题目描述
In Takahashi Kingdom, which once existed, there are N cities, and some pairs of cities are connected bidirectionally by roads. The following are known about the road network:
- People traveled between cities only through roads. It was possible to reach any city from any other city, via intermediate cities if necessary.
- Different roads may have had different lengths, but all the lengths were positive integers.
Snuke the archeologist found a table with N rows and N columns, A, in the ruin of Takahashi Kingdom. He thought that it represented the shortest distances between the cities along the roads in the kingdom.
Determine whether there exists a road network such that for each u and v, the integer A_{u, v} at the u-th row and v-th column of A is equal to the length of the shortest path from City u to City v. If such a network exist, find the shortest possible total length of the roads.
Constraints
- $1 \leq N \leq 300$
- If i ≠ $j, 1 \leq A_{i, j} = A_{j, i} \leq 10^9$.
- A_{i, i} = 0
Inputs
Input is given from Standard Input in the following format:
N
A_{1, 1} A_{1, 2} ... A_{1, N}
A_{2, 1} A_{2, 2} ... A_{2, N}
...
A_{N, 1} A_{N, 2} ... A_{N, N}
Outputs
If there exists no network that satisfies the condition, print -1. If it exists, print the shortest possible total length of the roads.
Examples
Input
3
0 1 3
1 0 2
3 2 0
Output
3
Input
3
0 1 3
1 0 1
3 1 0
Output
-1
Input
5
0 21 18 11 28
21 0 13 10 26
18 13 0 23 13
11 10 23 0 17
28 26 13 17 0
Output
82
Input
3
0 1000000000 1000000000
1000000000 0 1000000000
1000000000 1000000000 0
Output
3000000000
输入格式
见题面
输出格式
见题面