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

We have discussed Dijkstra’s Shortest Path algorithm in below posts.

- Dijkstra’s shortest path for adjacency matrix representation
- Dijkstra’s shortest path for adjacency list representation

The implementations discussed above only find shortest distances, but do not print paths. In this post printing of paths is discussed.

For example, consider below graph andsource as 0, Output should be Vertex Distance Path 0 -> 1 4 0 1 0 -> 2 12 0 1 2 0 -> 3 19 0 1 2 3 0 -> 4 21 0 7 6 5 4 0 -> 5 11 0 7 6 5 0 -> 6 9 0 7 6 0 -> 7 8 0 7 0 -> 8 14 0 1 2 8

The idea is to create a separate array parent[]. Value of parent[v] for a vertex v stores parent vertex of v in shortest path tree. Parent of root (or source vertex) is -1. Whenever we find shorter path through a vertex u, we make u as parent of current vertex.

Once we have parent array constructed, we can print path using below recursive function.

void printPath(int parent[], int j) { // Base Case : If j is source if (parent[j]==-1) return; printPath(parent, parent[j]); printf("%d ", j); }

Below is complete implementation

## C/C++

// A C / C++ program for Dijkstra's single source shortest // path algorithm. The program is for adjacency matrix // representation of the graph. #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; } // Function to print shortest path from source to j // using parent array void printPath(int parent[], int j) { // Base Case : If j is source if (parent[j]==-1) return; printPath(parent, parent[j]); printf("%d ", j); } // A utility function to print the constructed distance // array int printSolution(int dist[], int n, int parent[]) { int src = 0; printf("Vertex\t Distance\tPath"); for (int i = 1; i < V; i++) { printf("\n%d -> %d \t\t %d\t\t%d ", src, i, dist[i], src); printPath(parent, 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 // sptSet[i] will true if vertex i is included / in shortest // path tree or shortest distance from src to i is finalized bool sptSet[V]; // Parent array to store shortest path tree int parent[V]; // Initialize all distances as INFINITE and stpSet[] as false for (int i = 0; i < V; i++) { parent[0] = -1; 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] + graph[u][v] < dist[v]) { parent[v] = u; dist[v] = dist[u] + graph[u][v]; } } // print the constructed distance array printSolution(dist, V, parent); } // 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; }

## Python

# Python program for Dijkstra's single source shortest # path algorithm. The program is for adjacency matrix # representation of the graph from collections import defaultdict #Class to represent a graph class Graph: # A utility function to find the vertex with minimum dist value, from # the set of vertices still in queue def minDistance(self,dist,queue): # Initialize min value and min_index as -1 minimum = float("Inf") min_index = -1 #from the dist array,pick one which has min value and is till in queue for i in range(len(dist)): if dist[i] < minimum and i in queue: minimum = dist[i] min_index = i return min_index # Function to print shortest path from source to j # using parent array def printPath(self, parent, j): if parent[j] == -1 : #Base Case : If j is source print j, return self.printPath(parent , parent[j]) print j, # A utility function to print the constructed distance # array def printSolution(self, dist, parent): src = 0 print("Vertex \t\tDistance from Source\tPath") for i in range(1, len(dist)): print("\n%d --> %d \t\t%d \t\t\t\t\t" % (src, i, dist[i])), self.printPath(parent,i) '''Function that implements Dijkstra's single source shortest path algorithm for a graph represented using adjacency matrix representation''' def dijkstra(self, graph, src): row = len(graph) col = len(graph[0]) # The output array. dist[i] will hold the shortest distance from src to i # Initialize all distances as INFINITE dist = [float("Inf")] * row #Parent array to store shortest path tree parent = [-1] * row # Distance of source vertex from itself is always 0 dist[src] = 0 # Add all vertices in queue queue = [] for i in range(row): queue.append(i) #Find shortest path for all vertices while queue: # Pick the minimum dist vertex from the set of vertices # still in queue u = self.minDistance(dist,queue) # remove min element queue.remove(u) # Update dist value and parent index of the adjacent vertices of # the picked vertex. Consider only those vertices which are still in # queue for i in range(col): '''Update dist[i] only if it is in queue, there is an edge from u to i, and total weight of path from src to i through u is smaller than current value of dist[i]''' if graph[u][i] and i in queue: if dist[u] + graph[u][i] < dist[i]: dist[i] = dist[u] + graph[u][i] parent[i] = u # print the constructed distance array self.printSolution(dist,parent) g= Graph() graph = [[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, 14, 10, 0, 2, 0, 0], [0, 0, 0, 0, 0, 2, 0, 1, 6], [8, 11, 0, 0, 0, 0, 1, 0, 7], [0, 0, 2, 0, 0, 0, 6, 7, 0] ] #Print the solution g.dijkstra(graph,0) #This code is contributed by Neelam Yadav

Output:

Vertex Distance Path 0 -> 1 4 0 1 0 -> 2 12 0 1 2 0 -> 3 19 0 1 2 3 0 -> 4 21 0 7 6 5 4 0 -> 5 11 0 7 6 5 0 -> 6 9 0 7 6 0 -> 7 8 0 7 0 -> 8 14 0 1 2 8

This article is contributed by **Aditya Goel**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above