102645: [AtCoder]ABC264 F - Monochromatic Path

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

问题描述

我们有一个有$H$行和$W$列的网格。每个方块都被涂成白色或黑色。对于每对整数$(i, j)$，如果$1 \leq i \leq H$和$1 \leq j \leq W$，则位于从顶部数第$i$行，从左数第$j$列的方块（我们简单地将这个方块表示为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。

您可以按任意顺序在任意次数（包括$0$次）上执行以下操作：

• 选择一个整数$i$，使$1 \leq i \leq H$，支付$R_i$日元（日本的货币），并翻转网格中从顶部数第$i$行中每个方块的颜色。 （白色方块被涂成黑色，黑色方块被涂成白色。）
• 选择一个整数$j$，使$1 \leq j \leq W$，支付$C_j$日元，并翻转网格中从左数第$j$列中每个方块的颜色。

打印满足以下条件的最小总成本：

• 存在一条从Square $(1, 1)$到Square $(H, W)$的路径，可以通过反复向右或向下移动来获得，使得路径中的所有方块（包括Square $(1, 1)$和Square $(H, W)$）具有相同的颜色。

我们可以证明，在本问题的约束下，总是在有限数量的操作中可以满足条件。

约束

• $2 \leq H, W \leq 2000$
• $1 \leq R_i \leq 10^9$
• $1 \leq C_j \leq 10^9$
• $A_{i, j} \in \lbrace 0, 1\rbrace$
• 输入中的所有值都是整数。

输入

输入从标准输入以以下格式给出：

$H$ $W$
$R_1$ $R_2$ $\ldots$ $R_H$
$C_1$ $C_2$ $\ldots$ $C_W$
$A_{1, 1}A_{1, 2}\ldots A_{1, W}$
$A_{2, 1}A_{2, 2}\ldots A_{2, W}$
$\vdots$
$A_{H, 1}A_{H, 2}\ldots A_{H, W}$

输出

打印答案。

样例输入 1

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

样例输出 1

9

我们将白色方块表示为0，将黑色方块表示为1。 在初始网格上，您可以支付$R_2 = 3$日元来翻转网格中从顶部数第$2$行中每个方块的颜色，使网格变成：

0100
0100
1010

然后，您可以支付$C_2 = 6$日元来翻转网格中从左数第$2$列中每个方块的颜色，使网格变成：

0000
0000
1110

现在，存在一条从Square $(1, 1)$到Square $(3, 4)$的路径，使得路径中的所有方块都具有相同的颜色（例如，路径$(1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (2, 4) \rightarrow (3, 4)$）。支付的总成本为$3+6 = 9$日元，这是可能的最小值。

样例输入 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

样例输出 2

125