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C - RANDOM

Score : $300$ points

Problem Statement

You are given patterns $S$ and $T$ consisting of # and ., each with $H$ rows and $W$ columns.
The pattern $S$ is given as $H$ strings, and the $j$-th character of $S_i$ represents the element at the $i$-th row and $j$-th column. The same goes for $T$.

Determine whether $S$ can be made equal to $T$ by rearranging the columns of $S$.

Here, rearranging the columns of a pattern $X$ is done as follows.

• Choose a permutation $P=(P_1,P_2,\dots,P_W)$ of $(1,2,\dots,W)$.
• Then, for every integer $i$ such that $1 \le i \le H$, simultaneously do the following.
  • For every integer $j$ such that $1 \le j \le W$, simultaneously replace the element at the $i$-th row and $j$-th column of $X$ with the element at the $i$-th row and $P_j$-th column.

Constraints

• $H$ and $W$ are integers.
• $1 \le H,W$
• $1 \le H \times W \le 4 \times 10^5$
• $S_i$ and $T_i$ are strings of length $W$ consisting of # and ..

Input

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

$H$ $W$

$S_1$

$S_2$

$\vdots$

$S_H$

$T_1$

$T_2$

$\vdots$

$T_H$

Output

If $S$ can be made equal to $T$, print Yes; otherwise, print No.

3 4
##.#
##..
#...
.###
..##
...#

Yes

If you, for instance, arrange the $3$-rd, $4$-th, $2$-nd, and $1$-st columns of $S$ in this order from left to right, $S$ will be equal to $T$.

3 3
#.#
.#.
#.#
##.
##.
.#.

No

In this input, $S$ cannot be made equal to $T$.

2 1
#
.
#
.

Yes

It is possible that $S=T$.

8 7
#..#..#
.##.##.
#..#..#
.##.##.
#..#..#
.##.##.
#..#..#
.##.##.
....###
####...
....###
####...
....###
####...
....###
####...

Yes