Score : 500 points
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:
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.
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}
If there exists no network that satisfies the condition, print -1
.
If it exists, print the shortest possible total length of the roads.
3 0 1 3 1 0 2 3 2 0
3
The network below satisfies the condition:
3 0 1 3 1 0 1 3 1 0
-1
As there is a path of length 1 from City 1 to City 2 and City 2 to City 3, there is a path of length 2 from City 1 to City 3. However, according to the table, the shortest distance between City 1 and City 3 must be 3.
Thus, we conclude that there exists no network that satisfies the condition.
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
82
3 0 1000000000 1000000000 1000000000 0 1000000000 1000000000 1000000000 0
3000000000