Dijkstra’s single source shortest path algorithm

Dijkstra’s single source shortest path algorithm

Given a graph and a source vertex in the graph, find shortest paths from source to all vertices in the given graph.

Below are the detailed steps used in Dijkstra’s algorithm to find the shortest path from a single source vertex to all other vertices in the given graph.

Algorithm

1) Create a set sptSet (shortest path tree set) that keeps track of vertices included in shortest path tree, i.e., whose minimum distance from source is calculated and finalized. Initially, this set is empty.

2) Assign a distance value to all vertices in the input graph. Initialize all distance values as 

INFINITE. Assign distance value as 0 for the source vertex so that it is picked first.

3) While sptSet doesn’t include all vertices

     a) Pick a vertex u which is not there in sptSet and has minimum distance value.
     b) Include u to sptSet.
     c) Update distance value of all adjacent vertices of u. To update the distance values, iterate           through all adjacent vertices. For every adjacent vertex v, if sum of distance value of u (from source) and weight of edge u-v, is less than the distance value of v, then update the distance value of v.

Let us consider the following example:

The set sptSet is initially empty and distances assigned to vertices are {0, INF, INF, INF, INF, INF, INF, INF} where INF indicates infinite.

The vertex 0 is picked, include it in sptSet. So sptSet becomes {0}. After including 0 to sptSet, update distance values of its adjacent vertices. Adjacent vertices of 0 are 1 and 7. The distance values of 1 and 7 are updated as 4 and 8.

Pick the vertex with minimum distance value and not already included in SPT (not in sptSET). The vertex 1 is picked and added to sptSet.

So sptSet now becomes {0, 1}. Update the distance values of adjacent vertices of 1.

Pick the vertex with minimum distance value and not already included in SPT (not in sptSET). Vertex 7 is picked. So sptSet now becomes {0, 1, 7}. Update the distance values of adjacent vertices of 7. 

Pick the vertex with minimum distance value and not already included in SPT (not in sptSET). Vertex 6 is picked. So sptSet now becomes {0, 1, 7, 6}. Update the distance values of adjacent vertices of 6. The distance value of vertex 5 and 8 are updated.

We repeat the above steps until sptSet does include all vertices of given graph. Finally, we get the following Shortest Path Tree (SPT).

Time Complexity of the implementation is O(V²). If the input graph is represented using adjacency list, it can be reduced to O(E log V) with the help of binary heap.

Dijkstra’s algorithm doesn’t work for graphs with negative weight edges. For graphs with negative weight edges, Bellman–Ford algorithm can be used.