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 and source 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 the complete implementation
C/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 Distance Path"); for (int i = 1; i < V; i++) { printf("
%d -> %d %d %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) { // The output array. dist[i] // will hold the shortest // distance from src to i int dist[V]; // 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 Code 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; } |
Java
// A Java program for Dijkstra's // single source shortest path // algorithm. The program is for // adjacency matrix representation // of the graph. class DijkstrasAlgorithm { private static final int NO_PARENT = -1; // Function that implements Dijkstra's // single source shortest path // algorithm for a graph represented // using adjacency matrix // representation private static void dijkstra(int[][] adjacencyMatrix, int startVertex) { int nVertices = adjacencyMatrix[0].length; // shortestDistances[i] will hold the // shortest distance from src to i int[] shortestDistances = new int[nVertices]; // added[i] will true if vertex i is // included / in shortest path tree // or shortest distance from src to // i is finalized boolean[] added = new boolean[nVertices]; // Initialize all distances as // INFINITE and added[] as false for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { shortestDistances[vertexIndex] = Integer.MAX_VALUE; added[vertexIndex] = false; } // Distance of source vertex from // itself is always 0 shortestDistances[startVertex] = 0; // Parent array to store shortest // path tree int[] parents = new int[nVertices]; // The starting vertex does not // have a parent parents[startVertex] = NO_PARENT; // Find shortest path for all // vertices for (int i = 1; i < nVertices; i++) { // Pick the minimum distance vertex // from the set of vertices not yet // processed. nearestVertex is // always equal to startNode in // first iteration. int nearestVertex = -1; int shortestDistance = Integer.MAX_VALUE; for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { if (!added[vertexIndex] && shortestDistances[vertexIndex] < shortestDistance) { nearestVertex = vertexIndex; shortestDistance = shortestDistances[vertexIndex]; } } // Mark the picked vertex as // processed added[nearestVertex] = true; // Update dist value of the // adjacent vertices of the // picked vertex. for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { int edgeDistance = adjacencyMatrix[nearestVertex][vertexIndex]; if (edgeDistance > 0 && ((shortestDistance + edgeDistance) < shortestDistances[vertexIndex])) { parents[vertexIndex] = nearestVertex; shortestDistances[vertexIndex] = shortestDistance + edgeDistance; } } } printSolution(startVertex, shortestDistances, parents); } // A utility function to print // the constructed distances // array and shortest paths private static void printSolution(int startVertex, int[] distances, int[] parents) { int nVertices = distances.length; System.out.print("Vertex Distance Path"); for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { if (vertexIndex != startVertex) { System.out.print("
" + startVertex + " -> "); System.out.print(vertexIndex + " "); System.out.print(distances[vertexIndex] + " "); printPath(vertexIndex, parents); } } } // Function to print shortest path // from source to currentVertex // using parents array private static void printPath(int currentVertex, int[] parents) { // Base case : Source node has // been processed if (currentVertex == NO_PARENT) { return; } printPath(parents[currentVertex], parents); System.out.print(currentVertex + " "); } // Driver Code public static void main(String[] args) { int[][] adjacencyMatrix = { { 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(adjacencyMatrix, 0); } } // This code is contributed by Harikrishnan Rajan |
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): #Base Case : If j is source if parent[j] == -1 : 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 Distance from Source Path") for i in range(1, len(dist)): print("
%d --> %d %d " % (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 |
C#
// C# program for Dijkstra's // single source shortest path // algorithm. The program is for // adjacency matrix representation // of the graph. using System; public class DijkstrasAlgorithm { private static readonly int NO_PARENT = -1; // Function that implements Dijkstra's // single source shortest path // algorithm for a graph represented // using adjacency matrix // representation private static void dijkstra(int[,] adjacencyMatrix, int startVertex) { int nVertices = adjacencyMatrix.GetLength(0); // shortestDistances[i] will hold the // shortest distance from src to i int[] shortestDistances = new int[nVertices]; // added[i] will true if vertex i is // included / in shortest path tree // or shortest distance from src to // i is finalized bool[] added = new bool[nVertices]; // Initialize all distances as // INFINITE and added[] as false for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { shortestDistances[vertexIndex] = int.MaxValue; added[vertexIndex] = false; } // Distance of source vertex from // itself is always 0 shortestDistances[startVertex] = 0; // Parent array to store shortest // path tree int[] parents = new int[nVertices]; // The starting vertex does not // have a parent parents[startVertex] = NO_PARENT; // Find shortest path for all // vertices for (int i = 1; i < nVertices; i++) { // Pick the minimum distance vertex // from the set of vertices not yet // processed. nearestVertex is // always equal to startNode in // first iteration. int nearestVertex = -1; int shortestDistance = int.MaxValue; for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { if (!added[vertexIndex] && shortestDistances[vertexIndex] < shortestDistance) { nearestVertex = vertexIndex; shortestDistance = shortestDistances[vertexIndex]; } } // Mark the picked vertex as // processed added[nearestVertex] = true; // Update dist value of the // adjacent vertices of the // picked vertex. for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { int edgeDistance = adjacencyMatrix[nearestVertex,vertexIndex]; if (edgeDistance > 0 && ((shortestDistance + edgeDistance) < shortestDistances[vertexIndex])) { parents[vertexIndex] = nearestVertex; shortestDistances[vertexIndex] = shortestDistance + edgeDistance; } } } printSolution(startVertex, shortestDistances, parents); } // A utility function to print // the constructed distances // array and shortest paths private static void printSolution(int startVertex, int[] distances, int[] parents) { int nVertices = distances.Length; Console.Write("Vertex Distance Path"); for (int vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) { if (vertexIndex != startVertex) { Console.Write("
" + startVertex + " -> "); Console.Write(vertexIndex + " "); Console.Write(distances[vertexIndex] + " "); printPath(vertexIndex, parents); } } } // Function to print shortest path // from source to currentVertex // using parents array private static void printPath(int currentVertex, int[] parents) { // Base case : Source node has // been processed if (currentVertex == NO_PARENT) { return; } printPath(parents[currentVertex], parents); Console.Write(currentVertex + " "); } // Driver Code public static void Main(String[] args) { int[,] adjacencyMatrix = { { 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(adjacencyMatrix, 0); } } // This code has been contributed by 29AjayKumar |
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
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