i\) topics \(S_{i,1}\) ... \(S_{i,M_i}\), each a string of lowercase letters. Each player starts on a unique page, and will race to reach page \(1\) (the target page) by repeatedly clicking to neighboring pages. + +Unfortunately, someone out there has already published the shortest path from each page to page \(1\) for the whole internet to see. This forms a tree structure with page \(1\) as the root, and edges directed upwards from each descendent page. In particular, you and your friends all know that at each page \(i > 1\), one should click to the neighboring page \(P_i\) for the fastest route to page \(1\). + +In light of this, all players have agreed to a new rule that each page should be visited by exactly one person per game. This means that once a player hits a page someone has already visited, their upward path ends there and the player loses. Formally, we can express each game's outcome as a partition of the tree into vertex-disjoint paths, with each path's upper endpoint always having an edge to a node on another player's path (except the winner, whose upper endpoint is page \(1\)). + +After playing the game for so long, you're all starting to realize that "*it's not about winning, but the knowledge acquired along the way!*" + +For a given partition of the tree into paths, a topic is _mutually-learned_ by all players if every path has at least one page covering that topic. You'd like to know, across all possible partitions of paths, all topics that could be mutually-learned. + + +# Constraints + +\(1 \leq T \leq 30\) +\(2 \leq N \leq 1{,}000{,}000\) +\(1 \leq P_i \leq N\) +\(1 \leq M_i \leq 100{,}000\) +\(1 \leq |S_{i,j}| \leq 10\) +\(S_{i,j}\) consists only of lowercase letters ('`a`'..'`z`'). +The sum of \(N\) across all test cases is at most \(3{,}000{,}000\). +The sum of \(M_i\) across all test cases is at most \(8{,}000{,}000\). + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. For each case, there is first a line containing a single integer \(N\). Then, there is a line containing \(N - 1\) integers \(P_2, ..., P_N\). Then, \(N\) lines follow, the \(i\)th of which contains the integer \(M_i\), followed by strings \(S_{i,1}\) ... \(S_{i,M_i}\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by the number of topics that can be mutually-learned. + + +# Sample Explanation + +The three sample cases are depicted below: + +{{PHOTO_ID:636025968710743|WIDTH:600}} + +In the first case: +* "gouda" is mutually-learned in the partition \(\{\{4\}, \{6, 5, 2\}, \{3, 1\}\}\) +* "cheddar" is mutually-learned in the partition \(\{\{4, 2\}, \{6, 5\}, \{3, 1\}\}\) +* "edam" is mutually-learned in the same partition as "cheddar" as well as in other partitions such as \(\{\{4, 2, 1\}, \{6\}, \{5\}, \{3\}\}\) +* "gjetost", "gruyere", and "mozzarella" can never be mutually-learned + +In the second case, the only topic that can be mutually-learned is "earth". + +In the third case, the partition \(\{\{1, 2, 3\}\}\) makes all three topics mutually-learned. + diff --git a/2023/round2/wiki_race.out b/2023/round2/wiki_race.out new file mode 100644 index 0000000000000000000000000000000000000000..a6905b580ee61ddb04fa678da6d25637c445b3f9 --- /dev/null +++ b/2023/round2/wiki_race.out @@ -0,0 +1,32 @@ +Case #1: 3 +Case #2: 1 +Case #3: 3 +Case #4: 2 +Case #5: 1 +Case #6: 3 +Case #7: 5 +Case #8: 9 +Case #9: 3 +Case #10: 0 +Case #11: 0 +Case #12: 26 +Case #13: 11 +Case #14: 3 +Case #15: 15 +Case #16: 136 +Case #17: 76 +Case #18: 38265 +Case #19: 165 +Case #20: 44 +Case #21: 146 +Case #22: 62 +Case #23: 8 +Case #24: 275 +Case #25: 9 +Case #26: 9 +Case #27: 4322 +Case #28: 0 +Case #29: 1315 +Case #30: 232673 +Case #31: 3351 +Case #32: 115 diff --git a/2023/round2/wiki_race_sol.md b/2023/round2/wiki_race_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..ea4e1c3b6c906e5bfcf5b7021812896c4e49093a --- /dev/null +++ b/2023/round2/wiki_race_sol.md @@ -0,0 +1,9 @@ +Let \(L\) be the number of leaves in the tree. A partition that optimally learns topics for all players might as well start each player's path from a leaf. Therefore there can be at most \(L\) paths to consider in a given partition. + +Let \(F(s)\) be the frequency of topic \(s\) in the input. If a topic \(s\) is mutually-learnable, then it must be that \(F(s) \ge L\). This means there can be at most \(\mathcal{O}((\sum M_i) / L)\) candidate words. Assuming we have an \(\mathcal{O}(L)\) algorithm to verify that a candidate word is mutually-learnable, then we would have an overall running time of \(\mathcal{O}((\sum M_i) / L) * \mathcal{O}(L) = \mathcal{O}(\sum M_i)\), which is sufficient. + +We can compress the tree by merging "lines", or sequences of nodes with only one child each. This can be done with a DFS, where if we reach a node \(u\) with \(1\) child, we repeatedly "reel up" the single-child chain (merging the topic lists into node \(u\)), Once the last single-child's children has become \(u\)'s children, we branch out from \(u\). + +With this compression, we guarantee that each tree node has at least \(2\) children. This means that the numbers of nodes at each level from the bottom to the root will be limited by \(L\), \(\frac{L}{2}\), \(\frac{L}{4}\), …, \(1\). The total sum of this geometric series, as we should know, is \(\mathcal{O}(L)\). + +To check each candidate word, we can do a DFS on the compressed tree, resolving the problem bottom-up. If we find the string in the root of the subtree, we pair it with any leaf needing it. However, if at any time a subtree has two leaves or more with no matches, that topic can’t be mutually-learned. diff --git a/2023/round3/1286569645378322.jpg b/2023/round3/1286569645378322.jpg new file mode 100644 index 0000000000000000000000000000000000000000..892ad4e556e8c977e31c68aa54d5db5bde64148e --- /dev/null +++ b/2023/round3/1286569645378322.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4080a0ca1f64115d9f3e5d568d250973b075689a82417d16a5ca2a61ed71f757 +size 50090 diff --git a/2023/round3/1573931960080150.jpg b/2023/round3/1573931960080150.jpg new file mode 100644 index 0000000000000000000000000000000000000000..1a52b3d27d1850f6a72843777636d4d5456181c8 --- /dev/null +++ b/2023/round3/1573931960080150.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:286a8d0a747b76970e0fc53185e37e2efaaeae3aa09b2cca7e0bd8d2de7010dc +size 36474 diff --git a/2023/round3/216354288143884.jpg b/2023/round3/216354288143884.jpg new file mode 100644 index 0000000000000000000000000000000000000000..660b8b619a663f0644fc783c8b89cbfe0a613816 --- /dev/null +++ b/2023/round3/216354288143884.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:266478eadbadc9471b0b83086ab8f00597d360bc5318155b0422a77879d0cc7c +size 57297 diff --git a/2023/round3/720027639598818.jpg b/2023/round3/720027639598818.jpg new file mode 100644 index 0000000000000000000000000000000000000000..6f30823e3148490fcbc6d1b366b5413da4a7c735 --- /dev/null +++ b/2023/round3/720027639598818.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b6170b8d14641bbf7c4dee4eec50e92645ec42c644985616c5a2e3cc9f1ea719 +size 16710 diff --git a/2023/round3/854841726123175.jpg b/2023/round3/854841726123175.jpg new file mode 100644 index 0000000000000000000000000000000000000000..a1211a29ee32b989a72ee7229fbfbb657cf8e666 --- /dev/null +++ b/2023/round3/854841726123175.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8042c39857e03d48a30e7c7f6601f3be8bc4b80c830d4e9a272d0566b031022a +size 51818 diff --git a/2023/round3/double_stars.cpp b/2023/round3/double_stars.cpp new file mode 100644 index 0000000000000000000000000000000000000000..8b7efbdb466cfacadaa9f9416a7e6721d7e387a6 --- /dev/null +++ b/2023/round3/double_stars.cpp @@ -0,0 +1,126 @@ +#include +#include +#include +using namespace std; + +// Solution: +// - For each edge u-v, find d(u, v) = longest arm length starting with u-v-.. +// - For each edge u-v, consider all double stars (x, y) with u-v as the center. +// - Iterate through f[u] and f[v] with 2-pointer, with d' initially 1: +// - For the first (d, x), sum min(degree(u) - 1, degree(v) - 1) for every +// [d', d] (those d's, we can make 2-star with that number of arms). +// - Remove the first and decrease x from degree(u) or degree(v), depending +// on what the pair belongs to. +// - Set d' = d + 1 + +vector> adj; +vector> d; // d[u][i] = greatest distance reachable from adj[u][i] +vector> f; // f[u] = map of (d, x), representing d is the + // greatest distance of an arm (u, v) for x edges v + +int pre(int u, int par = -1, int l = 0) { + int mai = 0; + for (int i = 0; i < (int)adj[u].size(); ++i) { + int v = adj[u][i]; + d[u].push_back(-1); + if (v == par) { + continue; + } + int ret = pre(v, u); + mai = max(mai, ret); + d[u][i] = ret; + } + return 1 + mai; +} + +void pre2(int u, int par = -1, int farthest_up = 0) { + vector pref(adj[u].size()), suf(adj[u].size()); + for (int i = 0; i < (int)adj[u].size(); ++i) { + pref[i] = max(i ? pref[i - 1] : 0, d[u][i]); + } + for (int i = (int)adj[u].size() - 1; i >= 0; --i) { + suf[i] = max(i + 1 < (int)adj[u].size() ? suf[i + 1] : 0, d[u][i]); + } + for (int i = 0; i < (int)adj[u].size(); ++i) { + int v = adj[u][i]; + if (v == par) { + d[u][i] = farthest_up; + continue; + } + // Farthest going down here. + int tmp = 1 + max(i ? pref[i - 1] : 0, + i + 1 < (int)adj[u].size() ? suf[i + 1] : 0); + pre2(v, u, max(1 + farthest_up, tmp)); + } +} + +long long rec(int u, int par = -1) { + long long ans = 0; + for (int i = 0; i < (int)adj[u].size(); ++i) { + int v = adj[u][i]; + if (v == par) { + // Calculate for DST centered at (u, par). + f[u][d[u][i]]--; + + auto it1 = f[v].begin(); + auto it2 = f[u].begin(); + int tot1 = (int)adj[v].size() - 1; + int tot2 = (int)adj[u].size() - 1; + + int l = 1; + while (it1 != f[v].end() || it2 != f[u].end()) { + if (it1 != f[v].end() && + (it2 == f[u].end() || it1->first < it2->first)) { + ans += (long long)(it1->first - l + 1) * min(tot1, tot2); + tot1 -= it1->second; + l = it1->first + 1; + it1++; + } else { + ans += (long long)(it2->first - l + 1) * min(tot1, tot2); + tot2 -= it2->second; + l = it2->first + 1; + it2++; + } + } + // assert(tot1 == 0 && tot2 == 0); + f[u][d[u][i]]++; + continue; + } + f[u][d[u][i]]--; + ans += rec(v, u); + f[u][d[u][i]]++; + } + return ans; +} + +long long solve() { + int N; + cin >> N; + adj.assign(N + 1, {}); + d.assign(N + 1, {}); + f.assign(N + 1, {}); + for (int i = 0, p; i < N - 1; i++) { + cin >> p; + adj[i + 2].push_back(p); + adj[p].push_back(i + 2); + } + pre(1); + pre2(1); + for (int u = 1; u <= N; ++u) { + for (auto dd : d[u]) { + f[u][dd]++; + } + } + return rec(1); +} + +int main() { + ios_base::sync_with_stdio(0); + cin.tie(0); + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round3/double_stars.in b/2023/round3/double_stars.in new file mode 100644 index 0000000000000000000000000000000000000000..20fc671a91e5432dfbd86951b77a28c9d6bcf8ec --- /dev/null +++ b/2023/round3/double_stars.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8bef6093725fcf8971cb1a0aae2d74bf88d3a159f8f60386c7bde8fd5a69545f +size 51687186 diff --git a/2023/round3/double_stars.md b/2023/round3/double_stars.md new file mode 100644 index 0000000000000000000000000000000000000000..d557aae11abac50c49b7ba02e5891d0a5c53d099 --- /dev/null +++ b/2023/round3/double_stars.md @@ -0,0 +1,32 @@ +Patrick's big sister Sam is visiting for Halloween, and Sandy wants to impress the two Stars by dressing as an astronaut. Unfortunately her regular outfit already resembles an astronaut, so she'll have to go as a computer scientist instead. Sandy knows a lot about [double stars](https://en.wikipedia.org/wiki/Double_star) in astronomy, but will now have to learn about double stars in graph theory. + +There is an unrooted tree with \(N\) nodes, and an edge between nodes \(i\) and \(P_i\) (for each \(i \ge 2\)). A *double star* is a tree centered at some edge \(u \leftrightarrow v\), where \(u\) and \(v\) each have \(x\) chains connected to them, each chain consisting of \(y\) nodes (\(x, y \ge 1\)). Alternatively, you can think of a double star as two identical star graphs (each with \(x\) chains of length \(y\)) connected by their centers. Thus, a distinct double star is specified by a tuple \((u, v, x, y)\), where \(u \lt v\). + +Please help Sandy count the number of distinct tuples \((u, v, x, y)\) for which the tree contains the double star described by that tuple as a subgraph. + +# Constraints + +\(1 \leq T \leq 110\) +\(2 \leq N \leq 1{,}000{,}000\) +\(1 \leq P_i \leq N\) + +The sum of \(N\) across all test cases is at most \(8{,}000{,}000\). + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case has two lines. The first contains the integer \(N\), and the second contains the \(N-1\) integers \(P_2\) through \(P_N\). + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by the number of distinct double star tuples \((u, v, x, y)\), where \(u \lt v\). + +# Sample Explanation +The first sample tree is depicted below. There are \(5\) distinct double star tuples: \((1, 2, 1, 1)\), \((1, 2, 1, 2)\), \((1, 2, 2, 1)\), \((1, 4, 1, 1)\) and \((2, 6, 1, 1)\). + +{{PHOTO_ID:1573931960080150|WIDTH:350}} + +The second sample tree is depicted below. There are \(5\) distinct double star tuples: \((1, 2, 1, 1)\), \((1, 2, 1, 2)\), \((1, 8, 1, 1)\), \((1, 8, 2, 1)\) and \((2, 7, 1, 1)\). + +{{PHOTO_ID:1286569645378322|WIDTH:500}} + +In the third sample tree, no double stars exist because arm lengths of \(0\) are not allowed \((y \ge 1 )\). diff --git a/2023/round3/double_stars.out b/2023/round3/double_stars.out new file mode 100644 index 0000000000000000000000000000000000000000..d54c6159f462e5abef41bd154750b093abb0a71d --- /dev/null +++ b/2023/round3/double_stars.out @@ -0,0 +1,134 @@ +Case #1: 5 +Case #2: 5 +Case #3: 0 +Case #4: 9 +Case #5: 16 +Case #6: 8 +Case #7: 16 +Case #8: 6 +Case #9: 10 +Case #10: 9 +Case #11: 8 +Case #12: 8 +Case #13: 12 +Case #14: 9 +Case #15: 8 +Case #16: 7 +Case #17: 7 +Case #18: 5 +Case #19: 6 +Case #20: 7 +Case #21: 6 +Case #22: 7 +Case #23: 8 +Case #24: 6 +Case #25: 6 +Case #26: 10 +Case #27: 7 +Case #28: 9 +Case #29: 7 +Case #30: 4 +Case #31: 9 +Case #32: 8 +Case #33: 10 +Case #34: 12 +Case #35: 9 +Case #36: 10 +Case #37: 914 +Case #38: 2544 +Case #39: 465 +Case #40: 3442 +Case #41: 757 +Case #42: 3270 +Case #43: 1074 +Case #44: 2675 +Case #45: 618 +Case #46: 2903 +Case #47: 4739 +Case #48: 1216 +Case #49: 4734 +Case #50: 808 +Case #51: 3346 +Case #52: 1107 +Case #53: 3353 +Case #54: 585 +Case #55: 4438 +Case #56: 994 +Case #57: 5281 +Case #58: 5083 +Case #59: 4242 +Case #60: 5871 +Case #61: 4809 +Case #62: 1003182 +Case #63: 12582273 +Case #64: 81 +Case #65: 38809 +Case #66: 2209 +Case #67: 55225 +Case #68: 25 +Case #69: 37249 +Case #70: 1849 +Case #71: 53361 +Case #72: 249001 +Case #73: 249999000001 +Case #74: 209 +Case #75: 5797 +Case #76: 101700 +Case #77: 6216 +Case #78: 640 +Case #79: 738 +Case #80: 23 +Case #81: 840 +Case #82: 190 +Case #83: 2008 +Case #84: 241 +Case #85: 233 +Case #86: 324 +Case #87: 274 +Case #88: 456 +Case #89: 55 +Case #90: 494901 +Case #91: 494901 +Case #92: 94539 +Case #93: 499499001 +Case #94: 69 +Case #95: 57 +Case #96: 1201 +Case #97: 73 +Case #98: 61 +Case #99: 909 +Case #100: 1091 +Case #101: 249901 +Case #102: 227511 +Case #103: 499999 +Case #104: 1568 +Case #105: 27 +Case #106: 2739 +Case #107: 252 +Case #108: 1003 +Case #109: 15 +Case #110: 1058 +Case #111: 225 +Case #112: 967 +Case #113: 30 +Case #114: 1852 +Case #115: 1706 +Case #116: 247 +Case #117: 1512 +Case #118: 398 +Case #119: 1084 +Case #120: 116 +Case #121: 1821 +Case #122: 195 +Case #123: 1418 +Case #124: 2886 +Case #125: 287 +Case #126: 1905 +Case #127: 455 +Case #128: 41456 +Case #129: 23833 +Case #130: 201426 +Case #131: 95185 +Case #132: 3708265 +Case #133: 4140864 +Case #134: 2801448 diff --git a/2023/round3/double_stars_sol.md b/2023/round3/double_stars_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..65006db88b6c892983f1aa01a631aefad88a090c --- /dev/null +++ b/2023/round3/double_stars_sol.md @@ -0,0 +1,10 @@ +For each vertex \(u\), and for each incident edge \(e\), we first have to calculate the longest distance to another vertex coming from this edge \(e\). It can be observed that for each vertex, there can be at most \(\mathcal{O}(\sqrt N)\) distinct distances. + +We'll build a table \(d(u, v)\) storing the longest arm length starting with \(u \to v \to ...\), which can be computed in \(\mathcal{O}(N)\) using two DFS's. For each vertex \(v\), we have a list of pairs \((d, x)\) indicating that from vertex \(v\) there are \(x\) different edges incident to \(v\) in which the furthest distance you can reach is \(d\). We'll store this in a table \(f(u)\) holding a map of \((d, x)\), denoting that \(d\) is the greatest distance of an arm \(u \to v\) for \(x\) edges \(v\). \(f(u)\) can be computed using just the table \(d(u, v)\), + +From here, the problem can be solved using the two-pointer technique. We'll want consider every edge \(u \leftrightarrow v\), and efficiently generate every \((x, y)\) that defines a double star tree with \(u \leftrightarrow v\) as its center. For each \((u, v, x, y)\), we iterate through \(f(u)\) and \(f(v)\) with two-pointers, with \(d'\) initially set to \(1\): + - For the first \((d, x)\), take the sum of \(\min(\text{degree}(u) - 1, \text{degree}(v) - 1)\) over every \([d', d]\) (on those \(d\)'s, we can make a double star with that number of arms). + - Remove the first and decrease \(x\) from \(\text{degree}(u)\) or \(\text{degree}(v)\), depending on what the pair belongs to. + - Set \(d' = d + 1\). + +The are \(\mathcal{O}(N)\) possible edges for the center of the star, each of which we'll examine \(\mathcal{O}(\sqrt N)\) distinct distances using the two-pointers. The overall time complexity is \(\mathcal{O}(N \sqrt N)\). diff --git a/2023/round3/hash_slinger.cpp b/2023/round3/hash_slinger.cpp new file mode 100644 index 0000000000000000000000000000000000000000..06e9afd096401291aeab337eec377f2927fde4c9 --- /dev/null +++ b/2023/round3/hash_slinger.cpp @@ -0,0 +1,59 @@ +#include +#include +#include +using namespace std; + +const int MAX = 1 << 13; + +int solve() { + int N, K; + cin >> N >> K; + vector A(N); + for (int i = 0; i < N; i++) { + cin >> A[i]; + A[i] %= K; + } + vector first_sofar(K, N + 1); + vector> first(N, vector(K)); + for (int i = N - 1; i >= 0; --i) { + int d = 0; + for (int j = i; j < N; ++j) { + d = (d + A[j]) % K; + first_sofar[d] = min(first_sofar[d], j + 1); + } + first[i] = first_sofar; + } + vector dist(MAX, N + 1); + vector visit(MAX); + dist[0] = 0; + for (;;) { + int x = -1; + for (int j = 0; j < MAX; ++j) { + if (!visit[j] && dist[j] < N && (x == -1 || dist[x] > dist[j])) { + x = j; + } + } + if (x == -1) { + break; + } + visit[x] = 1; + for (int d = 0; d < K; d++) { + dist[x ^ d] = min(dist[x ^ d], first[dist[x]][d]); + } + } + int res = 0; + for (int i = 0; i < MAX; ++i) { + res += (dist[i] != N + 1); + } + return res; + +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/round3/hash_slinger.in b/2023/round3/hash_slinger.in new file mode 100644 index 0000000000000000000000000000000000000000..ea243841e99e2d160ddef8331824a925cf378476 --- /dev/null +++ b/2023/round3/hash_slinger.in @@ -0,0 +1,391 @@ +195 +3 10 +2 2 0 +5 2 +8 8 4 0 1 +4 100 +6 3 5 9 +8 100 +35 53 3 56 58 14 10 61 +8 64 +48 16 64 16 19 16 32 48 +10 10 +259051541 65887488 530604911 813205302 907139220 531875291 795733712 135498985 303233570 953669086 +1000 1000 +152842180 75519060 226871641 141204543 671215795 382478787 77472318 434336944 242822668 634091931 766468477 85696666 20967306 551018907 757239215 768368225 156003699 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--- /dev/null +++ b/2023/round3/hash_slinger.md @@ -0,0 +1,51 @@ +Spongebob wants to go out trick-or-treating on Halloween, but has to work a graveyard shift at the Krusty Krab. Luckily, his ghostly fry cook friend, the hash-slinging slasher, is here to cover. + +At the start of the shift, there are \(N\) patties on the grill (numbered from \(1\) to \(N\)), the \(i\)th of which weighs \(A_i\) grams. For each order that comes in, the slasher removes from the grill a non-empty sequence of patties at contiguous indices \(i..j\) (patties \(i..j\) must all be on the grill at that moment). The *deliciousness* of the order is defined as \((A_i + ... + A_j)\), modulo \(M\) because too much meat can be overpowering! Note that if a patty is removed, a future order's range cannot span across that empty spot. Also, there may be patties left over at the end. + +As proof of his hard work at the end of the shift, the slasher will compute a hash by taking the [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation) of the deliciousnesses of all the orders. How many distinct hashes are possible across all valid sequences of \(0\) or more orders? + + +# Constraints + +\(1 \leq T \leq 125\) +\(0 \leq A_i \leq 1{,}000{,}000{,}000\) +\(1 \leq N \leq 9{,}000\) +\(1 \leq M \leq 5{,}000\) + +The sum of \(N + M\) across all test cases is at most \(200{,}000\). + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case begins with a line containing the two integers \(N\) and \(M\). Then, there is a single line containing \(N\) integers, \(A_1, ..., A_N\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by the number of different possible hashes. + + +# Sample Explanation + +In the first sample case, there are \(3\) patties with \(A = [2, 2, 0]\) and \(M = 10\). The possible valid order sequences (patties removed) and hashes are: + +- No patties removed \(\to\) hash \(0\) +- Patties \(1..1\) \(\to\) hash \(2\) +- Patties \(2..2\) \(\to\) hash \(2\) +- Patties \(3..3\) \(\to\) hash \(0\) +- Patties \(1..2\) \(\to\) hash \(4\) +- Patties \(2..3\) \(\to\) hash \(2\) +- Patties \(1..3\) \(\to\) hash \(4\) +- Patties \(1..1\) and patties \(2..2\) \(\to\) hash \(0\) +- Patties \(1..1\) and patties \(3..3\) \(\to\) hash \(2\) +- Patties \(2..2\) and patties \(3..3\) \(\to\) hash \(2\) +- Patties \(1..1\) and patties \(2..3\) \(\to\) hash \(0\) +- Patties \(1..2\) and patties \(3..3\) \(\to\) hash \(4\) +- Patties \(1..1\), patties \(2..2\), and patties \(3..3\) \(\to\) hash \(0\) + +There are \(3\) distinct hashes the slasher can make: \(0\), \(2\), and \(4\). + +In the second case, the slasher can make hashes \(0\) (for example, with no orders) and \(1\) (for example, with \(5\) orders, each including one of the patties). + + + diff --git a/2023/round3/hash_slinger.out b/2023/round3/hash_slinger.out new file mode 100644 index 0000000000000000000000000000000000000000..7081e2831dd228d3be4fe3b6434cf6339ef9a944 --- /dev/null +++ b/2023/round3/hash_slinger.out @@ -0,0 +1,195 @@ +Case #1: 3 +Case #2: 2 +Case #3: 16 +Case #4: 128 +Case #5: 8 +Case #6: 16 +Case #7: 1024 +Case #8: 4096 +Case #9: 8192 +Case #10: 2 +Case #11: 8192 +Case #12: 8192 +Case #13: 512 +Case #14: 1280 +Case #15: 1664 +Case #16: 2 +Case #17: 1 +Case #18: 2 +Case #19: 4 +Case #20: 7 +Case #21: 2 +Case #22: 2 +Case #23: 7 +Case #24: 2 +Case #25: 8 +Case #26: 2 +Case #27: 4 +Case #28: 4 +Case #29: 4 +Case #30: 4 +Case #31: 8 +Case #32: 4 +Case #33: 4 +Case #34: 8 +Case #35: 1 +Case #36: 8 +Case #37: 1 +Case #38: 4 +Case #39: 4 +Case #40: 1 +Case #41: 4 +Case #42: 8 +Case #43: 2 +Case #44: 2 +Case #45: 6 +Case #46: 2 +Case #47: 9 +Case #48: 16 +Case #49: 3 +Case #50: 6 +Case #51: 2 +Case #52: 2 +Case #53: 1 +Case #54: 8 +Case #55: 2 +Case #56: 7 +Case #57: 4 +Case #58: 2 +Case #59: 1 +Case #60: 1 +Case #61: 8 +Case #62: 2 +Case #63: 2 +Case #64: 2 +Case #65: 8 +Case #66: 10 +Case #67: 1 +Case #68: 4 +Case #69: 8 +Case #70: 1 +Case #71: 2 +Case #72: 7 +Case #73: 1 +Case #74: 4 +Case #75: 14 +Case #76: 4 +Case #77: 1 +Case #78: 2 +Case #79: 2 +Case #80: 1 +Case #81: 4 +Case #82: 1 +Case #83: 10 +Case #84: 16 +Case #85: 8 +Case #86: 2 +Case #87: 1 +Case #88: 2 +Case #89: 4 +Case #90: 1 +Case #91: 2 +Case #92: 8 +Case #93: 8 +Case #94: 16 +Case #95: 12 +Case #96: 512 +Case #97: 1024 +Case #98: 256 +Case #99: 1024 +Case #100: 512 +Case #101: 16 +Case #102: 1024 +Case #103: 1024 +Case #104: 1024 +Case #105: 1024 +Case #106: 1024 +Case #107: 1024 +Case #108: 1024 +Case #109: 256 +Case #110: 512 +Case #111: 128 +Case #112: 64 +Case #113: 256 +Case #114: 1024 +Case #115: 1024 +Case #116: 1024 +Case #117: 1024 +Case #118: 512 +Case #119: 1024 +Case #120: 256 +Case #121: 1024 +Case #122: 64 +Case #123: 1024 +Case #124: 256 +Case #125: 1024 +Case #126: 1024 +Case #127: 512 +Case #128: 1024 +Case #129: 1024 +Case #130: 512 +Case #131: 256 +Case #132: 1024 +Case #133: 1024 +Case #134: 16 +Case #135: 1024 +Case #136: 1024 +Case #137: 1024 +Case #138: 1024 +Case #139: 256 +Case #140: 64 +Case #141: 512 +Case #142: 1024 +Case #143: 1024 +Case #144: 256 +Case #145: 512 +Case #146: 1024 +Case #147: 1024 +Case #148: 512 +Case #149: 512 +Case #150: 1024 +Case #151: 512 +Case #152: 1024 +Case #153: 256 +Case #154: 1024 +Case #155: 1024 +Case #156: 512 +Case #157: 1024 +Case #158: 1024 +Case #159: 1024 +Case #160: 1024 +Case #161: 64 +Case #162: 1024 +Case #163: 512 +Case #164: 1024 +Case #165: 1024 +Case #166: 128 +Case #167: 1024 +Case #168: 1024 +Case #169: 1024 +Case #170: 1024 +Case #171: 1024 +Case #172: 256 +Case #173: 256 +Case #174: 512 +Case #175: 512 +Case #176: 1024 +Case #177: 1024 +Case #178: 256 +Case #179: 1024 +Case #180: 1024 +Case #181: 1024 +Case #182: 512 +Case #183: 1024 +Case #184: 1024 +Case #185: 256 +Case #186: 512 +Case #187: 1024 +Case #188: 128 +Case #189: 512 +Case #190: 128 +Case #191: 1024 +Case #192: 512 +Case #193: 512 +Case #194: 512 +Case #195: 128 diff --git a/2023/round3/hash_slinger_sol.md b/2023/round3/hash_slinger_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..72aeccdb7499142a1cdb3d85261030df1e159fad --- /dev/null +++ b/2023/round3/hash_slinger_sol.md @@ -0,0 +1,11 @@ +Another way to think of the hash is partitioning the array \(A\) into segments, choosing only a subset of segments to XOR together for a hash. + +Consider a dynamic programming approach where we build a table \(dp(i, x)\) storing whether it's possible to attain an XOR value of \(x\) by choosing a set of non-overlapping subarrays in \(A_{1..i}\). We initialize \(dp(0, 0)\) as true (no orders \(\to\) hash \(0\)). For each true \(d(i, x)\), we'll consider each \(j > i\), corresponding to segment \(A_{(i+1)..j}\). We can either choose to exclude this segment (not changing \(x\)) and set \(dp(j, x)\) to true, or include it and set \(dp(j, x \oplus [(A_{i+1} + ... + A_j) \text{ mod } M])\) to true. In the end, we want to output the number of \(x\)'s where \(d(N, x)\) is true. + +We can first pre-compute a modded prefix sum array on \(A\), allowing us to later obtain the deliciousness of any range in constant time. However, the number of states in the approach above is \(\mathcal{O}(N)\) times the \(\mathcal{O}(M)\) possible XOR values. Transitioning adds another factor of \(N\), for an overall running time of \(\mathcal{O}(N^2 M)\), which won't be fast enough. + +Instead, let's imagine starting with a hash \(x = 0\), and repeatedly transitioning \(x\) to \(x \oplus d\) by applying a deliciousness score \(d\) of a segment \(A_{j..k}\) *starting after the minimum* \(i\) such that \(A_{1..i}\) can produce \(x\) (where \(i < j \le k\)). If here are many such segments for a given \(i\), we might as well always choose one with the minimum \(k\), as it will leave us with more options later on. Using \(\mathcal{O}(N^2)\) time, we can precompute a table \(\text{first}[i][d]\) storing the first \(j \ge i\) such that \(A_{i..j}\) yields deliciousness \(d\) (infinity if there's no such \(j\)). + +To search the \(\mathcal{O}(M)\) hash space, we will apply an algorithm similar to Dijkstra's, except we'll let the distance array \(\text{dist}[x]\) denote the minimum \(i\) such that \(A_{1..i}\) can produce hash \(x\) (\(\text{dist}[x]\) initialized to infinity). In the main loop, we repeatedly find the unvisited hash \(x\) with minimum \(\text{dist}[x]\), mark it as visited, then consider each deliciousness score \(d = 0..(K-1)\) for which we perform the relaxation of neighbor \(x \oplus d\) as \(\text{dist}[x \oplus d] = \text{min}(\text{dist}[x \oplus d], \text{first}[\text{dist}[x]][d])\). + +In the end, we report the number of hashes \(x\) with non-infinite \(\text{dist}[x]\). Analogous to quadratic Dijkstra, with \(\mathcal{O}(M)\) states we get an algorithm that runs in time \(\mathcal{O}(M^2)\). diff --git a/2023/round3/krabotage.cpp b/2023/round3/krabotage.cpp new file mode 100644 index 0000000000000000000000000000000000000000..5acbda47f8357929abd913e616bde813bb6a2d8f --- /dev/null +++ b/2023/round3/krabotage.cpp @@ -0,0 +1,117 @@ +#include +#include +#include +using namespace std; + +using int64 = long long; + +int N, M; +vector> A; +vector> dpl, dpr; +vector>> dp; + +int64 rec(int dir, int x, int y) { + if (dp[dir][x][y] != -1LL) { + return dp[dir][x][y]; + } + int64 res = dpr[0][0]; + if (dir) { + int64 extra_exit = (y + 2 < M) ? dpr[x][y + 2] : 0; + int64 max_cost = extra_exit; + for (int ty = y; ty >= 0; --ty) { + int64 best_entrance = 0LL; + if (x) { + best_entrance = max(best_entrance, dpl[x - 1][ty]); + if (ty != y) { + best_entrance = max(best_entrance, dpl[x - 1][ty + 1]); + } + } + int64 best_exit = extra_exit; + if (x + 1 != N && ty + 1 != M) { + best_exit = max(best_exit, dpr[x + 1][ty + 1]); + } + max_cost = max(max_cost, best_entrance + best_exit); + if (ty) { + best_entrance = max(best_entrance, dpl[x][ty - 1]); + } + int64 local_max_cost = max(max_cost, best_entrance + best_exit); + if (x + 1 != N || !ty) { + int64 cur = (x + 1 == N) ? 0LL : rec(dir ^ 1, x + 1, ty); + cur = max(cur, local_max_cost); + res = min(res, cur); + } + } + } else { + int64 extra_entrance = (x - 2 >= 0) ? dpl[x - 2][y] : 0; + int64 max_cost = extra_entrance; + for (int tx = x; tx < N; ++tx) { + int64 best_exit = 0LL; + if (y + 1 != M) { + best_exit = max(best_exit, dpr[tx][y + 1]); + if (tx != x) { + best_exit = max(best_exit, dpr[tx - 1][y + 1]); + } + } + int64 best_entrance = extra_entrance; + if (y && tx) { + best_entrance = max(best_entrance, dpl[tx - 1][y - 1]); + } + max_cost = max(max_cost, best_entrance + best_exit); + if (tx + 1 != N) { + best_exit = max(best_exit, dpr[tx + 1][y]); + } + int64 local_max_cost = max(max_cost, best_entrance + best_exit); + if (y || tx + 1 == N) { + int64 cur = y ? rec(dir ^ 1, tx, y - 1) : 0LL; + cur = max(cur, local_max_cost); + res = min(res, cur); + } + } + } + return dp[dir][x][y] = res; +} + +int64 solve() { + cin >> N >> M; + A.assign(N, vector(M)); + for (int i = 0; i < N; i++) { + for (int j = 0; j < M; j++) { + cin >> A[i][j]; + } + } + dpl.assign(N, vector(M)); + dpr.assign(N, vector(M)); + for (int i = 0; i < N; i++) { + for (int j = 0; j < M; j++) { + dpl[i][j] = A[i][j]; + if (i) { + dpl[i][j] = max(dpl[i][j], A[i][j] + dpl[i - 1][j]); + } + if (j) { + dpl[i][j] = max(dpl[i][j], A[i][j] + dpl[i][j - 1]); + } + } + } + for (int i = N - 1; i >= 0; i--) { + for (int j = M - 1; j >= 0; j--) { + dpr[i][j] = A[i][j]; + if (i + 1 != N) { + dpr[i][j] = max(dpr[i][j], A[i][j] + dpr[i + 1][j]); + } + if (j + 1 != M) { + dpr[i][j] = max(dpr[i][j], A[i][j] + dpr[i][j + 1]); + } + } + } + dp.assign(2, vector>(N, vector(M, -1LL))); + return min(rec(0, 0, M - 1), rec(1, 0, M - 1)); +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/round3/krabotage.in b/2023/round3/krabotage.in new file mode 100644 index 0000000000000000000000000000000000000000..1aa0d465ebc3df4dc44482c7cc9ebe51a66a5405 --- /dev/null +++ b/2023/round3/krabotage.in @@ -0,0 +1,3044 @@ +118 +3 4 +2 5 3 6 +1 7 2 5 +4 5 1 3 +2 2 +2 2 +2 1 +3 3 +3 4 5 +5 3 1 +7 6 4 +3 3 +1 1 1 +3 3 3 +3 2 1 +3 3 +1 1 3 +3 3 1 +1 3 2 +2 2 +1 9 +9 0 +2 2 +0 9 +9 1 +5 5 +0 0 0 0 9 +0 0 0 9 7 +0 0 9 0 7 +0 9 0 0 7 +9 0 0 0 0 +10 10 +0 0 0 0 0 0 0 0 0 9 +0 0 0 0 0 0 0 0 2 0 +0 0 0 0 0 0 0 9 0 0 +0 0 0 0 0 0 1 0 0 0 +0 0 0 0 0 6 0 0 0 0 +0 0 0 0 6 0 0 0 0 0 +0 0 0 2 0 0 0 0 0 0 +0 0 6 0 0 0 0 0 0 0 +0 9 0 0 0 0 0 0 0 0 +8 0 0 0 0 0 0 0 0 0 +11 7 +8 0 2 7 6 1 3 +6 5 2 7 9 6 7 +7 9 4 7 9 4 8 +7 2 9 1 2 9 3 +4 6 4 2 4 4 6 +8 3 2 6 3 6 9 +5 0 4 8 8 6 5 +2 7 0 2 8 6 0 +7 4 3 5 5 0 7 +6 9 4 9 0 6 2 +2 7 2 9 9 6 2 +19 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674 114 524 998 939 844 +733 439 509 950 921 601 402 93 484 249 462 378 664 486 149 147 494 +433 223 748 864 614 971 453 155 106 74 36 186 21 525 373 777 318 +683 125 926 15 982 634 386 947 623 668 24 660 546 892 551 17 693 +41 119 705 389 181 305 655 248 753 510 681 456 441 940 812 633 17 +135 497 962 628 195 625 791 811 205 122 214 217 774 841 494 721 30 +316 249 997 509 497 883 75 846 524 590 385 291 184 793 223 374 821 +93 611 572 18 454 600 706 326 441 80 469 404 453 435 855 823 112 diff --git a/2023/round3/krabotage.md b/2023/round3/krabotage.md new file mode 100644 index 0000000000000000000000000000000000000000..735876ef0bfc3043d61ad0eeeaddc991b74c6ade --- /dev/null +++ b/2023/round3/krabotage.md @@ -0,0 +1,33 @@ +Despite his age, Mr. Krabs still goes trick-or-treating. Free candy (free anything, really) is irresistible to him. Bikini Bottom is an \(R \times C\) grid of houses, with house \((i, j)\) having \(A_{i,j}\) pieces of candy. + +Mr. Krabs plans to travel from house \((1, 1)\), the top-left corner, to house \((R, C)\), the bottom-right corner, taking as much candy as possible along the way. When Mr. Krabs is at house \((i, j)\), he takes all the candy, and then either moves down to house \((i + 1, j)\) or right to house \((i, j + 1)\). + +Plankton wants to sabotage him, and has decided to complete his own candy run before Mr. Krabs even starts his trip. Plankton will go from house \((1, C)\), the top-right corner, to house \((R, 1)\), the bottom-left corner. When Plankton is at house \((i, j)\), he takes all its candy (so there's none left for Mr. Krabs later) and either moves down to house \((i + 1, j)\) or left to house \((i, j - 1)\). + +What's the maximum amount of candy that Mr. Krabs can collect, assuming that Plankton strives to minimize this value by completing his trip before Mr. Krabs starts? + + +# Constraints + +\(1 \leq T \leq 100\) +\(2 \leq R, C \leq 300\) +\(0 \leq A_{i,j} \leq 1{,}000{,}000{,}000\) + +The sum of \(R * C\) across all test cases is at most \(500{,}000\). + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case begins with a line containing the two integers \(R\) and \(C\). Then, \(R\) lines follow, the \(i\)th of which contains the \(C\) integers \(A_{i,1}\) through \(A_{i,C}\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by a single integer, the maximum pieces of candy that Mr. Krabs can get if Plankton optimally sabotages. + + +# Sample Explanation + +The following depicts sample cases 1 through 5. Plankton's paths to minimize Mr. Krabs's candy are greyed out, while Mr. Krabs's optimal paths through the remaining candy (in spite of Plankton's sabotage) are drawn in red. + +{{PHOTO_ID:216354288143884|WIDTH:650}} diff --git a/2023/round3/krabotage.out b/2023/round3/krabotage.out new file mode 100644 index 0000000000000000000000000000000000000000..66a0308847f1b7869f610c805f7191f9bbf7bf4c --- /dev/null +++ b/2023/round3/krabotage.out @@ -0,0 +1,118 @@ +Case #1: 12 +Case #2: 1 +Case #3: 11 +Case #4: 5 +Case #5: 6 +Case #6: 0 +Case #7: 0 +Case #8: 9 +Case #9: 0 +Case #10: 87 +Case #11: 206 +Case #12: 179 +Case #13: 214 +Case #14: 159 +Case #15: 209 +Case #16: 141 +Case #17: 440232 +Case #18: 25366908261 +Case #19: 444607 +Case #20: 417249290158 +Case #21: 313166646562 +Case #22: 295333780498 +Case #23: 354329304294 +Case #24: 24851924182 +Case #25: 24025705279 +Case #26: 25117794775 +Case #27: 26020054053 +Case #28: 25472637052 +Case #29: 26520631012 +Case #30: 25225959731 +Case #31: 24666879819 +Case #32: 24312496221 +Case #33: 24867845273 +Case #34: 25489174917 +Case #35: 25580238667 +Case #36: 26498080148 +Case #37: 25817164330 +Case #38: 23954460111 +Case #39: 5427 +Case #40: 8079 +Case #41: 14932 +Case #42: 18290 +Case #43: 16392 +Case #44: 13505 +Case #45: 22053 +Case #46: 14391 +Case #47: 19661 +Case #48: 18098 +Case #49: 13747 +Case #50: 19186 +Case #51: 19375 +Case #52: 7739 +Case #53: 12270 +Case #54: 11031 +Case #55: 9912 +Case #56: 11486 +Case #57: 13253 +Case #58: 21162 +Case #59: 12168 +Case #60: 5390 +Case #61: 15716 +Case #62: 3770 +Case #63: 12272 +Case #64: 14164 +Case #65: 4885 +Case #66: 12192 +Case #67: 24479 +Case #68: 15730 +Case #69: 16155 +Case #70: 13967 +Case #71: 19828 +Case #72: 14494 +Case #73: 9371 +Case #74: 27278 +Case #75: 8831 +Case #76: 23260 +Case #77: 9895 +Case #78: 10460 +Case #79: 12116 +Case #80: 10706 +Case #81: 17251 +Case #82: 16376 +Case #83: 10205 +Case #84: 17270 +Case #85: 21699 +Case #86: 12617 +Case #87: 19710 +Case #88: 13490 +Case #89: 9594 +Case #90: 9407 +Case #91: 15393 +Case #92: 25730 +Case #93: 23225 +Case #94: 14376 +Case #95: 4357 +Case #96: 20558 +Case #97: 13625 +Case #98: 18466 +Case #99: 25232 +Case #100: 22035 +Case #101: 12370 +Case #102: 16765 +Case #103: 5222 +Case #104: 19733 +Case #105: 25853 +Case #106: 6856 +Case #107: 14272 +Case #108: 17719 +Case #109: 17962 +Case #110: 17166 +Case #111: 15669 +Case #112: 20264 +Case #113: 11044 +Case #114: 15291 +Case #115: 19593 +Case #116: 13407 +Case #117: 19510 +Case #118: 15559 diff --git a/2023/round3/krabotage_sol.md b/2023/round3/krabotage_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..740ce630a3eba9654ff0b04484d1bcbff0708d2b --- /dev/null +++ b/2023/round3/krabotage_sol.md @@ -0,0 +1,17 @@ +The key observation required is that the common part of Mr. Krabs and Plankton's paths will always be a single sub-row or sub-column of the input grid. We can use this observation to construct a dynamic programming approach. + +Let's denote \(dp(r, c, d)\) as the minimum possible number of candies Mr. Krabs can collect if: +- Plankton started his route from house \((r, c)\), +- Plankton's route's next segment's direction is \(d\) (\(0\) for vertical, \(1\) for horizontal), +- Mr. Krabs's path intersects Plankton's path, and +- Mr. Krabs strives to maximize the number of candies he gets. + +We see that the answer to the problem is equal to \(\min(dp(1, C, 0), dp(1, C, 1))\). To compute each \(dp(r, c, d)\), we will need to consider all valid lengths of the next path's segment. For each length, consider two scenarios and pick the maximum of those: +1. Mr. Krabs's path intersects the current segment. +2. Mr. Krabs's path will intersect one of the next segments. + +For case 1, we can precompute helper tables \(\text{max\_up\_left}(r, c)\) and \(\text{max\_down\_right}(r, c)\) for the maximum number of pieces of candy that can be collected on the untouched grid. These computations are a classic dynamic programming problem, which can be done in \(O(R* C)\) time. The implementation requires considering all possible ways of crossing the current segment and finding the most profitable way for Mr. Krabs. + +For case 2, we should simply check the value of \(dp(r', c', 1 - d)\), where \((r', c')\) is the first cell of the next segment. + +In total, there are \(\mathcal{O}(R*C)\) DP states. Since we need to consider all possible valid segment lengths for each state, the overall complexity of the solution is \(\mathcal{O}(R*C*(R + C))\). diff --git a/2023/round3/similar_ships.cpp b/2023/round3/similar_ships.cpp new file mode 100644 index 0000000000000000000000000000000000000000..75fae1f5d2275175019a50af1565ac32907f8414 --- /dev/null +++ b/2023/round3/similar_ships.cpp @@ -0,0 +1,54 @@ +#include +#include +#include +using namespace std; + +const long long MOD = 1000000007; + +int N; +vector> adj; + +void dfs(int u, vector& visit, int& farthest, int& maxd, int d) { + visit[u] = true; + if (d > maxd) { + farthest = u; + maxd = d; + } + for (int v : adj[u]) { + if (!visit[v]) { + dfs(v, visit, farthest, maxd, d + 1); + } + } +} + +int solve() { + cin >> N; + adj.assign(N, {}); + for (int i = 1, p; i < N; i++) { + cin >> p; + p--; + adj[i].push_back(p); + adj[p].push_back(i); + } + vector visit(N, false); + int farthest = 0, diameter = 0; + dfs(0, visit, farthest, diameter, 0); + visit.assign(N, false); + dfs(farthest, visit, farthest, diameter, 0); + int ans = 0; + for (int i = 0; i <= diameter; i++) { + ans = (ans + (N - i)) % MOD; + } + return ans; +} + +int main() { + ios_base::sync_with_stdio(0); + cin.tie(0); + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ": " << solve() << endl; + } + return 0; +} diff --git a/2023/round3/similar_ships.in b/2023/round3/similar_ships.in new file mode 100644 index 0000000000000000000000000000000000000000..46fdefcaa65e5cca16a6eac7b1234d3cbd03bcf2 --- /dev/null +++ b/2023/round3/similar_ships.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:36e4f6f52ac446b6cedb64ccea07a3761759b3b2c58e88d24bb14d6f0cce0d21 +size 56983631 diff --git a/2023/round3/similar_ships.md b/2023/round3/similar_ships.md new file mode 100644 index 0000000000000000000000000000000000000000..c36fa78baf99b3def7b5843f39d8c71fbbec5546 --- /dev/null +++ b/2023/round3/similar_ships.md @@ -0,0 +1,37 @@ +The Flying Dutchman is making a killing giving out haunted house tours of his ghost ship. The ship has \(N\) cabins, numbered from \(1\) to \(N\), connected by corridors in the shape of an unrooted tree \(t_1\). For each \(i=2..N\), there is a corridor that connects cabin \(i\) to cabin \(P_{i}\). + +Mr. Krabs noticed this business opportunity, and wants to make his own ship, also with \(N\) cabins, connected as a yet-to-be-determined tree \(t_2\). To avoid any lawsuits from the Dutchman, he would like to make his ship as different as possible. + +In particular, let \(S(t)\) be the multiset of all subtrees of tree \(t\), where a subtree is any non-empty subset of cabins that are connected. For two multisets of trees \(S_1\) and \(S_2\), let \(\text{similarity}(S_1, S_2)\) be the number of trees in \(S_2\) that are [isomorphic](https://en.wikipedia.org/wiki/Graph_isomorphism) to at least one tree in \(S_1\). Note that \(\text{similarity}(S_1, S_2)\) and \(\text{similarity}(S_2, S_1)\) are not necessarily equal. + +Help Mr. Krabs find the minimum possible value of \(\text{similarity}(S(t_1), S(t_2))\) over all trees \(t_2\) of size \(N\). As this value may be large, output it modulo \(1{,}000{,}000{,}007\). + + +# Constraints + +\(1 \leq T \leq 35\) +\(2 \leq N \leq 1{,}000{,}000\) +\(1 \leq P_i \leq N\) + +The sum of \(N\) across all test cases is at most \(10{,}000{,}000\). + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case has two lines. The first contains the integer \(N\), and the second contains the \(N-1\) integers \(P_2\) through \(P_N\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by the minimum possible similarity if \(t_2\) is chosen optimally, modulo \(1{,}000{,}000{,}007\). + + +# Sample Explanation + +The first sample case is depicted below. One possible tree that minimizes the similarity score is the same tree given in the input (i.e. choose \(t_2 = t_1\)). Then, \(S(t_2)\) consists of the \(6\) subtrees: \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1, 2\}\), \(\{2, 3\}\), and \(\{1, 2, 3\}\). Each subtree is isomorphic to a subtree of \(t_1\), so the similarity score is \(6\). + +{{PHOTO_ID:720027639598818|WIDTH:280}} + +In the second case, one possible \(t_2\) that minimizes the similarity score is depicted below. This tree has \(24\) different subtrees, of which \(21\) are isomorphic to a subtree of the given \(t_1\). For example, the subtree \(\{2, 3, 4, 6\}\) of this \(t_2\) is isomorphic to subtree \(\{2, 3, 4, 5\}\) of the given \(t_1\), but the subtree \(\{1, 2, 3, 6\}\) of this \(t_2\) is not isomorphic to any subtree of \(t_1\). + +{{PHOTO_ID:854841726123175|WIDTH:350}} diff --git a/2023/round3/similar_ships.out b/2023/round3/similar_ships.out new file mode 100644 index 0000000000000000000000000000000000000000..9b616f7f835da6da2c7a6d64f0eac0eb5768ea4c --- /dev/null +++ b/2023/round3/similar_ships.out @@ -0,0 +1,40 @@ +Case #1: 6 +Case #2: 21 +Case #3: 52 +Case #4: 45 +Case #5: 27 +Case #6: 182 +Case #7: 132 +Case #8: 57 +Case #9: 376249 +Case #10: 18829 +Case #11: 2997 +Case #12: 349892 +Case #13: 6999405 +Case #14: 599997 +Case #15: 34174 +Case #16: 496500 +Case #17: 967824784 +Case #18: 747375 +Case #19: 38999259 +Case #20: 2999997 +Case #21: 58998289 +Case #22: 59998230 +Case #23: 62997984 +Case #24: 27998460 +Case #25: 27 +Case #26: 25 +Case #27: 18 +Case #28: 27 +Case #29: 135 +Case #30: 124 +Case #31: 54 +Case #32: 145 +Case #33: 57155807 +Case #34: 332964 +Case #35: 37032 +Case #36: 468407 +Case #37: 658144464 +Case #38: 3258717 +Case #39: 296292 +Case #40: 4937025 diff --git a/2023/round3/similar_ships_sol.md b/2023/round3/similar_ships_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..5677a596ea81b8b1600826aa70b1f80ee4aead10 --- /dev/null +++ b/2023/round3/similar_ships_sol.md @@ -0,0 +1,10 @@ +Choosing a tree \(t_2\) that's a line (that is, \(N\) nodes connected as a chain\) always minimizes \(\text{similarity}(S(t_1), S(t_2))\). + +Proof: Let \(t_\text{line}\) be a line of size-\(N\), and \(t_x\) be any tree of size \(N\). We'll show for any \(t_1\) and any \(t_x\), \(\text{similarity}(S(t_1), S(t_\text{line})) \le \text{similarity}(S(t_1), S(t_x))\)\). Let’s also not consider subtrees of size \(1\) (because all trees of size \(n\) have \(n\) subtrees of size \(1\)). There are two cases: + +*Case 1*: Consider \(t_x\) such that \(\text{diameter}(t_x) \le \text{diameter}(t_1)\). We can choose any two nodes from \(t_x\) and get the path between them. This will be a subtree of \(t_1\). However, \(\text{similarity}(S(t_1), S(t_\text{line}))\) is at most \(\binom{n}{2}\), because \(S(t_\text{line})\) itself has only \(\binom{n}{2}\) elements. + +*Case 2*: Consider \(t_x\) such that \(\text{diameter}(t_x) > \text{diameter}(t_1)\). For any pair of nodes \((u, v) \in t_x\), either the path from \(u\) to \(v\) is a subtree of \(t_1\), or there is a corresponding pair in \(t_\text{line}\) that is not a subtree of \(t_1\). + +Therefore, \(\text{similarity}(S(t_1), S(t_\text{line})) \le \text{similarity}(S(t_1), S(t_x))\). To compute the former, simply calculate the diameter of \(t_1\) using two DFS's and use combinatorics. In particular, each length \(i\) line (of \(i + 1\) nodes) appears in \(S(t_\text{line})\) exactly \(N - i\) times, and all of these sub-lines are isomorphic to a subtree of \(t_1\). The answer will be: +\[ \sum_{i=0}^{\text{diameter}(t_\text{line})} N - i \] diff --git a/2023/round3/spooky_splits.cpp b/2023/round3/spooky_splits.cpp new file mode 100644 index 0000000000000000000000000000000000000000..82deb66005184314f23002ef7049d5c16ded9a6c --- /dev/null +++ b/2023/round3/spooky_splits.cpp @@ -0,0 +1,98 @@ +#include +#include +#include +using namespace std; + +int N, M; +vector> adj; +vector visit; +vector sizes; + +void rec(int u) { + sizes.back()++; + visit[u] = true; + for (int v : adj[u]) { + if (!visit[v]) { + rec(v); + } + } +} + +int tgt; +vector groups; +multiset state; +set> seen; + +bool can_partition(int i) { + if (i == (int)sizes.size()) { + for (int sz : groups) { + if (sz != tgt) { + return false; + } + } + return true; + } + if (seen.count(state)) { + return false; + } + seen.insert(state); + for (int &g : groups) { + if (g + sizes[i] <= tgt) { + state.erase(state.find(g)); + g += sizes[i]; + state.insert(g); + if (can_partition(i + 1)) { + return true; + } + state.erase(state.find(g)); + g -= sizes[i]; + state.insert(g); + } + } + return false; +} + +void solve() { + cin >> N >> M; + adj.assign(N, {}); + visit.assign(N, false); + sizes.clear(); + for (int i = 0, a, b; i < M; i++) { + cin >> a >> b; + a--; + b--; + adj[a].push_back(b); + adj[b].push_back(a); + } + for (int i = 0; i < N; i++) { + if (!visit[i]) { + sizes.push_back(0); + rec(i); + } + } + for (int K = 1; K <= N; K++) { + if (N % K == 0) { + tgt = N / K; + groups.assign(K, 0); + state.clear(); + for (int i = 0; i < K; i++) { + state.insert(0); + } + seen.clear(); + if (can_partition(0)) { + cout << " " << K; + } + } + } +} + +int main() { + int T; + cin >> T; + for (int t = 1; t <= T; t++) { + cout << "Case #" << t << ":"; + solve(); + cout << endl; + } + return 0; +} \ No newline at end of file diff --git a/2023/round3/spooky_splits.in b/2023/round3/spooky_splits.in new file mode 100644 index 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Her \(N\) students, numbered from \(1\) to \(N\), are dressed up in spooky outfits and ready to collect some candy. To divide and conquer every house in Bikini Bottom, she would like to break up the class into equal-sized groups. + +\(M\) pairs of students have similar-themed costumes that amplify their spookiness, the \(i\)th being students \(A_i\) and \(B_i\). Ms. Puff's would like to know: for which values of \(K\) can the class be divided into \(K\) groups of equal size without splitting up any of these pairs? + +# Constraints + +\(1 \leq T \leq 75\) +\(2 \leq N \leq 100\) +\(0 \leq M \leq \frac{N(N-1)}{2}\) +\(1 \leq A_i, B_i \leq N\) +\(A_i \neq B_i\) + +Each unordered pair of employees occurs at most once in a given test case. + + +# Input Format + +Input begins with an integer \(T\), the number of test cases. Each case begins with a line containing the two integers \(N\) and \(M\). Then, \(M\) lines follow, the \(i\)th of which contains the two integers \(A_i\) and \(B_i\). + + +# Output Format + +For the \(i\)th test case, print "`Case #i:` " followed by all possible values of \(K\) in ascending order, separated by spaces. + + +# Sample Explanation + +In the first sample case, there are three possible values of \(K\). Here's a possible arrangement: +- \(K = 1\): All students can be put into a single group. +- \(K = 2\): Groups \(\{1, 2, 4, 5, 6, 7\}\) and \(\{3, 8, 9, 10, 11, 12\}\). +- \(K = 3\): Groups \(\{1, 2, 4, 6\}\), \(\{3, 5, 8, 12\}\) and \(\{7, 9, 10, 11\}\). + +In the second case, either all students can be in a single group, or each should form a separate group. + +In the third test case, the valid values of \(K\), with an example of each, are: +- \(K = 1\): All students can be put into a single group. +- \(K = 2\): Groups \(\{1, 2, 5, 7\}\) and \(\{3, 4, 6, 8\}\). +- \(K = 4\): Groups \(\{1, 7\}\), \(\{2, 5\}\), \(\{3, 6\}\) and \(\{4, 8\}\). diff --git a/2023/round3/spooky_splits.out b/2023/round3/spooky_splits.out new file mode 100644 index 0000000000000000000000000000000000000000..289afc9b2930455a4cbae1c2b64fbb0d91a3ca5f --- /dev/null +++ b/2023/round3/spooky_splits.out @@ -0,0 +1,80 @@ +Case #1: 1 2 4 5 +Case #2: 1 2 4 5 10 +Case #3: 1 2 4 5 10 20 25 +Case #4: 1 3 9 11 33 +Case #5: 1 3 +Case #6: 1 2 4 5 10 +Case #7: 1 2 4 5 10 +Case #8: 1 2 4 5 +Case #9: 1 2 4 5 +Case #10: 1 2 4 5 +Case #11: 1 2 4 5 +Case #12: 1 3 9 11 +Case #13: 1 3 9 +Case #14: 1 3 +Case #15: 1 3 +Case #16: 1 2 7 +Case #17: 1 3 +Case #18: 1 2 4 5 +Case #19: 1 2 4 5 10 +Case #20: 1 3 9 +Case #21: 1 2 4 +Case #22: 1 2 3 4 6 8 12 +Case #23: 1 2 3 4 6 8 +Case #24: 1 5 +Case #25: 1 2 4 +Case #26: 1 47 +Case #27: 1 2 3 4 6 7 +Case #28: 1 2 3 4 6 7 +Case #29: 1 2 3 4 6 8 +Case #30: 1 2 3 5 6 +Case #31: 1 +Case #32: 1 2 4 5 10 20 25 50 100 +Case #33: 1 2 +Case #34: 1 +Case #35: 1 97 +Case #36: 1 2 4 5 10 +Case #37: 1 2 4 5 +Case #38: 1 +Case #39: 1 3 +Case #40: 1 3 9 11 +Case #41: 1 2 4 5 10 20 25 +Case #42: 1 2 4 5 10 25 +Case #43: 1 2 4 5 10 20 +Case #44: 1 2 +Case #45: 1 2 3 4 6 +Case #46: 1 2 3 4 6 8 +Case #47: 1 2 3 4 6 8 12 +Case #48: 1 2 4 5 10 20 +Case #49: 1 2 4 5 +Case #50: 1 5 19 +Case #51: 1 2 5 +Case #52: 1 5 +Case #53: 1 2 5 +Case #54: 1 2 +Case #55: 1 3 11 +Case #56: 1 3 +Case #57: 1 3 +Case #58: 1 2 3 4 6 8 12 +Case #59: 1 2 4 +Case #60: 1 2 3 4 6 8 12 +Case #61: 1 +Case #62: 1 2 3 6 +Case #63: 1 2 4 11 +Case #64: 1 +Case #65: 1 2 3 +Case #66: 1 5 +Case #67: 1 2 5 10 25 +Case #68: 1 2 5 +Case #69: 1 2 4 5 10 +Case #70: 1 2 5 10 +Case #71: 1 2 3 4 6 8 9 12 +Case #72: 1 2 3 4 6 8 9 +Case #73: 1 2 4 5 +Case #74: 1 2 4 5 10 +Case #75: 1 2 4 5 10 +Case #76: 1 2 4 5 10 +Case #77: 1 3 9 11 +Case #78: 1 +Case #79: 1 3 9 11 +Case #80: 1 5 diff --git a/2023/round3/spooky_splits_sol.md b/2023/round3/spooky_splits_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..5fd2d405a345533770ffaa715c24b4df68baf8aa --- /dev/null +++ b/2023/round3/spooky_splits_sol.md @@ -0,0 +1,10 @@ +If we draw an edge for each \((A_i, B_i)\), we'll get a graph of the \(N\) students. It's clear that to avoid splitting up any of the \(M\) pairs, we have to avoid splitting up any connected component in the graph. From here, we see that what only the component sizes matter, which can be found with a simple breadth- or depth-first search. + +Fixing \(K\), the problem reduces to checking whether the array of component sizes can be partitioned into \(K\) subsets of equal sum. First, if \(N\) is not a multiple of \(K\), then we cannot partition \(N\) into equal-sized groups of size \(N / K\). Otherwise, we'll need to find a valid partition. + +Consider a backtracking algorithm that attempts to recursively assign each component to one of the \(K\) groups. We will maintain a size \(K\) array of current group sizes, and only assign to a group if the new size doesn't exceed the target size of \(N / K\). Clearly, this naive method is exponential, and will not be fast enough as we have up to \(100\) components/groups. + + +One optimization is to observe that for a current state (array of partially-assigned group sizes), e.g. \([1, 1, 1, 2, 0, 3]\), adding to any one of identical groups is the same, so we only have to recurse for one of them. Instead of an array, we can think of the state as a *multiset* of numbers with sum not exceeding \(N\). We can maintain a set of multisets to track states we've already seen and avoid recomputing them. Each group can take on sizes \(0...\frac{N}{K}\) (i.e. \(\frac{N}{K} - 1\) different values). The number of states is the number of ways to choose \(\frac{N}{K} + 1\) elements to fill \(K\) slots with repetition allowed. This is \((\frac{N}{K} + 1)\) [multichoose](https://mathworld.wolfram.com/Multichoose.html) \(K\), equal to \(\bigl( \binom{N/K + 1}{K} \bigr) = \binom{N/K + K}{K} \). At worst, this comes out to \(\approx185{,}000\) states when \(\frac{N}{K}=K=10\). + +The overall time complexity will be bounded by the number of \(K\)s to check (a.k.a the number of divisors of \(N\), here at most \(12\) for \(N = 60 \le 100\)) multiplied by the number of states, multiplied by the time to process each state. In practice, the jury's solution solves the full dataset in under a few seconds on a modest system.