Feature/floyd warshall pr#365
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Floyd–Warshall Algorithm — All-Pairs Shortest Path
Problem:
Compute the shortest distances between every pair of vertices in a weighted graph (directed or undirected). Handles graphs with non-negative and negative edge weights, but no negative cycles.
Approach:
Represent the graph as an adjacency matrix dist[V][V].
Initialize distances:
dist[i][i] = 0 (self-loops)
dist[u][v] = weight(u,v) if an edge exists
dist[u][v] = INF if no direct edge
For each intermediate vertex k:
For each source i and destination j:
dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
After all iterations, dist[i][j] contains the shortest distance from i to j.
Dry Run Example:
Initial matrix set with edges and INF for no connection
Step-by-step updates after considering each intermediate vertex k
Final matrix shows shortest distances between all pairs
Complexity:
Time: O(V³)
Space: O(V²)
Input:
Number of vertices V
Number of edges E
Edge list {u, v, w}
Output:
Matrix of shortest distances between all pairs of vertices