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D - Restoring Road Network

Score : $500$ points

Problem Statement

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 $i ≠ j$, $1 \leq A_{i, j} = A_{j, i} \leq 10^9$.
• $A_{i, i} = 0$

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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.

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3
0 1 3
1 0 2
3 2 0

3

The network below satisfies the condition:

• City $1$ and City $2$ is connected by a road of length $1$.
• City $2$ and City $3$ is connected by a road of length $2$.
• City $3$ and City $1$ is not connected by a road.

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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.

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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

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3
0 1000000000 1000000000
1000000000 0 1000000000
1000000000 1000000000 0

3000000000