Score: $525$ points

问题陈述

有一个 $N \times N$ 的网格，其中第 $i$ 行从顶部开始，第 $j$ 列从左侧开始的单元格 $(1 \leq i, j \leq N)$ 包含整数 $A_{i,j}$。

给定一个整数 $M$。当选择三个不重叠的 $M \times M$ 网格时，找出所选网格中写入的整数的最大可能和。

问题的形式定义：一个整数的 $6$-元组 $(i_1, j_1, i_2, j_2, i_3, j_3)$ 当满足以下三个条件时被称为一个好的 $6$-元组：

• $1 \leq i_k \leq N-M+1 \ (k=1,2,3)$
• $1 \leq j_k \leq N-M+1 \ (k=1,2,3)$
• 如果 $k \neq l \ (k,l \in \lbrace1,2,3\rbrace)$，集合 $\lbrace(i,j) \mid i_k \leq i < i_k+M \wedge j_k \leq j < j_k+M\rbrace$ 和 $\lbrace(i,j) \mid i_l \leq i < i_l+M \wedge j_l \leq j < j_l+M\rbrace$ 不相交。

对于一个好的 $6$-元组 $(i_1, j_1, i_2, j_2, i_3, j_3)$，找出 $\displaystyle \sum_{k=1}^{3}\sum_{i=i_k}^{i_k+M-1}\sum_{j=j_k}^{j_k+M-1}A_{i,j}$ 的最大值。可以证明，在这个问题的约束条件下，存在一个好的 $6$-元组。

以上为大语言模型 kimi 翻译，仅供参考。

Problem Statement

There is an $N\times N$ grid, and the cell at the $i$-th row from the top and the $j$-th column from the left $(1\leq i,j\leq N)$ contains the integer $A _ {i,j}$.

You are given an integer $M$. When choosing three non-overlapping $M\times M$ grids, find the maximum possible sum of the integers written in the chosen grids.

Formal definition of the problem A $6$-tuple of integers $(i _ 1,j _ 1,i _ 2,j _ 2,i _ 3,j _ 3)$ is called a good $6$-tuple when it satisfies the following three conditions:

• $1\leq i _ k\leq N-M+1\ (k=1,2,3)$
• $1\leq j _ k\leq N-M+1\ (k=1,2,3)$
• If $k\neq l\ (k,l\in\lbrace1,2,3\rbrace)$, the sets $\lbrace(i,j)\mid i _ k\leq i\lt i _ k+M\wedge j _ k\leq j\lt j _ k+M\rbrace$ and $\lbrace(i,j)\mid i _ l\leq i\lt i _ l+M\wedge j _ l\leq j\lt j _ l+M\rbrace$ do not intersect.

Find the maximum value of $\displaystyle \sum _ {k=1} ^ 3\sum _ {i=i _ k} ^ {i _ k+M-1}\sum _ {j=j _ k} ^ {j _ k+M-1}A _ {i,j}$ for a good $6$-tuple $(i _ 1,j _ 1,i _ 2,j _ 2,i _ 3,j _ 3)$. It can be shown that a good $6$-tuple exists under the constraints of this problem.

Constraints

• $2\leq N\leq 1000$
• $1\leq M\leq N/2$
• $0\leq A _ {i,j}\leq10 ^ 9$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

$A _ {1,1}$ $A _ {1,2}$ $\ldots$ $A _ {1,N}$

$A _ {2,1}$ $A _ {2,2}$ $\ldots$ $A _ {2,N}$

$\vdots$ $\ \vdots$ $\ddots$ $\vdots$

$A _ {N,1}$ $A _ {N,2}$ $\ldots$ $A _ {N,N}$

Output

Print the answer.

Sample Input 1

7 3
3 1 4 1 5 9 2
6 5 3 5 8 9 7
9 3 2 3 8 4 6
2 6 4 3 3 8 3
2 7 9 5 0 2 8
8 4 1 9 7 1 6
9 3 9 9 3 7 5

Sample Output 1

154

From the given grid, if we choose three $3\times3$ grids as shown in the figure below (this corresponds to setting $(i _ 1,j _ 1,i _ 2,j _ 2,i _ 3,j _ 3)=(1,5,2,1,5,2)$), the sum of the numbers written in the chosen grids will be $154$.

There is no way to make the sum $155$ or greater while satisfying the conditions in the problem statement, so print $154$.

Sample Input 2

7 1
3 1 4 1 5 9 2
6 5 3 5 8 9 7
9 3 2 3 8 4 6
2 6 4 3 3 8 3
2 7 9 5 0 2 8
8 4 1 9 7 1 6
9 3 9 9 3 7 5

Sample Output 2

27

The following choice is optimal.

Sample Input 3

16 4
74 16 58 32 97 52 43 51 40 58 13 24 65 11 63 29
98 75 40 77 15 50 83 85 35 46 38 37 56 38 63 55
95 42 10 70 53 40 25 10 70 32 33 19 52 79 74 58
33 91 53 11 65 63 78 77 81 46 81 63 11 82 55 62
39 95 92 69 77 89 14 84 53 78 71 81 66 39 96 29
74 26 60 55 89 35 32 64 17 26 74 92 84 33 59 82
23 69 10 95 94 14 58 58 97 95 62 58 72 55 71 43
93 77 27 87 74 72 91 37 53 80 51 71 37 35 97 46
81 88 26 79 78 30 53 68 83 28 59 28 74 55 20 86
93 13 25 19 53 53 17 24 69 14 67 81 10 19 69 90
88 83 62 92 22 31 27 34 67 48 42 32 68 14 96 87
44 69 25 48 68 42 53 82 44 42 96 31 13 56 68 83
63 87 24 75 16 70 63 99 95 10 63 26 56 12 77 49
94 83 69 95 48 41 40 97 45 61 26 38 83 91 44 31
43 69 54 64 20 60 17 15 62 25 58 50 59 63 88 70
72 95 21 28 41 14 77 22 64 78 33 55 67 51 78 40

Sample Output 3

3295

The following choice is optimal.