| You've got yourself an unrooted tree with **N** nodes — that is, a connected, | |
| undirected graph with **N** nodes numbered from 1 to **N**, and **N** \- 1 | |
| edges. The **i**th edge connects nodes **Ai** and **Bi**. | |
| You'd like to spend as little money as possible to label each node with a | |
| number from 1 to **K**, inclusive. It costs **Ci,j** dollars to label the | |
| **i**th node with the number **j**. | |
| Additionally, after the whole tree has been labelled, you must pay **P** 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 **u**, you must pay | |
| **P** dollars if there exist two other nodes **v** and **w** which are both | |
| adjacent to node **u**, such that the labels on nodes **v** and **w** are | |
| equal (note that node **u**'s label is irrelevant). You only pay the penalty | |
| of **P** dollars once for a given central node **u**, even if it has multiple | |
| pairs of neighbours which satisfy the above condition. | |
| What's the minimum cost (in dollars) to label all **N** nodes? | |
| ### Input | |
| Input begins with an integer **T**, the number of trees. For each tree, there | |
| is first a line containing the space-separated integers **N**, **K**, and | |
| **P**. Then, **N** lines follow, the **i**th of which contains the space- | |
| separated integers **Ci,1** through **Ci,K** in order. Then, **N** \- 1 lines | |
| follow, the **i**th of which contains the space-separated integers **Ai** and | |
| **Bi** | |
| ### Output | |
| For the **i**th tree, print a line containing "Case #**i**: " followed by the | |
| minimum cost to label all of the tree's nodes. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 30 | |
| 1 ≤ **N** ≤ 1,000 | |
| 1 ≤ **K** ≤ 30 | |
| 0 ≤ **P** ≤ 1,000,000 | |
| 0 ≤ **Ci,j** ≤ 1,000,000 | |
| 1 ≤ **Ai**, **Bi** ≤ **N** | |
| ### Explanation of Sample | |
| 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. | |