F. Earnest Matrix Complementtime limit per test5 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard output3, 2, 1, ... We are the — RiOI Team!— Felix & All, Special Thanks 3 Peter: Good news: My problem T311013 is approved! $$$\delta$$$: I'm glad my computer had gone out of battery so that I wouldn't have participated in wyrqwq's round and gained a negative delta. Felix: [thumbs_up] The problem statement concerning a removed song! Aquawave: Do I mourn my Chemistry? E.Space: ahh? Trine: Bread. Iris: So why am I always testing problems? Time will pass, and we might meet again. Looking back at the past, everybody has lived the life they wanted.Aquawave has a matrix $$$A$$$ of size $$$n\times m$$$, whose elements can only be integers in the range $$$[1, k]$$$, inclusive. In the matrix, some cells are already filled with an integer, while the rest are currently not filled, denoted by $$$-1$$$.You are going to fill in all the unfilled places in $$$A$$$. After that, let $$$c_{u,i}$$$ be the number of occurrences of element $$$u$$$ in the $$$i$$$-th row. Aquawave defines the beauty of the matrix as$$$$$$\sum_{u=1}^k \sum_{i=1}^{n-1} c_{u,i} \cdot c_{u,i+1}.$$$$$$You have to find the maximum possible beauty of $$$A$$$ after filling in the blanks optimally.InputThe first line of input contains a single integer $$$t$$$ ($$$1 \leq t \leq 2\cdot 10^4$$$) — the number of test cases. The description of test cases follows.The first line of each test case contains three integers $$$n$$$, $$$m$$$, and $$$k$$$ ($$$2 \leq n \leq 2\cdot 10^5$$$, $$$2 \leq m \leq 2\cdot 10^5$$$, $$$n \cdot m \leq 6\cdot 10^5$$$, $$$1 \leq k \leq n\cdot m$$$) — the number of rows and columns of the matrix $$$A$$$, and the range of the integers in the matrix, respectively.Then $$$n$$$ lines follow, the $$$i$$$-th line containing $$$m$$$ integers $$$A_{i,1},A_{i,2},\ldots,A_{i,m}$$$ ($$$1 \leq A_{i,j} \leq k$$$ or $$$A_{i,j} = -1$$$) — the elements in $$$A$$$.It is guaranteed that the sum of $$$n\cdot m$$$ over all test cases does not exceed $$$6\cdot 10^5$$$.OutputFor each test case, output a single integer — the maximum possible beauty.ExampleInput93 3 31 2 23 1 33 2 12 3 3-1 3 32 2 -13 3 6-1 -1 11 2 -1-1 -1 43 4 51 3 2 3-1 -1 2 -13 1 5 15 3 85 -1 21 8 -1-1 5 67 7 -14 4 46 6 5-1 -1 5 -1 -1 -1-1 -1 -1 -1 2 -1-1 1 3 3 -1 -1-1 1 -1 -1 -1 44 2 -1 -1 -1 4-1 -1 1 2 -1 -16 6 4-1 -1 -1 -1 1 -13 -1 2 2 4 -13 1 2 2 -1 -13 3 3 3 -1 2-1 3 3 -1 1 33 -1 2 2 3 -15 5 31 1 3 -1 12 2 -1 -1 3-1 -1 -1 2 -13 -1 -1 -1 2-1 1 2 3 -16 2 7-1 7-1 67 -1-1 -1-1 -12 2Output4
4
10
10
8
102
93
58
13
NoteIn the first test case, the matrix $$$A$$$ is already determined. Its beauty is$$$$$$\sum_{u=1}^k \sum_{i=1}^{n-1} c_{u,i} \cdot c_{u,i+1} = c_{1,1}\cdot c_{1,2} + c_{1,2}\cdot c_{1,3} + c_{2,1}\cdot c_{2,2} + c_{2,2}\cdot c_{2,3} + c_{3,1}\cdot c_{3,2} + c_{3,2}\cdot c_{3,3} = 1\cdot 1 + 1\cdot 1 + 2\cdot 0 + 0\cdot 1 + 0\cdot 2 + 2\cdot 1 = 4.$$$$$$In the second test case, one can fill the matrix as follows:$$$$$$ \begin{bmatrix} 2 &3 &3 \\ 2 &2 &3 \end{bmatrix}, $$$$$$and get the value $$$4$$$. It can be proven this is the maximum possible answer one can get.In the third test case, one of the possible optimal configurations is:$$$$$$ \begin{bmatrix} 1 &1 &1 \\ 1 &2 &1 \\ 1 &1 &4 \end{bmatrix}. $$$$$$In the fourth test case, one of the possible optimal configurations is:$$$$$$ \begin{bmatrix} 1 &3 &2 &3 \\ 1 &3 &2 &1 \\ 3 &1 &5 &1 \end{bmatrix}. $$$$$$In the fifth test case, one of the possible optimal configurations is:$$$$$$ \begin{bmatrix} 5 &5 &2 \\ 1 &8 &5 \\ 7 &5 &6 \\ 7 &7 &4 \\ 4 &4 &4 \end{bmatrix}. $$$$$$