Problem p03600 - Generation 1

Orig Description

Score : 500 points
Problem StatementIn 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
InputsInput 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}
OutputsIf there exists no network that satisfies the condition, print -1.
If it exists, print the shortest possible total length of the roads.
Sample Input 13
0 1 3
1 0 2
3 2 0
Sample Output 13
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.
Sample Input 23
0 1 3
1 0 1
3 1 0
Sample Output 2-1
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.
Sample Input 35
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
Sample Output 382
Sample Input 43
0 1000000000 1000000000
1000000000 0 1000000000
1000000000 1000000000 0
Sample Output 43000000000

Extracted Specification

An integer N (1 ≤ N ≤ 300), representing some quantity or size.
A list of N lists, where each inner list contains N integers between 0 and 10^9, with the diagonal elements being 0 and the matrix being symmetric.

### Example Input:

```
3
0 1 3
1 0 2
3 2 0
```

### Function Signature:
Write a function f(N, A) that takes in the input.
```python
def f(N: int, A: List[List[int]]):
    ''' 
    N: an integer 
    A: a list of lists of integers
    '''
```

Test Case Generator

Certainly! Let's create a test case generator for the given problem statement in Python.

The problem involves generating an \(N \times N\) matrix \(A\) where \(A[i][j]\) represents the shortest distances between cities. The constraints and conditions given need to be respected.

Here's a test case generator for this problem:

```python
import random

def generate_matrix(N):
    A = [[0] * N for _ in range(N)]
    
    # Ensure the diagonal is zero
    for i in range(N):
        A[i][i] = 0
    
    # Populate the upper triangle and mirror it to the lower triangle
    for i in range(N):
        for j in range(i + 1, N):
            if random.choice([True, False]):
                dist = random.randint(1, 10**9)
                A[i][j] = dist
                A[j][i] = dist
            else:
                # For ensuring some -1 cases, let's create some inconsistent paths
                dist = random.randint(1, 10**9)
                A[i][j] = dist
                A[j][i] = dist + random.randint(1, 100)  # Inconsistent distance

    return A

def tcgen():
    N = random.randint(1, 300)
    A = generate_matrix(N)
    
    # Convert matrix to the input format
    inputs = [str(N)]
    for row in A:
        inputs.append(" ".join(map(str, row)))
    
    return "\n".join(inputs)

# Example usage:
if __name__ == "__main__":
    print(tcgen())
```

### Explanation:
1. **Generating N**: Generate a random integer \(N\) between 1 and 300.
2. **Generating Matrix**:
   - Create an \(N \times N\) matrix initialized with zeros.
   - Ensure the diagonal elements are zero since \(A[i][i] = 0\).
   - Populate the upper triangle of the matrix with random distances.
   - Mirror the upper triangle to the lower triangle to maintain symmetry.
   - Introduce some inconsistencies deliberately to ensure some test cases will fail to meet the shortest path conditions.
3. **Formatting the Output**: Convert the matrix into the required input format.

This generator will provide various test cases, some of which will have valid road networks, while others will not, helping to thoroughly test the solution.

Feel free to modify the generator as needed or add more constraints to better fit specific testing scenarios.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    A = []
    for _ in range(N):
        A.append(list(map(int, fh.readline().strip().split())))
    return N, A