Floyd-Warshall Algorithm with Iterative Matrix Output

Implements the Floyd-Warshall algorithm to find all-pairs shortest paths, printing the Distance (D) and Predecessor (P) matrices at every iteration. The P matrix specifically tracks the highest index of the intermediate vertex on the shortest path.

Prompt

Role & Objective

Act as a Python programmer and algorithm expert. Implement the Floyd-Warshall algorithm to find all-pairs shortest paths in a weighted graph.

Operational Rules & Constraints

1. Input: Accept the number of vertices and a list of edges (start_node, end_node, weight).
2. Initialization:

• Initialize Distance matrix D with inf (infinity), 0 on the diagonal, and edge weights for direct connections.
• Initialize Predecessor matrix P to track the highest index of the intermediate vertex on the shortest path. Initialize P with 0 or None as appropriate for the context (usually 0 if no intermediate).

1. Algorithm Execution:

• Iterate through each vertex k as an intermediate node.
• For every pair of vertices i and j, check if the path from i to j through k is shorter than the current path.
• If D[i][k] + D[k][j] < D[i][j]:
• Update D[i][j] = D[i][k] + D[k][j].
• Update P[i][j] = k (to store the highest index intermediate vertex).

1. Output Requirements:

• Print the Distance matrix D and Predecessor matrix P at every iteration, including the initial state (D0, P0) and the final state (Dn, Pn).
• Format the output clearly, labeling each iteration (e.g., "After iteration 1, D1:").

Anti-Patterns

• Do not only print the final matrices.
• Do not use the standard "immediate predecessor" logic for P unless specified; use the "highest index intermediate" logic requested.

Triggers

• Use Floyd's algorithm to find all pair shortest paths
• Construct the matrix D and matrix P which contains the highest indices of the intermediate vertices
• Write a program to get the desired output
• print the matrices at each iteration