UnWeighted or Equal Weighted Graph

The shortest path from s to t in an unweighted graph can be constructed using a breadth-first search from s. The minimum-link path is recorded in the breadth-first search tree, and it provides the shortest path when all edges have equal weight.

General Weighted Graph

Single Source Shortest Path

1. Dijkstra'S Algorithm (Negative Edge Weight Not Allowed)

2. BellMan Ford Algorithm

We can find Shortest Path using BellMan Ford algorithm even when there are negative weight edges, but there should not be any negative weight cycle in the Graph.

Input: Graph and a source vertex src
Output: Shortest distance to all vertices from src. If there is a negative weight cycle, then shortest distances are not calculated, negative weight cycle is reported.

1. This step initializes distances from source to all vertices as infinite and distance to source itself as 0. Create an array dist[] of size |V| with all values as infinite except dist[src] where src is source vertex.

2. This step calculates shortest distances. Do following |V|-1 times where |V| is the number of vertices in given graph.

   Do following for each edge u-v  
     If dist[v] > dist[u] + weight of edge uv, then update dist[v]  
         dist[v] = dist[u] + weight of edge uv  

3. This step reports if there is a negative weight cycle in graph. Do following for each edge u-v ……If dist[v] > dist[u] + weight of edge uv, then “Graph contains negative weight cycle” The idea of step 3 is, step 2 guarantees shortest distances if graph doesn’t contain negative weight cycle. If we iterate through all edges one more time and get a shorter path for any vertex, then there is a negative weight cycle
   Time Complexity: O(VE)

All Pair Shorted Path

1. Floyd-Warshall all-pairs shortest path(Negative Edge Weight Allowed)

2. Johnson's algorithm

• Johnson's Algo can also be used for finding Transitive Closure