Score : $500$ points

问题描述

你得到一个矩阵 $A$，它有 $H$ 行和 $W$ 列。其每个元素的值为 $0$ 或 $1$。对于满足条件 $1 \leq i \leq H$ 和 $1 \leq j \leq W$ 的整数对 $(i, j)$，我们用 $A_{i,j}$ 表示第 $i$ 行第 $j$ 列的元素。

你可以对矩阵 $A$ 执行以下操作任意多次（可能为零次）：

• 选择一个满足 $1 \leq i \leq H$ 的整数 $i$。对于每一个满足 $1 \leq j \leq W$ 的整数 $j$，将 $A_{i,j}$ 的值替换为 $1-A_{i,j}$。

若 $A_{i,j}$ 满足以下条件，则称其为 孤立元素：不存在与其相邻且具有相同值的元素；换句话说，如果不存在四个整数对 $(x,y) = (i-1,j),(i+1,j),(i,j-1),(i,j+1)$ 满足 $1 \leq x \leq H, 1 \leq y \leq W$ 以及 $A_{i,j} = A_{x,y}$。

确定是否可以通过重复执行上述操作，使得矩阵 $A$ 处于没有孤立元素的状态。如果可能，请找到完成此目标所需的最少操作次数。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a matrix $A$ with $H$ rows and $W$ columns. The value of each of its elements is $0$ or $1$. For an integer pair $(i, j)$ such that $1 \leq i \leq H$ and $1 \leq j \leq W$, we denote by $A_{i,j}$ the element at the $i$-th row and $j$-th column.

You can perform the following operation on the matrix $A$ any number of times (possibly zero):

• Choose an integer $i$ such that $1 \leq i \leq H$. For every integer $j$ such that $1 \leq j \leq W$, replace the value of $A_{i,j}$ with $1-A_{i,j}$.

$A_{i,j}$ is said to be isolated if and only if there is no adjacent element with the same value; in other words, if and only if none of the four integer pairs $(x,y) = (i-1,j),(i+1,j),(i,j-1),(i,j+1)$ satisfies $1 \leq x \leq H, 1 \leq y \leq W$, and $A_{i,j} = A_{x,y}$.

Determine if you can make the matrix $A$ in such a state that no element is isolated by repeating the operation. If it is possible, find the minimum number of operations required to do so.

Constraints

• $2 \leq H,W \leq 1000$
• $A_{i,j} = 0$ or $A_{i,j} = 1$
• All values in the input are integers.

Input

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

$H$ $W$

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

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

$\vdots$

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

Output

If you can make it in such a state that no element is isolated by repeating the operation, print the minimum number of operations required to do so; otherwise, print -1.

Sample Input 1

3 3
1 1 0
1 0 1
1 0 0

Sample Output 1

1

An operation with $i = 1$ makes $A = ((0,0,1),(1,0,1),(1,0,0))$, where there is no longer an isolated element.

Sample Input 2

4 4
1 0 0 0
0 1 1 1
0 0 1 0
1 1 0 1

Sample Output 2

2

Sample Input 3

2 3
0 1 0
0 1 1

Sample Output 3

-1

update @ 2024/3/10 11:52:37