Increasing Matrix

2000ms

262144K

Description:

In this problem, a $$$n \times m$$$ rectangular matrix $$$a$$$ is called increasing if, for each row of $$$i$$$, when go from left to right, the values strictly increase (that is, $$$a_{i,1}<a_{i,2}<\dots<a_{i,m}$$$) and for each column $$$j$$$, when go from top to bottom, the values strictly increase (that is, $$$a_{1,j}<a_{2,j}<\dots<a_{n,j}$$$).

In a given matrix of non-negative integers, it is necessary to replace each value of $$$0$$$ with some positive integer so that the resulting matrix is increasing and the sum of its elements is maximum, or find that it is impossible.

It is guaranteed that in a given value matrix all values of $$$0$$$ are contained only in internal cells (that is, not in the first or last row and not in the first or last column).

Input:

The first line contains integers $$$n$$$ and $$$m$$$ ($$$3 \le n, m \le 500$$$) — the number of rows and columns in the given matrix $$$a$$$.

The following lines contain $$$m$$$ each of non-negative integers — the values in the corresponding row of the given matrix: $$$a_{i,1}, a_{i,2}, \dots, a_{i,m}$$$ ($$$0 \le a_{i,j} \le 8000$$$).

It is guaranteed that for all $$$a_{i,j}=0$$$, $$$1 < i < n$$$ and $$$1 < j < m$$$ are true.

Output:

If it is possible to replace all zeros with positive numbers so that the matrix is increasing, print the maximum possible sum of matrix elements. Otherwise, print -1.

Sample Input:

4 5
1 3 5 6 7
3 0 7 0 9
5 0 0 0 10
8 9 10 11 12

Sample Output:

Sample Input:

Sample Output:

Sample Input:

Sample Output:

-1

Sample Input:

Sample Output:

-1

Note:

In the first example, the resulting matrix is as follows:


1 3 5 6 7
3 6 7 8 9
5 7 8 9 10
8 9 10 11 12

In the second example, the value $$$3$$$ must be put in the middle cell.

In the third example, the desired resultant matrix does not exist.

Informação

Provedor Codeforces

Código CF1231C