2016/round2/costly_labels.html

<p>
You've got yourself an unrooted tree with <strong>N</strong> nodes &mdash; that is, a connected, undirected graph with 
<strong>N</strong> nodes numbered from 1 to <strong>N</strong>, and <strong>N</strong> - 1 edges. 
The <strong>i</strong>th edge connects nodes <strong>A<sub>i</sub></strong> and <strong>B<sub>i</sub></strong>.
</p>

<p>
You'd like to spend as little money as possible to label each node with a number from 1 to <strong>K</strong>, inclusive. 
It costs <strong>C<sub>i,j</sub></strong> dollars to label the <strong>i</strong>th node with the number <strong>j</strong>.
</p>

<p>
Additionally, after the whole tree has been labelled, you must pay <strong>P</strong> more dollars for each node which has at least one pair of 
neighbours that share the same label as each other. In other words, for each node <strong>u</strong>, you must pay <strong>P</strong> dollars 
if there exist two other nodes <strong>v</strong> and <strong>w</strong> which are both adjacent to node <strong>u</strong>, 
such that the labels on nodes <strong>v</strong> and <strong>w</strong> are equal (note that node <strong>u</strong>'s label is irrelevant). 
You only pay the penalty of <strong>P</strong> dollars once for a given central node <strong>u</strong>, 
even if it has multiple pairs of neighbours which satisfy the above condition.
</p>

<p>
What's the minimum cost (in dollars) to label all <strong>N</strong> nodes?
</p>


<h3>Input</h3>
<p>
Input begins with an integer <strong>T</strong>, the number of trees.
For each tree, there is first a line containing the space-separated integers <strong>N</strong>, <strong>K</strong>, and <strong>P</strong>.
Then, <strong>N</strong> lines follow, the <strong>i</strong>th of which contains the space-separated integers
<strong>C<sub>i,1</sub></strong> through <strong>C<sub>i,K</sub></strong> in order.
Then, <strong>N</strong> - 1 lines follow, the <strong>i</strong>th of which contains the space-separated integers
<strong>A<sub>i</sub></strong> and <strong>B<sub>i</sub></strong>
</p>


<h3>Output</h3>
<p>
For the <strong>i</strong>th tree, print a line containing "Case #<strong>i</strong>: " followed by the minimum cost to label all of the tree's nodes.
</p>


<h3>Constraints</h3>
<p>
1 &le; <strong>T</strong> &le; 30 <br />
1 &le; <strong>N</strong> &le; 1,000 <br />
1 &le; <strong>K</strong> &le; 30 <br />
0 &le; <strong>P</strong> &le; 1,000,000 <br />
0 &le; <strong>C<sub>i,j</sub></strong> &le; 1,000,000 <br />
1 &le; <strong>A<sub>i</sub></strong>, <strong>B<sub>i</sub></strong> &le; <strong>N</strong> <br />
</p>


<h3>Explanation of Sample</h3>
<p>
In the first case, there is only one node which must be painted the only possible color for 111 dollars. In the second case, there is only one color, so a penalty of 8 dollars must be paid since node 2 has two neighbors with the same color. In total we pay 1 + 2 + 4 + 8 = 15 dollars. In the third case, it's optimal to paint nodes 1 and 2 with color 1, and node 3 with color 2. The total cost is 4 + 8 + 3 = 15 dollars.
</p>