102775: [AtCoder]ABC277 F - Sorting a Matrix

Problem Statement

You are given a matrix $A$ whose elements are non-negative integers. For a pair of integers $(i, j)$ such that $1 \leq i \leq H$ and $1 \leq j \leq W$, let $A_{i, j}$ denote the element at the $i$-th row and $j$-th column of $A$.

Let us perform the following procedure on $A$.

• First, replace each element of $A$ that is $0$ with an arbitrary positive integer (if multiple elements are $0$, they may be replaced with different positive integers).

• Then, repeat performing one of the two operations below, which may be chosen each time, as many times as desired (possibly zero).

  • Choose a pair of integers $(i, j)$ such that $1 \leq i \lt j \leq H$ and swap the $i$-th and $j$-th rows of $A$.
  • Choose a pair of integers $(i, j)$ such that $1 \leq i \lt j \leq W$ and swap the $i$-th and $j$-th columns of $A$.

Determine whether $A$ can be made to satisfy the following condition.

• $A_{1, 1} \leq A_{1, 2} \leq \cdots \leq A_{1, W} \leq A_{2, 1} \leq A_{2, 2} \leq \cdots \leq A_{2, W} \leq A_{3, 1} \leq \cdots \leq A_{H, 1} \leq A_{H, 2} \leq \cdots \leq A_{H, W}$.

• In other words, for every two pairs of integers $(i, j)$ and $(i', j')$ such that $1 \leq i, i' \leq H$ and $1 \leq j, j' \leq W$, both of the following conditions are satisfied.

  • If $i \lt i'$, then $A_{i, j} \leq A_{i', j'}$.
  • If $i = i'$ and $j \lt j'$, then $A_{i, j} \leq A_{i', j'}$.

Constraints

• $2 \leq H, W$
• $H \times W \leq 10^6$
• $0 \leq A_{i, j} \leq H \times W$
• 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 $A$ can be made to satisfy the condition in the problem statement, print Yes; otherwise, print No.

Sample Input 1

3 3
9 6 0
0 4 0
3 0 3

Sample Output 1

Yes

One can perform the operations as follows to make $A$ satisfy the condition in the problem statement, so you should print Yes.

• First, replace the elements of $A$ that are $0$, as shown below:

9 6 8
5 4 4
3 1 3

• Swap the second and third columns. Then, $A$ becomes:

9 8 6
5 4 4
3 3 1

• Swap the first and third rows. Then, $A$ becomes:

3 3 1
5 4 4
9 8 6

• Swap the first and third columns. Then, $A$ becomes the following and satisfies the condition in the problem statement.

1 3 3
4 4 5
6 8 9

Sample Input 2

2 2
2 1
1 2

Sample Output 2

No

There is no way to perform the operations to make $A$ satisfy the condition in the problem statement, so you should print No.

问题描述

给你一个矩阵$A$，其元素为非负整数。对于一对整数$(i, j)$，满足$1 \leq i \leq H$和$1 \leq j \leq W$，记$A_{i, j}$表示矩阵$A$的第$i$行第$j$列的元素。

在$A$上执行以下操作。

• 首先，将矩阵$A$中的每个$0$元素替换为任意一个正整数（如果多个元素为$0$，可以替换成不同的正整数）。

• 然后，按需求重复执行以下两个操作之一（可以任意选择），次数任意（包括$0$次）。

  • 选择一对整数$(i, j)$，满足$1 \leq i \lt j \leq H$，将矩阵$A$的第$i$行和第$j$行交换。
  • 选择一对整数$(i, j)$，满足$1 \leq i \lt j \leq W$，将矩阵$A$的第$i$列和第$j$列交换。

判断是否可以将$A$调整为满足以下条件。

• $A_{1, 1} \leq A_{1, 2} \leq \cdots \leq A_{1, W} \leq A_{2, 1} \leq A_{2, 2} \leq \cdots \leq A_{2, W} \leq A_{3, 1} \leq \cdots \leq A_{H, 1} \leq A_{H, 2} \leq \cdots \leq A_{H, W}$。

• 换句话说，对于任意两对整数$(i, j)$和$(i', j')$，满足$1 \leq i, i' \leq H$和$1 \leq j, j' \leq W$，都满足以下两个条件。

  • 如果$i \lt i'$，则$A_{i, j} \leq A_{i', j'}$。
  • 如果$i = i'$且$j \lt j'$，则$A_{i, j} \leq A_{i', j'}$。

约束条件

• $2 \leq H, W$
• $H \times W \leq 10^6$
• $0 \leq A_{i, j} \leq H \times W$
• 输入中的所有值均为整数。

输入

输入从标准输入以以下格式给出：

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

输出

如果可以将$A$调整为满足题目描述中的条件，输出Yes；否则，输出No。

样例输入 1

3 3
9 6 0
0 4 0
3 0 3

样例输出 1

Yes

可以通过以下操作使$A$满足题目描述中的条件，所以应该输出Yes。

• 首先，将矩阵$A$中为$0$的元素替换为如下所示的值：

9 6 8
5 4 4
3 1 3

• 交换第二列和第三列。然后，$A$变为：

9 8 6
5 4 4
3 3 1

• 交换第一行和第三行。然后，$A$变为：

3 3 1
5 4 4
9 8 6

• 交换第一列和第三列。然后，$A$变为以下形式，满足题目描述中的条件。

1 3 3
4 4 5
6 8 9

样例输入 2

2 2
2 1
1 2

样例输出 2

No

无法通过操作使$A$满足题目描述中的条件，所以应该输出No。