Printing Paths in Dijkstra’s Shortest Path Algorithm

Difficulty Level : Medium
Last Updated : 14 Sep, 2021

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.

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 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);
    }
}
   
// Function 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\t Distance\tPath");
         
        for (int vertexIndex = 0;
                 vertexIndex < nVertices;
                 vertexIndex++)
        {
            if (vertexIndex != startVertex)
            {
                System.out.print("\n" + startVertex + " -> ");
                System.out.print(vertexIndex + " \t\t ");
                System.out.print(distances[vertexIndex] + "\t\t");
                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 \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

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\t Distance\tPath");
         
        for (int vertexIndex = 0;
                vertexIndex < nVertices;
                vertexIndex++)
        {
            if (vertexIndex != startVertex)
            {
                Console.Write("\n" + startVertex + " -> ");
                Console.Write(vertexIndex + " \t\t ");
                Console.Write(distances[vertexIndex] + "\t\t");
                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

Javascript

<script>
// A Javascript program for Dijkstra's
// single source shortest path
// algorithm. The program is for
// adjacency matrix representation
// of the graph.
let NO_PARENT = -1;
 
function dijkstra(adjacencyMatrix,startVertex)
{
    let nVertices = adjacencyMatrix[0].length;
   
        // shortestDistances[i] will hold the
        // shortest distance from src to i
        let shortestDistances = new Array(nVertices);
   
        // added[i] will true if vertex i is
        // included / in shortest path tree
        // or shortest distance from src to
        // i is finalized
        let added = new Array(nVertices);
   
        // Initialize all distances as
        // INFINITE and added[] as false
        for (let vertexIndex = 0; vertexIndex < nVertices;
                                            vertexIndex++)
        {
            shortestDistances[vertexIndex] = Number.MAX_VALUE;
            added[vertexIndex] = false;
        }
           
        // Distance of source vertex from
        // itself is always 0
        shortestDistances[startVertex] = 0;
   
        // Parent array to store shortest
        // path tree
        let parents = new Array(nVertices);
   
        // The starting vertex does not
        // have a parent
        parents[startVertex] = NO_PARENT;
   
        // Find shortest path for all
        // vertices
        for (let 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.
            let nearestVertex = -1;
            let shortestDistance = Number.MAX_VALUE;
            for (let 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 (let vertexIndex = 0;
                     vertexIndex < nVertices;
                     vertexIndex++)
            {
                let edgeDistance = adjacencyMatrix[nearestVertex][vertexIndex];
                   
                if (edgeDistance > 0
                    && ((shortestDistance + edgeDistance) <
                        shortestDistances[vertexIndex]))
                {
                    parents[vertexIndex] = nearestVertex;
                    shortestDistances[vertexIndex] = shortestDistance +
                                                       edgeDistance;
                }
            }
        }
   
        printSolution(startVertex, shortestDistances, parents);
}
 
function printSolution(startVertex,distances,parents)
{
     let nVertices = distances.length;
        document.write("Vertex     Distance   Path");
           
        for (let vertexIndex = 0;
                 vertexIndex < nVertices;
                 vertexIndex++)
        {
            if (vertexIndex != startVertex)
            {
                document.write("<br>" + startVertex + " -> ");
                document.write(vertexIndex + "        ");
                document.write(distances[vertexIndex] + "      ");
                printPath(vertexIndex, parents);
            }
        }
}
 
function printPath(currentVertex,parents)
{
     // Base case : Source node has
        // been processed
        if (currentVertex == NO_PARENT)
        {
            return;
        }
        printPath(parents[currentVertex], parents);
        document.write(currentVertex + " ");
}
 
 
let 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]
        ];
dijkstra(graph,0)   
 
// This code is contributed by rag2127.
</script>

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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