Printing Paths in Dijkstra's Shortest Path Algorithm

Given a graph and a source vertex in the graph, find the shortest paths from the source to all vertices in the given graph.
We have discussed Dijkstra's Shortest Path algorithm in the below posts.

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

Example:

Input: Consider below graph and source as 0,

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

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. The parent of the root (or source vertex) is -1. Whenever we find a shorter path through a vertex u, we make u as a parent of the current vertex.

Once we have the parent array constructed, we can print the path using the 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++

#include <bits/stdc++.h>
using namespace std;
// A C++ program for Dijkstra's
// single source shortest path
// algorithm. The program is for
// adjacency matrix representation
// of the graph.

int NO_PARENT = -1;

// Function to print shortest path
// from source to currentVertex
// using parents array
void printPath(int currentVertex, vector<int> parents)
{

    // Base case : Source node has
    // been processed
    if (currentVertex == NO_PARENT) {
        return;
    }
    printPath(parents[currentVertex], parents);
    cout << currentVertex << " ";
}

// A utility function to print
// the constructed distances
// array and shortest paths
void printSolution(int startVertex, vector<int> distances,
                   vector<int> parents)
{
    int nVertices = distances.size();
    cout << "Vertex\t Distance\tPath";

    for (int vertexIndex = 0; vertexIndex < nVertices;
         vertexIndex++) {
        if (vertexIndex != startVertex) {
            cout << "\n" << startVertex << " -> ";
            cout << vertexIndex << " \t\t ";
            cout << distances[vertexIndex] << "\t\t";
            printPath(vertexIndex, parents);
        }
    }
}

// Function that implements Dijkstra's
// single source shortest path
// algorithm for a graph represented
// using adjacency matrix
// representation

void dijkstra(vector<vector<int> > adjacencyMatrix,
              int startVertex)
{
    int nVertices = adjacencyMatrix[0].size();

    // shortestDistances[i] will hold the
    // shortest distance from src to i
    vector<int> shortestDistances(nVertices);

    // added[i] will true if vertex i is
    // included / in shortest path tree
    // or shortest distance from src to
    // i is finalized
    vector<bool> added(nVertices);

    // Initialize all distances as
    // INFINITE and added[] as false
    for (int vertexIndex = 0; vertexIndex < nVertices;
         vertexIndex++) {
        shortestDistances[vertexIndex] = INT_MAX;
        added[vertexIndex] = false;
    }

    // Distance of source vertex from
    // itself is always 0
    shortestDistances[startVertex] = 0;

    // Parent array to store shortest
    // path tree
    vector<int> parents(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_MAX;
        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);
}

// Driver Code
int main()
{
    vector<vector<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, 3);
    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

Python3

import sys

NO_PARENT = -1

def dijkstra(adjacency_matrix, start_vertex):
    n_vertices = len(adjacency_matrix[0])

    # shortest_distances[i] will hold the
    # shortest distance from start_vertex to i
    shortest_distances = [sys.maxsize] * n_vertices

    # added[i] will true if vertex i is
    # included in shortest path tree
    # or shortest distance from start_vertex to 
    # i is finalized
    added = [False] * n_vertices

    # Initialize all distances as 
    # INFINITE and added[] as false
    for vertex_index in range(n_vertices):
        shortest_distances[vertex_index] = sys.maxsize
        added[vertex_index] = False
        
    # Distance of source vertex from
    # itself is always 0
    shortest_distances[start_vertex] = 0

    # Parent array to store shortest
    # path tree
    parents = [-1] * n_vertices

    # The starting vertex does not 
    # have a parent
    parents[start_vertex] = NO_PARENT

    # Find shortest path for all 
    # vertices
    for i in range(1, n_vertices):
        # Pick the minimum distance vertex
        # from the set of vertices not yet
        # processed. nearest_vertex is 
        # always equal to start_vertex in 
        # first iteration.
        nearest_vertex = -1
        shortest_distance = sys.maxsize
        for vertex_index in range(n_vertices):
            if not added[vertex_index] and shortest_distances[vertex_index] < shortest_distance:
                nearest_vertex = vertex_index
                shortest_distance = shortest_distances[vertex_index]

        # Mark the picked vertex as
        # processed
        added[nearest_vertex] = True

        # Update dist value of the
        # adjacent vertices of the
        # picked vertex.
        for vertex_index in range(n_vertices):
            edge_distance = adjacency_matrix[nearest_vertex][vertex_index]
            
            if edge_distance > 0 and shortest_distance + edge_distance < shortest_distances[vertex_index]:
                parents[vertex_index] = nearest_vertex
                shortest_distances[vertex_index] = shortest_distance + edge_distance

    print_solution(start_vertex, shortest_distances, parents)


# A utility function to print 
# the constructed distances
# array and shortest paths
def print_solution(start_vertex, distances, parents):
    n_vertices = len(distances)
    print("Vertex\t Distance\tPath")
    
    for vertex_index in range(n_vertices):
        if vertex_index != start_vertex:
            print("\n", start_vertex, "->", vertex_index, "\t\t", distances[vertex_index], "\t\t", end="")
            print_path(vertex_index, parents)


# Function to print shortest path
# from source to current_vertex
# using parents array
def print_path(current_vertex, parents):
    # Base case : Source node has
    # been processed
    if current_vertex == NO_PARENT:
        return
    print_path(parents[current_vertex], parents)
    print(current_vertex, end=" ")


# Driver code
if __name__ == '__main__':
    adjacency_matrix = [[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(adjacency_matrix, 0)

// 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

const NO_PARENT = -1;

function dijkstra(adjacencyMatrix, startVertex) {
  const nVertices = adjacencyMatrix[0].length;

  // shortestDistances[i] will hold the shortest distance from startVertex to i
  const shortestDistances = new Array(nVertices).fill(Number.MAX_SAFE_INTEGER);

  // added[i] will true if vertex i is included in shortest path tree
  // or shortest distance from startVertex to i is finalized
  const added = new Array(nVertices).fill(false);

  // Initialize all distances as infinite and added[] as false
  for (let vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) {
    shortestDistances[vertexIndex] = Number.MAX_SAFE_INTEGER;
    added[vertexIndex] = false;
  }

  // Distance of source vertex from itself is always 0
  shortestDistances[startVertex] = 0;

  // Parent array to store shortest path tree
  const parents = new Array(nVertices).fill(NO_PARENT);

  // 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 startVertex in first iteration.
    let nearestVertex = -1;
    let shortestDistance = Number.MAX_SAFE_INTEGER;

    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++) {
      const 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
function printSolution(startVertex, distances, parents) {
  const nVertices = distances.length;
  console.log("Vertex\t Distance\tPath");

  for (let vertexIndex = 0; vertexIndex < nVertices; vertexIndex++) {
    if (vertexIndex !== startVertex) {
      process.stdout.write(`\n ${startVertex} -> ${vertexIndex}\t\t ${distances[vertexIndex]}\t\t`);
      printPath(vertexIndex, parents);
    }
  }
}

// Function to print shortest path from source to currentVertex using parents array
function printPath(currentVertex, parents) {
  // Base case: Source node has been processed
  if (currentVertex === NO_PARENT) {
    return;
  }

  printPath(parents[currentVertex], parents);
  process.stdout.write(`${currentVertex} `);
}

// Driver code

const 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, 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(adjacencyMatrix, 0);

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

Time Complexity:- O(V^2)
Space Complexity:- O(V^2)

Printing Paths in Dijkstra's Shortest Path Algorithm

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