Floyd-Warshall Algorithm

Write a C++ program to implement the Floyd-Warshall algorithm to find the shortest paths between all pairs of vertices in a weighted graph.

Example 1

Input: graph = 
   {{0, 5, INF, 10},
    {INF, 0, 3, INF},
    {INF, INF, 0, 1},
    {INF, INF, INF, 0}}
Output: 
   0 5 8 9
   INF 0 3 4
   INF INF 0 1
   INF INF INF 0
Explanation:
   

   
Step 1: Initialize the distance matrix with the given graph.
   
Step 2: For each vertex k as an intermediate vertex, update the shortest path between every pair of vertices (i,j) if path i→k→j is shorter than the direct path i→j.
   
Step 3: After processing all vertices as intermediates, the matrix represents the shortest path distances between all pairs of vertices.
   
Step 4: Return the final distance matrix.
   

Example 2

Input: graph = 
   {{0, 3, INF, 7},
    {8, 0, 2, INF},
    {5, INF, 0, 1},
    {2, INF, INF, 0}}
Output: 
   0 3 5 6
   5 0 2 3
   3 6 0 1
   2 5 7 0
Explanation:
   

   
Step 1: Initialize the distance matrix with the given graph.
   
Step 2: For vertex 0 as intermediate, update distances if path i→0→j is shorter.
   
Step 3: For vertex 1 as intermediate, update distances if path i→1→j is shorter.
   
Step 4: Continue for vertices 2 and 3 as intermediates.
   
Step 5: After processing all vertices as intermediates, the matrix represents the shortest path distances between all pairs of vertices.
   
Step 6: Return the final distance matrix.
   

Constraints

1 ≤ number of vertices ≤ 100
Time Complexity: O(n^3), where n is the number of vertices
Space Complexity: O(n^2)