Printing Paths in Dijkstra's Shortest Path Algorithm

Last Updated : 23 Jul, 2025

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,

Graph Used in the problem
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)

kartik

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