Dijkstra’s algorithm
Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum spanning tree. Like Prim’s MST, we generate a SPT (shortest path tree) with given source as root. We maintain two sets, one set contains vertices included in shortest path tree, other set includes vertices not yet included in shortest path tree. At every step of the algorithm, we find a vertex which is in the other set (set of not yet included) and has minimum distance from source.
Given below are the detailed steps used in Dijkstra’s algorithm is 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) set a distance value to all vertices in the input graph. Initialize all distance values as INFINITE. set distance value as 0 for the source vertex so that it is picked first.
3) While sptSet doesn’t include all vertices
a) select a vertex u which is not there in sptSetand 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 understand with the given example:
The set sptSetis initially empty and distances assigned to vertices are {0, INF, INF, INF, INF, INF, INF, INF} where INF indicates infinite. Now select the vertex with minimum distance value. 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. Below subgraph shows vertices and their distance values, only the vertices with finite distance values are figured. The vertices included in SPT are shown green in color.
Select 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. The distance value of vertex 2 becomes 12.
Select 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. The distance value of vertex 6 and 8 becomes finite (15 and 9 respectively).
Select 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 doesn’t include all vertices of given graph. Finally, we get the following Shortest Path Tree (SPT).
How to implement the above algorithm?
We use a boolean array sptSet[] to represent the set of vertices included in SPT. If a value sptSet[v] is true, then vertex v is included in SPT, otherwise not. Array dist[] is used to store shortest distance values of all vertices.
|
// The program is for adjacency matrix graph representation. #include <stdio.h> #include <limits.h> // Number of vertices in the graph #define V 9 // A utility function to find the vertex with minimum distance value, from // the set of vertices not yet included in shortest path tree int minDistance(int dist[], bool sptSet[]) { // Initialize min value int min = INT_MAX, min_index; for (int v = 0; v < V; v++) if (sptSet[v] == false && dist[v] <= min) min = dist[v], min_index = v; return min_index; } // A utility function to print the constructed distance array int printSolution(int dist[], int n) { printf(“Vertex Distance from Source\n”); for (int i = 0; i < V; i++) printf(“%d \t\t %d\n”, i, dist[i]); } // Funtion that implements Dijkstra’s single source shortest path algorithm // for a graph represented using adjacency matrix representation void dijkstra(int graph[V][V], int src) { int dist[V]; // The output array. dist[i] will hold the shortest // distance from src to i bool sptSet[V]; // sptSet[i] will true if vertex i is included in shortest // path tree or shortest distance from src to i is finalized // Initialize all distances as INFINITE and stpSet[] as false for (int i = 0; i < V; i++) dist[i] = INT_MAX, sptSet[i] = false; // Distance of source vertex from itself is always 0 dist[src] = 0; // Find shortest path for all vertices for (int count = 0; count < V-1; count++) { // Pick the minimum distance vertex from the set of vertices not // yet processed. u is always equal to src in first iteration. int u = minDistance(dist, sptSet); // Mark the picked vertex as processed sptSet[u] = true; // Update dist value of the adjacent vertices of the picked vertex. for (int v = 0; v < V; v++) // Update dist[v] only if is not in sptSet, there is an edge from // u to v, and total weight of path from src to v through u is // smaller than current value of dist[v] if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX && dist[u]+graph[u][v] < dist[v]) dist[v] = dist[u] + graph[u][v]; } // print the constructed distance array printSolution(dist, V); } // driver program to test above function int main() { /* Let us create the example graph discussed above */ int graph[V][V] = {{0, 4, 0, 0, 0, 0, 0, 8, 0}, {4, 0, 8, 0, 0, 0, 0, 11, 0}, {0, 8, 0, 7, 0, 4, 0, 0, 2}, {0, 0, 7, 0, 9, 14, 0, 0, 0}, {0, 0, 0, 9, 0, 10, 0, 0, 0}, {0, 0, 4, 0, 10, 0, 2, 0, 0}, {0, 0, 0, 14, 0, 2, 0, 1, 6}, {8, 11, 0, 0, 0, 0, 1, 0, 7}, {0, 0, 2, 0, 0, 0, 6, 7, 0} }; dijkstra(graph, 0); return 0; } |
Output:
Vertex Distance from Source 0 0 1 4 2 12 3 19 4 21 5 11 6 9 7 8 8 14
