How to find Shortest Paths from Source to all Vertices using Dijkstra’s Algorithm
Last Updated :
06 Aug, 2024
Given a weighted graph and a source vertex in the graph, find the shortest paths from the source to all the other vertices in the given graph.
Note: The given graph does not contain any negative edge.
Examples:
Input: src = 0, the graph is shown below.
Output: 0 4 12 19 21 11 9 8 14
Explanation: The distance from 0 to 1 = 4.
The minimum distance from 0 to 2 = 12. 0->1->2
The minimum distance from 0 to 3 = 19. 0->1->2->3
The minimum distance from 0 to 4 = 21. 0->7->6->5->4
The minimum distance from 0 to 5 = 11. 0->7->6->5
The minimum distance from 0 to 6 = 9. 0->7->6
The minimum distance from 0 to 7 = 8. 0->7
The minimum distance from 0 to 8 = 14. 0->1->2->8
The idea is to generate a SPT (shortest path tree) with a given source as a root. Maintain an Adjacency Matrix with two sets,
- one set contains vertices included in the shortest-path tree,
- other set includes vertices not yet included in the shortest-path tree.
At every step of the algorithm, find a vertex that is in the other set (set not yet included) and has a minimum distance from the source.
Algorithm :
- Create a set sptSet (shortest path tree set) that keeps track of vertices included in the shortest path tree, i.e., whose minimum distance from the source is calculated and finalized. Initially, this set is empty.
- Assign a distance value to all vertices in the input graph. Initialize all distance values as INFINITE . Assign the distance value as 0 for the source vertex so that it is picked first.
- While sptSet doesn’t include all vertices
- Pick a vertex u that is not there in sptSet and has a minimum distance value.
- Include u to sptSet .
- Then update the distance value of all adjacent vertices of u .
- To update the distance values, iterate through all adjacent vertices.
- For every adjacent vertex v, if the sum of the distance value of u (from source) and weight of edge u-v , is less than the distance value of v , then update the distance value of v .
Note: We use a boolean array sptSet[] to represent the set of vertices included in SPT . If a value sptSet[v] is true, then vertex v is included in SPT , otherwise not. Array dist[] is used to store the shortest distance values of all vertices.
Illustration of Dijkstra Algorithm :
To understand the Dijkstra’s Algorithm lets take a graph and find the shortest path from source to all nodes.
Consider below graph and src = 0
Step 1:
- The set sptSet is initially empty and distances assigned to vertices are {0, INF, INF, INF, INF, INF, INF, INF} where INF indicates infinite.
- Now pick the vertex with a minimum distance value. The vertex 0 is picked, include it in sptSet . So sptSet becomes {0}. After including 0 to sptSet , update distance values of its adjacent vertices.
- Adjacent vertices of 0 are 1 and 7. The distance values of 1 and 7 are updated as 4 and 8.
The following subgraph shows vertices and their distance values, only the vertices with finite distance values are shown. The vertices included in SPT are shown in green colour.
Step 2:
- Pick the vertex with minimum distance value and not already included in SPT (not in sptSET ). The vertex 1 is picked and added to sptSet .
- So sptSet now becomes {0, 1}. Update the distance values of adjacent vertices of 1.
- The distance value of vertex 2 becomes 12 .
Step 3:
- Pick the vertex with minimum distance value and not already included in SPT (not in sptSET ). Vertex 7 is picked. So sptSet now becomes {0, 1, 7}.
- Update the distance values of adjacent vertices of 7. The distance value of vertex 6 and 8 becomes finite ( 15 and 9 respectively).
Step 4:
- Pick the vertex with minimum distance value and not already included in SPT (not in sptSET ). Vertex 6 is picked. So sptSet now becomes {0, 1, 7, 6} .
- Update the distance values of adjacent vertices of 6. The distance value of vertex 5 and 8 are updated.
.jpg)
We repeat the above steps until sptSet includes all vertices of the given graph. Finally, we get the following S hortest Path Tree (SPT).
.jpg)
Below is the implementation of the above approach:
C++
// C++ program for Dijkstra's single source shortest path
// algorithm. The program is for adjacency matrix
// representation of the graph
#include <iostream>
using namespace std;
#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;
}
// A utility function to print the constructed distance
// array
void printSolution(int dist[])
{
cout << "Vertex \t Distance from Source" << endl;
for (int i = 0; i < V; i++)
cout << i << " \t\t\t\t" << dist[i] << endl;
}
// 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)
{
int dist[V]; // The output array. dist[i] will hold the
// shortest
// distance from src to i
bool sptSet[V]; // sptSet[i] will be true if vertex i is
// included in shortest
// path tree or shortest distance from src to i is
// finalized
// Initialize all distances as INFINITE and stpSet[] as
// false
for (int i = 0; i < V; i++)
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 the 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] != INT_MAX
&& dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
// print the constructed distance array
printSolution(dist);
}
// driver's 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, 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 } };
// Function call
dijkstra(graph, 0);
return 0;
}
// This code is contributed by shivanisinghss2110
// C program for Dijkstra's single source shortest path
// algorithm. The program is for adjacency matrix
// representation of the graph
#include <limits.h>
#include <stdbool.h>
#include <stdio.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;
}
// A utility function to print the constructed distance
// array
void printSolution(int dist[])
{
printf("Vertex \t\t Distance from Source\n");
for (int i = 0; i < V; i++)
printf("%d \t\t\t\t %d\n", i, dist[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)
{
int dist[V]; // The output array. dist[i] will hold the
// shortest
// distance from src to i
bool sptSet[V]; // sptSet[i] will be true if vertex i is
// included in shortest
// path tree or shortest distance from src to i is
// finalized
// Initialize all distances as INFINITE and stpSet[] as
// false
for (int i = 0; i < V; i++)
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 the 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] != INT_MAX
&& dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
// print the constructed distance array
printSolution(dist);
}
// driver's 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, 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 } };
// Function call
dijkstra(graph, 0);
return 0;
}
// A Java program for Dijkstra's single source shortest path
// algorithm. The program is for adjacency matrix
// representation of the graph
import java.io.*;
import java.lang.*;
import java.util.*;
class ShortestPath {
// A utility function to find the vertex with minimum
// distance value, from the set of vertices not yet
// included in shortest path tree
static final int V = 9;
int minDistance(int dist[], Boolean sptSet[])
{
// Initialize min value
int min = Integer.MAX_VALUE, min_index = -1;
for (int v = 0; v < V; v++)
if (sptSet[v] == false && dist[v] <= min) {
min = dist[v];
min_index = v;
}
return min_index;
}
// A utility function to print the constructed distance
// array
void printSolution(int dist[])
{
System.out.println(
"Vertex \t\t Distance from Source");
for (int i = 0; i < V; i++)
System.out.println(i + " \t\t " + dist[i]);
}
// Function that implements Dijkstra's single source
// shortest path algorithm for a graph represented using
// adjacency matrix representation
void dijkstra(int graph[][], int src)
{
int dist[] = new int[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
Boolean sptSet[] = new Boolean[V];
// Initialize all distances as INFINITE and stpSet[]
// as false
for (int i = 0; i < V; i++) {
dist[i] = Integer.MAX_VALUE;
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] != 0
&& dist[u] != Integer.MAX_VALUE
&& dist[u] + graph[u][v] < dist[v])
dist[v] = dist[u] + graph[u][v];
}
// print the constructed distance array
printSolution(dist);
}
// Driver's code
public static void main(String[] args)
{
/* Let us create the example graph discussed above
*/
int graph[][]
= new int[][] { { 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 } };
ShortestPath t = new ShortestPath();
// Function call
t.dijkstra(graph, 0);
}
}
// This code is contributed by Aakash Hasija
# Python program for Dijkstra's single
# source shortest path algorithm. The program is
# for adjacency matrix representation of the graph
# Library for INT_MAX
import sys
class Graph():
def __init__(self, vertices):
self.V = vertices
self.graph = [[0 for column in range(vertices)]
for row in range(vertices)]
def printSolution(self, dist):
print("Vertex \tDistance from Source")
for node in range(self.V):
print(node, "\t", dist[node])
# A utility function to find the vertex with
# minimum distance value, from the set of vertices
# not yet included in shortest path tree
def minDistance(self, dist, sptSet):
# Initialize minimum distance for next node
min = sys.maxsize
# Search not nearest vertex not in the
# shortest path tree
for u in range(self.V):
if dist[u] < min and sptSet[u] == False:
min = dist[u]
min_index = u
return min_index
# Function that implements Dijkstra's single source
# shortest path algorithm for a graph represented
# using adjacency matrix representation
def dijkstra(self, src):
dist = [sys.maxsize] * self.V
dist[src] = 0
sptSet = [False] * self.V
for cout in range(self.V):
# Pick the minimum distance vertex from
# the set of vertices not yet processed.
# x is always equal to src in first iteration
x = self.minDistance(dist, sptSet)
# Put the minimum distance vertex in the
# shortest path tree
sptSet[x] = True
# Update dist value of the adjacent vertices
# of the picked vertex only if the current
# distance is greater than new distance and
# the vertex in not in the shortest path tree
for y in range(self.V):
if self.graph[x][y] > 0 and sptSet[y] == False and \
dist[y] > dist[x] + self.graph[x][y]:
dist[y] = dist[x] + self.graph[x][y]
self.printSolution(dist)
# Driver's code
if __name__ == "__main__":
g = Graph(9)
g.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]
]
g.dijkstra(0)
# This code is contributed by Divyanshu Mehta and Updated by Pranav Singh Sambyal
// C# program for Dijkstra's single
// source shortest path algorithm.
// The program is for adjacency matrix
// representation of the graph
using System;
class GFG {
// A utility function to find the
// vertex with minimum distance
// value, from the set of vertices
// not yet included in shortest
// path tree
static int V = 9;
int minDistance(int[] dist, bool[] sptSet)
{
// Initialize min value
int min = int.MaxValue, min_index = -1;
for (int v = 0; v < V; v++)
if (sptSet[v] == false && dist[v] <= min) {
min = dist[v];
min_index = v;
}
return min_index;
}
// A utility function to print
// the constructed distance array
void printSolution(int[] dist)
{
Console.Write("Vertex \t\t Distance "
+ "from Source\n");
for (int i = 0; i < V; i++)
Console.Write(i + " \t\t " + dist[i] + "\n");
}
// Function that implements Dijkstra's
// single source shortest path algorithm
// for a graph represented using adjacency
// matrix representation
void dijkstra(int[, ] graph, int src)
{
int[] dist
= new int[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 = new bool[V];
// Initialize all distances as
// INFINITE and stpSet[] as false
for (int i = 0; i < V; i++) {
dist[i] = int.MaxValue;
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] != 0
&& dist[u] != int.MaxValue
&& dist[u] + graph[u, v] < dist[v])
dist[v] = dist[u] + graph[u, v];
}
// print the constructed distance array
printSolution(dist);
}
// Driver's Code
public static void Main()
{
/* Let us create the example
graph discussed above */
int[, ] graph
= new int[, ] { { 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 } };
GFG t = new GFG();
// Function call
t.dijkstra(graph, 0);
}
}
// This code is contributed by ChitraNayal
// A Javascript program for Dijkstra's single
// source shortest path algorithm.
// The program is for adjacency matrix
// representation of the graph
let 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
function minDistance(dist,sptSet)
{
// Initialize min value
let min = Number.MAX_VALUE;
let min_index = -1;
for(let v = 0; v < V; v++)
{
if (sptSet[v] == false && dist[v] <= min)
{
min = dist[v];
min_index = v;
}
}
return min_index;
}
// A utility function to print
// the constructed distance array
function printSolution(dist)
{
console.log("Vertex \t\t Distance from Source<br>");
for(let i = 0; i < V; i++)
{
console.log(i + " \t\t " +
dist[i] + "<br>");
}
}
// Function that implements Dijkstra's
// single source shortest path algorithm
// for a graph represented using adjacency
// matrix representation
function dijkstra(graph, src)
{
let dist = new Array(V);
let sptSet = new Array(V);
// Initialize all distances as
// INFINITE and stpSet[] as false
for(let i = 0; i < V; i++)
{
dist[i] = Number.MAX_VALUE;
sptSet[i] = false;
}
// Distance of source vertex
// from itself is always 0
dist[src] = 0;
// Find shortest path for all vertices
for(let 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.
let 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(let 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] != 0 &&
dist[u] != Number.MAX_VALUE &&
dist[u] + graph[u][v] < dist[v])
{
dist[v] = dist[u] + graph[u][v];
}
}
}
// Print the constructed distance array
printSolution(dist);
}
// Driver code
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
OutputVertex Distance from Source
0 0
1 4
2 12
3 19
4 21
5 11
6 9
7 8
8 14
Time Complexity: O(V 2 )
Auxiliary Space: O(V)
Notes:
- The code calculates the shortest distance but doesn’t calculate the path information. Create a parent array, update the parent array when distance is updated and use it to show the shortest path from source to different vertices.
- The time Complexity of the implementation is O(V 2 ) . If the input graph is represented using adjacency list , it can be reduced to O(E * log V) with the help of a binary heap. Please see Dijkstra’s Algorithm for Adjacency List Representation for more details.
- Dijkstra’s algorithm doesn’t work for graphs with negative weight cycles.
Why Dijkstra’s Algorithms fails for the Graphs having Negative Edges ?
The problem with negative weights arises from the fact that Dijkstra’s algorithm assumes that once a node is added to the set of visited nodes, its distance is finalized and will not change. However, in the presence of negative weights, this assumption can lead to incorrect results.
Consider the following graph for the example:
In the above graph, A is the source node, among the edges A to B and A to C , A to B is the smaller weight and Dijkstra assigns the shortest distance of B as 2, but because of existence of a negative edge from C to B , the actual shortest distance reduces to 1 which Dijkstra fails to detect.
Note: We use Bellman Ford’s Shortest path algorithm in case we have negative edges in the graph.
Dijkstra’s Algorithm using Adjacency List in O(E logV):
For Dijkstra’s algorithm, it is always recommended to use Heap (or priority queue ) as the required operations (extract minimum and decrease key) match with the speciality of the heap (or priority queue). However, the problem is, that priority_queue doesn’t support the decrease key. To resolve this problem, do not update a key, but insert one more copy of it. So we allow multiple instances of the same vertex in the priority queue. This approach doesn’t require decreasing key operations and has below important properties.
- Whenever the distance of a vertex is reduced, we add one more instance of a vertex in priority_queue. Even if there are multiple instances, we only consider the instance with minimum distance and ignore other instances.
- The time complexity remains O(E * LogV) as there will be at most O(E) vertices in the priority queue and O(logE) is the same as O(logV)
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
// Define INF as a large value to represent infinity
#define INF 0x3f3f3f3f
// iPair ==> Integer Pair
typedef pair<int, int> iPair;
// Class representing a graph using adjacency list representation
class Graph {
int V; // Number of vertices
list<iPair> *adj; // Adjacency list
public:
Graph(int V); // Constructor
void addEdge(int u, int v, int w); // Function to add an edge
void shortestPath(int s); // Function to print shortest path from source
};
// Constructor to allocate memory for the adjacency list
Graph::Graph(int V) {
this->V = V;
adj = new list<iPair>[V];
}
// Function to add an edge to the graph
void Graph::addEdge(int u, int v, int w) {
adj[u].push_back(make_pair(v, w));
adj[v].push_back(make_pair(u, w)); // Since the graph is undirected
}
// Function to print shortest paths from source
void Graph::shortestPath(int src) {
// Create a priority queue to store vertices being processed
// Priority queue sorted by the first element of the pair (distance)
priority_queue<iPair, vector<iPair>, greater<iPair>> pq;
// Create a vector to store distances and initialize all distances as INF
vector<int> dist(V, INF);
// Insert source into priority queue and initialize its distance as 0
pq.push(make_pair(0, src));
dist[src] = 0;
// Process the priority queue
while (!pq.empty()) {
// Get the vertex with the minimum distance
int u = pq.top().second;
pq.pop();
// Iterate through all adjacent vertices of the current vertex
for (auto &neighbor : adj[u]) {
int v = neighbor.first;
int weight = neighbor.second;
// If a shorter path to v is found
if (dist[v] > dist[u] + weight) {
// Update distance and push new distance to the priority queue
dist[v] = dist[u] + weight;
pq.push(make_pair(dist[v], v));
}
}
}
// Print the shortest distances
cout << "Vertex Distance from Source" << endl;
for (int i = 0; i < V; ++i)
cout << i << " \t\t " << dist[i] << endl;
}
// Driver's code
int main() {
int V = 9; // Number of vertices
Graph g(V);
// Add edges to the graph
g.addEdge(0, 1, 4);
g.addEdge(0, 7, 8);
g.addEdge(1, 2, 8);
g.addEdge(1, 7, 11);
g.addEdge(2, 3, 7);
g.addEdge(2, 8, 2);
g.addEdge(2, 5, 4);
g.addEdge(3, 4, 9);
g.addEdge(3, 5, 14);
g.addEdge(4, 5, 10);
g.addEdge(5, 6, 2);
g.addEdge(6, 7, 1);
g.addEdge(6, 8, 6);
g.addEdge(7, 8, 7);
// Call the shortestPath function
g.shortestPath(0);
return 0;
}
import java.util.*;
class Graph {
private int V;
private List<List<iPair>> adj;
Graph(int V) {
this.V = V;
adj = new ArrayList<>();
for (int i = 0; i < V; i++) {
adj.add(new ArrayList<>());
}
}
void addEdge(int u, int v, int w) {
adj.get(u).add(new iPair(v, w));
adj.get(v).add(new iPair(u, w));
}
void shortestPath(int src) {
PriorityQueue<iPair> pq = new PriorityQueue<>(V, Comparator.comparingInt(o -> o.second));
int[] dist = new int[V];
Arrays.fill(dist, Integer.MAX_VALUE);
pq.add(new iPair(0, src));
dist[src] = 0;
while (!pq.isEmpty()) {
int u = pq.poll().second;
for (iPair v : adj.get(u)) {
if (dist[v.first] > dist[u] + v.second) {
dist[v.first] = dist[u] + v.second;
pq.add(new iPair(dist[v.first], v.first));
}
}
}
System.out.println("Vertex Distance from Source");
for (int i = 0; i < V; i++) {
System.out.println(i + "\t\t" + dist[i]);
}
}
static class iPair {
int first, second;
iPair(int first, int second) {
this.first = first;
this.second = second;
}
}
}
public class Main {
public static void main(String[] args) {
int V = 9;
Graph g = new Graph(V);
g.addEdge(0, 1, 4);
g.addEdge(0, 7, 8);
g.addEdge(1, 2, 8);
g.addEdge(1, 7, 11);
g.addEdge(2, 3, 7);
g.addEdge(2, 8, 2);
g.addEdge(2, 5, 4);
g.addEdge(3, 4, 9);
g.addEdge(3, 5, 14);
g.addEdge(4, 5, 10);
g.addEdge(5, 6, 2);
g.addEdge(6, 7, 1);
g.addEdge(6, 8, 6);
g.addEdge(7, 8, 7);
g.shortestPath(0);
}
}
import heapq
# iPair ==> Integer Pair
iPair = tuple
# This class represents a directed graph using
# adjacency list representation
class Graph:
def __init__(self, V: int): # Constructor
self.V = V
self.adj = [[] for _ in range(V)]
def addEdge(self, u: int, v: int, w: int):
self.adj[u].append((v, w))
self.adj[v].append((u, w))
# Prints shortest paths from src to all other vertices
def shortestPath(self, src: int):
# Create a priority queue to store vertices that
# are being preprocessed
pq = []
heapq.heappush(pq, (0, src))
# Create a vector for distances and initialize all
# distances as infinite (INF)
dist = [float('inf')] * self.V
dist[src] = 0
while pq:
# The first vertex in pair is the minimum distance
# vertex, extract it from priority queue.
# vertex label is stored in second of pair
d, u = heapq.heappop(pq)
# 'i' is used to get all adjacent vertices of a
# vertex
for v, weight in self.adj[u]:
# If there is shorted path to v through u.
if dist[v] > dist[u] + weight:
# Updating distance of v
dist[v] = dist[u] + weight
heapq.heappush(pq, (dist[v], v))
# Print shortest distances stored in dist[]
for i in range(self.V):
print(f"{i} \t\t {dist[i]}")
# Driver's code
if __name__ == "__main__":
# create the graph given in above figure
V = 9
g = Graph(V)
# making above shown graph
g.addEdge(0, 1, 4)
g.addEdge(0, 7, 8)
g.addEdge(1, 2, 8)
g.addEdge(1, 7, 11)
g.addEdge(2, 3, 7)
g.addEdge(2, 8, 2)
g.addEdge(2, 5, 4)
g.addEdge(3, 4, 9)
g.addEdge(3, 5, 14)
g.addEdge(4, 5, 10)
g.addEdge(5, 6, 2)
g.addEdge(6, 7, 1)
g.addEdge(6, 8, 6)
g.addEdge(7, 8, 7)
g.shortestPath(0)
using System;
using System.Collections.Generic;
// This class represents a directed graph using
// adjacency list representation
public class Graph
{
private const int INF = 2147483647;
private int V;
private List<int[]>[] adj;
public Graph(int V)
{
// No. of vertices
this.V = V;
// In a weighted graph, we need to store vertex
// and weight pair for every edge
this.adj = new List<int[]>[V];
for (int i = 0; i < V; i++)
{
this.adj[i] = new List<int[]>();
}
}
public void AddEdge(int u, int v, int w)
{
this.adj[u].Add(new int[] { v, w });
this.adj[v].Add(new int[] { u, w });
}
// Prints shortest paths from src to all other vertices
public void ShortestPath(int src)
{
// Create a priority queue to store vertices that
// are being preprocessed.
SortedSet<int[]> pq = new SortedSet<int[]>(new DistanceComparer());
// Create an array for distances and initialize all
// distances as infinite (INF)
int[] dist = new int[V];
for (int i = 0; i < V; i++)
{
dist[i] = INF;
}
// Insert source itself in priority queue and initialize
// its distance as 0.
pq.Add(new int[] { 0, src });
dist[src] = 0;
/* Looping till priority queue becomes empty (or all
distances are not finalized) */
while (pq.Count > 0)
{
// The first vertex in pair is the minimum distance
// vertex, extract it from priority queue.
// vertex label is stored in second of pair (it
// has to be done this way to keep the vertices
// sorted by distance)
int[] minDistVertex = pq.Min;
pq.Remove(minDistVertex);
int u = minDistVertex[1];
// 'i' is used to get all adjacent vertices of a
// vertex
foreach (int[] adjVertex in this.adj[u])
{
// Get vertex label and weight of current
// adjacent of u.
int v = adjVertex[0];
int weight = adjVertex[1];
// If there is a shorter path to v through u.
if (dist[v] > dist[u] + weight)
{
// Updating distance of v
dist[v] = dist[u] + weight;
pq.Add(new int[] { dist[v], v });
}
}
}
// Print shortest distances stored in dist[]
Console.WriteLine("Vertex Distance from Source");
for (int i = 0; i < V; ++i)
Console.WriteLine(i + "\t" + dist[i]);
}
private class DistanceComparer : IComparer<int[]>
{
public int Compare(int[] x, int[] y)
{
if (x[0] == y[0])
{
return x[1] - y[1];
}
return x[0] - y[0];
}
}
}
public class Program
{
// Driver Code
public static void Main()
{
// create the graph given in above figure
int V = 9;
Graph g = new Graph(V);
// making above shown graph
g.AddEdge(0, 1, 4);
g.AddEdge(0, 7, 8);
g.AddEdge(1, 2, 8);
g.AddEdge(1, 7, 11);
g.AddEdge(2, 3, 7);
g.AddEdge(2, 8, 2);
g.AddEdge(2, 5, 4);
g.AddEdge(3, 4, 9);
g.AddEdge(3, 5, 14);
g.AddEdge(4, 5, 10);
g.AddEdge(5, 6, 2);
g.AddEdge(6, 7, 1);
g.AddEdge(6, 8, 6);
g.AddEdge(7, 8, 7);
g.ShortestPath(0);
}
}
// this code is contributed by bhardwajji
// javascript Program to find Dijkstra's shortest path using
// priority_queue in STL
const INF = 2147483647;
// This class represents a directed graph using
// adjacency list representation
class Graph {
constructor(V){
// No. of vertices
this.V = V;
// In a weighted graph, we need to store vertex
// and weight pair for every edge
this.adj = new Array(V);
for(let i = 0; i < V; i++){
this.adj[i] = new Array();
}
}
addEdge(u, v, w)
{
this.adj[u].push([v, w]);
this.adj[v].push([u, w]);
}
// Prints shortest paths from src to all other vertices
shortestPath(src)
{
// Create a priority queue to store vertices that
// are being preprocessed. This is weird syntax in C++.
// Refer below link for details of this syntax
// https://www.geeksforgeeks.org/implement-min-heap-using-stl/
let pq = [];
// Create a vector for distances and initialize all
// distances as infinite (INF)
let dist = new Array(V).fill(INF);
// Insert source itself in priority queue and initialize
// its distance as 0.
pq.push([0, src]);
dist[src] = 0;
/* Looping till priority queue becomes empty (or all
distances are not finalized) */
while (pq.length > 0) {
// The first vertex in pair is the minimum distance
// vertex, extract it from priority queue.
// vertex label is stored in second of pair (it
// has to be done this way to keep the vertices
// sorted distance (distance must be first item
// in pair)
let u = pq[0][1];
pq.shift();
// 'i' is used to get all adjacent vertices of a
// vertex
for(let i = 0; i < this.adj[u].length; i++){
// Get vertex label and weight of current
// adjacent of u.
let v = this.adj[u][i][0];
let weight = this.adj[u][i][1];
// If there is shorted path to v through u.
if (dist[v] > dist[u] + weight) {
// Updating distance of v
dist[v] = dist[u] + weight;
pq.push([dist[v], v]);
pq.sort((a, b) =>{
if(a[0] == b[0]) return a[1] - b[1];
return a[0] - b[0];
});
}
}
}
// Print shortest distances stored in dist[]
console.log("Vertex Distance from Source");
for (let i = 0; i < V; ++i)
console.log(i, " ", dist[i]);
}
}
// Driver's code
// create the graph given in above figure
let V = 9;
let g = new Graph(V);
// making above shown graph
g.addEdge(0, 1, 4);
g.addEdge(0, 7, 8);
g.addEdge(1, 2, 8);
g.addEdge(1, 7, 11);
g.addEdge(2, 3, 7);
g.addEdge(2, 8, 2);
g.addEdge(2, 5, 4);
g.addEdge(3, 4, 9);
g.addEdge(3, 5, 14);
g.addEdge(4, 5, 10);
g.addEdge(5, 6, 2);
g.addEdge(6, 7, 1);
g.addEdge(6, 8, 6);
g.addEdge(7, 8, 7);
// Function call
g.shortestPath(0);
// The code is contributed by Nidhi goel.
OutputVertex Distance from Source
0 0
1 4
2 12
3 19
4 21
5 11
6 9
7 8
8 14
Time Complexity: O(E * logV), Where E is the number of edges and V is the number of vertices.
Auxiliary Space: O(V)
Applications of Dijkstra’s Algorithm:
- Google maps uses Dijkstra algorithm to show shortest distance between source and destination.
- In computer networking , Dijkstra’s algorithm forms the basis for various routing protocols, such as OSPF (Open Shortest Path First) and IS-IS (Intermediate System to Intermediate System).
- Transportation and traffic management systems use Dijkstra’s algorithm to optimize traffic flow, minimize congestion, and plan the most efficient routes for vehicles.
- Airlines use Dijkstra’s algorithm to plan flight paths that minimize fuel consumption, reduce travel time.
- Dijkstra’s algorithm is applied in electronic design automation for routing connections on integrated circuits and very-large-scale integration (VLSI) chips.
For a more detailed explanation refer to this article Dijkstra’s Shortest Path Algorithm using priority_queue of STL .