B - Mongeness

Score : $200$ points

Problem Statement

We have a grid with $H$ horizontal rows and $W$ vertical columns, where each square contains an integer. The integer written on the square at the $i$-th row from the top and $j$-th column from the left is $A_{i, j}$.

Determine whether the grid satisfies the condition below.

$A_{i_1, j_1} + A_{i_2, j_2} \leq A_{i_2, j_1} + A_{i_1, j_2}$ holds for every quadruple of integers $(i_1, i_2, j_1, j_2)$ such that $1 \leq i_1 < i_2 \leq H$ and $1 \leq j_1 < j_2 \leq W$.

Constraints

• $2 \leq H, W \leq 50$
• $1 \leq A_{i, j} \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$H$ $W$

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

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

$\vdots$

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

Output

If the grid satisfies the condition in the Problem Statement, print Yes; otherwise, print No.

3 3
2 1 4
3 1 3
6 4 1

Yes

There are nine quadruples of integers $(i_1, i_2, j_1, j_2)$ such that $1 \leq i_1 < i_2 \leq H$ and $1 \leq j_1 < j_2 \leq W$. For all of them, $A_{i_1, j_1} + A_{i_2, j_2} \leq A_{i_2, j_1} + A_{i_1, j_2}$ holds. Some examples follow.

• For $(i_1, i_2, j_1, j_2) = (1, 2, 1, 2)$, we have $A_{i_1, j_1} + A_{i_2, j_2} = 2 + 1 \leq 3 + 1 = A_{i_2, j_1} + A_{i_1, j_2}$.
• For $(i_1, i_2, j_1, j_2) = (1, 2, 1, 3)$, we have $A_{i_1, j_1} + A_{i_2, j_2} = 2 + 3 \leq 3 + 4 = A_{i_2, j_1} + A_{i_1, j_2}$.
• For $(i_1, i_2, j_1, j_2) = (1, 2, 2, 3)$, we have $A_{i_1, j_1} + A_{i_2, j_2} = 1 + 3 \leq 1 + 4 = A_{i_2, j_1} + A_{i_1, j_2}$.
• For $(i_1, i_2, j_1, j_2) = (1, 3, 1, 2)$, we have $A_{i_1, j_1} + A_{i_2, j_2} = 2 + 4 \leq 6 + 1 = A_{i_2, j_1} + A_{i_1, j_2}$.
• For $(i_1, i_2, j_1, j_2) = (1, 3, 1, 3)$, we have $A_{i_1, j_1} + A_{i_2, j_2} = 2 + 1 \leq 6 + 4 = A_{i_2, j_1} + A_{i_1, j_2}$.

We can also see that the property holds for the other quadruples: $(i_1, i_2, j_1, j_2) = (1, 3, 2, 3), (2, 3, 1, 2), (2, 3, 1, 3), (2, 3, 2, 3)$.
Thus, we should print Yes.

2 4
4 3 2 1
5 6 7 8

No

We should print No because the condition is not satisfied.
This is because, for example, we have $A_{i_1, j_1} + A_{i_2, j_2} = 4 + 8 > 5 + 1 = A_{i_2, j_1} + A_{i_1, j_2}$ for $(i_1, i_2, j_1, j_2) = (1, 2, 1, 4)$.