#include #include #include #include #include #include using namespace std; using int64 = long long; const int MAX_COORD = 300000; struct Rect { int x1, y1, x2, y2, id; int mappedX1, mappedY1, mappedX2, mappedY2; bool ends_alive; Rect() {} Rect(int _x1, int _y1, int _x2, int _y2, int _id) : x1(_x1), y1(_y1), x2(_x2), y2(_y2), id(_id) {} bool intersects(const Rect &o) const { return x2 >= o.x1 && x1 <= o.x2 && y1 <= o.y2 && y2 >= o.y1; } void merge_in(const Rect &o) { x1 = min(x1, o.x1); mappedX1 = min(mappedX1, o.mappedX1); y1 = min(y1, o.y1); mappedY1 = min(mappedY1, o.mappedY1); x2 = max(x2, o.x2); mappedX2 = max(mappedX2, o.mappedX2); y2 = max(y2, o.y2); mappedY2 = max(mappedY2, o.mappedY2); } bool operator <(const Rect &o) { return id < o.id; } }; class InnerST { int lo, hi, delta; unique_ptr lchild, rchild; void init_children() { if (!lchild) { int mid = lo + (hi - lo) / 2; lchild = make_unique(lo, mid); rchild = make_unique(mid + 1, hi); } } public: InnerST(int l, int h) : lo(l), hi(h), delta(0), lchild(nullptr), rchild(nullptr) {} void add(int l, int h, int d) { if (l > hi || h < lo) { return; } if (l <= lo && h >= hi) { delta += d; return; } init_children(); lchild->add(l, h, d); rchild->add(l, h, d); } int query(int at) { if (lo == hi || !lchild) { return delta; } init_children(); return delta + (at <= lchild->hi ? lchild : rchild)->query(at); } }; class OuterST { int lo, hi, inner_lo, inner_hi; unique_ptr lchild, rchild; InnerST inner; public: OuterST(int l, int h, int inner_l, int inner_h) : lo(l), hi(h), inner_lo(inner_l), inner_hi(inner_h), inner(inner_l, inner_h) { if (lo != hi) { int mid = lo + (hi - lo) / 2; lchild = make_unique(lo, mid, inner_lo, inner_hi); rchild = make_unique(mid + 1, hi, inner_lo, inner_hi); } } // Adds a line to the inner dimension spanning inner_lo to inner_hi on // outer_idx void add_inner(int64 outer_idx, int inner_lo, int inner_hi) { inner.add(inner_lo, inner_hi, 1); if (lo != hi) { if (outer_idx <= lchild->hi) { lchild->add_inner(outer_idx, inner_lo, inner_hi); } else { rchild->add_inner(outer_idx, inner_lo, inner_hi); } } } // Queries all y's that have something here. vector query_inner(int outer_lo, int outer_hi, int inner_idx) { if (outer_lo > hi || outer_hi < lo || inner.query(inner_idx) == 0) return {}; if (lo == hi) { return {lo}; } auto lres = lchild->query_inner(outer_lo, outer_hi, inner_idx); auto rres = rchild->query_inner(outer_lo, outer_hi, inner_idx); vector sum(lres.size() + rres.size()); for (int i = 0; i < (int)sum.size(); i++) { sum[i] = i < (int)lres.size() ? lres[i] : rres[i - lres.size()]; } return sum; } void remove_inner(int outer_idx, int inner_lo, int inner_hi) { inner.add(inner_lo, inner_hi, -1); if (lo != hi) { if (outer_idx <= lchild->hi) { lchild->remove_inner(outer_idx, inner_lo, inner_hi); } else { rchild->remove_inner(outer_idx, inner_lo, inner_hi); } } } }; class RectHolder { struct RectMapping { int value; Rect *r; bool v1; bool operator<(const RectMapping &r) const { return value < r.value; } }; int sz; vector mapsXMajor, mapsYMajor; unordered_set aliveRects; OuterST xMajorST, yMajorST; vector usingXMapping, usingYMapping; vector firstOfXMajor, lastOfXMajor; vector firstOfYMajor, lastOfYMajor; public: RectHolder(vector &rects) : sz(rects.size()), xMajorST(0, 2 * sz, 0, 2 * sz), yMajorST(0, 2 * sz, 0, 2 * sz), usingXMapping(2 * sz), usingYMapping(2 * sz), firstOfXMajor(2 * sz, (int)1e9), lastOfXMajor(2 * sz, -1), firstOfYMajor(2 * sz, (int)1e9), lastOfYMajor(2 * sz, -1) { vector xMappings(2 * sz), yMappings(2 * sz); for (int i = 0; i < sz; i++) { xMappings[i * 2] = (RectMapping){rects[i].x1, &rects[i], true}; xMappings[i * 2 + 1] = (RectMapping){rects[i].x2, &rects[i], false}; yMappings[i * 2] = (RectMapping){rects[i].y1, &rects[i], true}; yMappings[i * 2 + 1] = (RectMapping){rects[i].y2, &rects[i], false}; } sort(xMappings.begin(), xMappings.end()); sort(yMappings.begin(), yMappings.end()); for (int i = 0; i < (int)xMappings.size(); i++) { usingXMapping[i] = xMappings[i].r; usingYMapping[i] = yMappings[i].r; if (xMappings[i].v1) { xMappings[i].r->mappedX1 = i; int orig = xMappings[i].r->x1; firstOfXMajor[orig] = min(firstOfXMajor[orig], i); lastOfXMajor[orig] = max(lastOfXMajor[orig], i); } else { xMappings[i].r->mappedX2 = i; int orig = xMappings[i].r->x2; firstOfXMajor[orig] = min(firstOfXMajor[orig], i); lastOfXMajor[orig] = max(lastOfXMajor[orig], i); } if (yMappings[i].v1) { yMappings[i].r->mappedY1 = i; int orig = yMappings[i].r->y1; firstOfYMajor[orig] = min(firstOfYMajor[orig], i); lastOfYMajor[orig] = max(lastOfYMajor[orig], i); } else { yMappings[i].r->mappedY2 = i; int orig = yMappings[i].r->y2; firstOfYMajor[orig] = min(firstOfYMajor[orig], i); lastOfYMajor[orig] = max(lastOfYMajor[orig], i); } } } void add_rect(Rect *r) { aliveRects.insert(r); usingXMapping[r->mappedX1] = r; usingXMapping[r->mappedX2] = r; usingYMapping[r->mappedY1] = r; usingYMapping[r->mappedY2] = r; xMajorST.add_inner(r->mappedX1, r->y1, r->y2); yMajorST.add_inner(r->mappedY1, r->x1, r->x2); xMajorST.add_inner(r->mappedX2, r->y1, r->y2); yMajorST.add_inner(r->mappedY2, r->x1, r->x2); } void remove_rect(Rect *r) { aliveRects.erase(r); xMajorST.remove_inner(r->mappedX1, r->y1, r->y2); yMajorST.remove_inner(r->mappedY1, r->x1, r->x2); xMajorST.remove_inner(r->mappedX2, r->y1, r->y2); yMajorST.remove_inner(r->mappedY2, r->x1, r->x2); } unordered_set get_alive_rect_intersecting(Rect *r) { unordered_set res; auto xsToTake1 = xMajorST.query_inner(firstOfXMajor[r->x1], lastOfXMajor[r->x2], r->y1); auto xsToTake2 = xMajorST.query_inner(firstOfXMajor[r->x1], lastOfXMajor[r->x2], r->y2); auto ysToTake1 = yMajorST.query_inner(firstOfYMajor[r->y1], lastOfYMajor[r->y2], r->x1); auto ysToTake2 = yMajorST.query_inner(firstOfYMajor[r->y1], lastOfYMajor[r->y2], r->x2); for (int x : xsToTake1) res.insert(usingXMapping[x]); for (int x : xsToTake2) res.insert(usingXMapping[x]); for (int y : ysToTake1) res.insert(usingYMapping[y]); for (int y : ysToTake2) res.insert(usingYMapping[y]); return res; } unordered_set get_all_alive_rects() { return aliveRects; } }; struct Event { int x, y1, y2; bool adding; Rect *r; bool operator<(const Event &e) const { if (x != e.x) { return x < e.x; } return (y2 - y1) < (e.y2 - e.y1); } }; int64 solve() { int N; vector rects; vector xvals, yvals; unordered_map x_map, y_map, x_rev, y_rev; // Input. cin >> N; for (int i = 0, x, y; i < N; i++) { cin >> x >> y; rects.push_back(Rect(x - 1, y - 1, x + 1, y + 1, i)); xvals.push_back(x - 1); xvals.push_back(x + 1); yvals.push_back(y - 1); yvals.push_back(y + 1); } // Compress rectangles. sort(xvals.begin(), xvals.end()); xvals.resize(distance(xvals.begin(), unique(xvals.begin(), xvals.end()))); sort(yvals.begin(), yvals.end()); yvals.resize(distance(yvals.begin(), unique(yvals.begin(), yvals.end()))); int i = 0; for (int x : xvals) { x_rev[i] = x; x_map[x] = i++; } i = 0; for (int y : yvals) { y_rev[i] = y; y_map[y] = i++; } for (Rect &r : rects) { r.x1 = x_map[r.x1]; r.y1 = y_map[r.y1]; r.x2 = x_map[r.x2]; r.y2 = y_map[r.y2]; } // Add to rect holder. RectHolder holder(rects); for (Rect &to_add : rects) { while (true) { auto kill_now = holder.get_alive_rect_intersecting(&to_add); if (kill_now.empty()) { break; } for (Rect *to_kill : kill_now) { holder.remove_rect(to_kill); to_add.merge_in(*to_kill); } } holder.add_rect(&to_add); } // Line sweep to remove any rectangles that are inside another rectangle sweep // left to right. On an X, process bigger then smaller. auto alive = holder.get_all_alive_rects(); InnerST st(0, MAX_COORD); vector events; for (Rect *r : alive) { r->ends_alive = true; events.push_back((Event){r->x1, r->y1, r->y2, true, r}); events.push_back((Event){r->x2, r->y1, r->y2, false, r}); } sort(events.begin(), events.end()); for (Event &e : events) { if (e.adding) { if (st.query(e.y1) == 0) { st.add(e.y1, e.y2, 1); } else { e.r->ends_alive = false; } } else { if (e.r->ends_alive) { st.add(e.y1, e.y2, -1); } } } // Uncompress and calculate areas. int64 ans = 0; for (const Rect &r : rects) { if (r.ends_alive) { int x1 = x_rev[r.x1], x2 = x_rev[r.x2]; int y1 = y_rev[r.y1], y2 = y_rev[r.y2]; ans += (int64)(x2 - x1) * (y2 - y1); } } 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; }