Score : $500$ points

问题描述

我们有一个包含 $H$ 行和 $W$ 列的网格。每个格子要么涂成白色，要么涂成黑色。对于满足条件 $1 \leq i \leq H$ 和 $1 \leq j \leq W$ 的每一对整数 $(i, j)$，第 $i$ 行从上到下、第 $j$ 列从左至右的格子（我们简单地记作 Square $(i, j)$）的颜色由 $A_{i, j}$ 表示。如果 $A_{i, j} = 0$，则 Square $(i, j)$ 为白色；如果 $A_{i, j} = 1$，则为黑色。

你可以按任意顺序执行以下操作任意次（可能为 $0$ 次）：

• 选择一个满足 $1 \leq i \leq H$ 的整数 $i$，支付 $R_i$ 日元（日本货币），然后将网格中第 $i$ 行从上到下的每个格子颜色反转。（白色格子变为黑色，黑色格子变为白色。）
• 选择一个满足 $1 \leq j \leq W$ 的整数 $j$，支付 $C_j$ 日元，然后将网格中第 $j$ 列从左至右的每个格子颜色反转。

请输出满足以下条件所需的最小总费用：

• 存在一个从 Square $(1, 1)$ 到 Square $(H, W)$ 的路径，该路径可以通过重复向下或向右移动得到，并且路径中所有格子（包括 Square $(1, 1)$ 和 Square $(H, W)$）颜色相同。

在本题的约束条件下，我们可以证明总是可以通过有限次操作满足上述条件。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a grid with $H$ rows and $W$ columns. Each square is painted either white or black. For each integer pair $(i, j)$ such that $1 \leq i \leq H$ and $1 \leq j \leq W$, the color of the square at the $i$-th row from the top and $j$-th column from the left (we simply denote this square by Square $(i, j)$) is represented by $A_{i, j}$. Square $(i, j)$ is white if $A_{i, j} = 0$, and black if $A_{i, j} = 1$.

You may perform the following operations any number of (possibly $0$) times in any order:

• Choose an integer $i$ such that $1 \leq i \leq H$, pay $R_i$ yen (the currency in Japan), and invert the color of each square in the $i$-th row from the top in the grid. (White squares are painted black, and black squares are painted white.)
• Choose an integer $j$ such that $1 \leq j \leq W$, pay $C_j$ yen, and invert the color of each square in the $j$-th column from the left in the grid.

Print the minimum total cost to satisfy the following condition:

• There exists a path from Square $(1, 1)$ to Square $(H, W)$ that can be obtained by repeatedly moving down or to the right, such that all squares in the path (including Square $(1, 1)$ and Square $(H, W)$) have the same color.

We can prove that it is always possible to satisfy the condition in a finite number of operations under the Constraints of this problem.

Constraints

• $2 \leq H, W \leq 2000$
• $1 \leq R_i \leq 10^9$
• $1 \leq C_j \leq 10^9$
• $A_{i, j} \in \lbrace 0, 1\rbrace$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$H$ $W$

$R_1$ $R_2$ $\ldots$ $R_H$

$C_1$ $C_2$ $\ldots$ $C_W$

$A_{1, 1}A_{1, 2}\ldots A_{1, W}$

$A_{2, 1}A_{2, 2}\ldots A_{2, W}$

$\vdots$

$A_{H, 1}A_{H, 2}\ldots A_{H, W}$

Output

Print the answer.

Sample Input 1

3 4
4 3 5
2 6 7 4
0100
1011
1010

Sample Output 1

9

We denote a white square by 0 and a black square by 1. On the initial grid, you can pay $R_2 = 3$ yen to invert the color of each square in the $2$-nd row from the top to make the grid:

0100
0100
1010

Then, you can pay $C_2 = 6$ yen to invert the color of each square in the $2$-nd row from the left to make the grid:

0000
0000
1110

Now, there exists a path from Square $(1, 1)$ to Square $(3, 4)$ such that all squares in the path have the same color (such as the path $(1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (2, 4) \rightarrow (3, 4)$). The total cost paid is $3+6 = 9$ yen, which is the minimum possible.

Sample Input 2

15 20
29 27 79 27 30 4 93 89 44 88 70 75 96 3 78
39 97 12 53 62 32 38 84 49 93 53 26 13 25 2 76 32 42 34 18
01011100110000001111
10101111100010011000
11011000011010001010
00010100011111010100
11111001101010001011
01111001100101011100
10010000001110101110
01001011100100101000
11001000100101011000
01110000111011100101
00111110111110011111
10101111111011101101
11000011000111111001
00011101011110001101
01010000000001000000

Sample Output 2

125

update @ 2024/3/10 11:10:13