Given a graph and a source vertex in the graph, find the shortest paths from the source to all vertices in the given graph.
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
Dijkstra shortest path algorithm for Adjacency Matrix in O(V2):
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.
Follow the steps below to solve the problem:
- 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.
Below is the illustration of the above approach:
Illustration:
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.
We repeat the above steps until sptSet includes all vertices of the given graph. Finally, we get the following Shortest Path Tree (SPT).
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 <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 treeint 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// arrayvoid 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 representationvoid 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 codeint 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;} |
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 treeint 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// arrayvoid 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 representationvoid 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 codeint 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 |
Java
// A Java program for Dijkstra's single source shortest path// algorithm. The program is for adjacency matrix// representation of the graphimport 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
# Python program for Dijkstra's single# source shortest path algorithm. The program is# for adjacency matrix representation of the graph# Library for INT_MAXimport sysclass 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 codeif __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#
// C# program for Dijkstra's single// source shortest path algorithm.// The program is for adjacency matrix// representation of the graphusing 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 examplegraph 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 |
Javascript
// 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){ document.write("Vertex \t\t Distance from Source<br>"); for(let i = 0; i < V; i++) { document.write(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 codelet 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 |
Output
Vertex Distance from Source 0 0 1 4 2 12 3 19 4 21 5 11 6 9 7 8 8 14
Time Complexity: O(V2)
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 code is for undirected graphs, the same Dijkstra function can be used for directed graphs also.
- The code finds the shortest distances from the source to all vertices. If we are interested only in the shortest distance from the source to a single target, break them for a loop when the picked minimum distance vertex is equal to the target.
- The time Complexity of the implementation is O(V2). 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. It may give correct results for a graph with negative edges but you must allow a vertex can be visited multiple times and that version will lose its fast time complexity. For graphs with negative weight edges and cycles, the Bellman-Ford algorithm can be used, we will soon be discussing it as a separate post.
Dijkstra’s shortest path algorithm for Adjacency List using Heap 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 specialty 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++
// C++ Program to find Dijkstra's shortest path using// priority_queue in STL#include <bits/stdc++.h>using namespace std;#define INF 0x3f3f3f3f// iPair ==> Integer Pairtypedef pair<int, int> iPair;// This class represents a directed graph using// adjacency list representationclass Graph { int V; // No. of vertices // In a weighted graph, we need to store vertex // and weight pair for every edge list<pair<int, int> >* adj;public: Graph(int V); // Constructor // function to add an edge to graph void addEdge(int u, int v, int w); // prints shortest path from s void shortestPath(int s);};// Allocates memory for adjacency listGraph::Graph(int V){ this->V = V; adj = new list<iPair>[V];}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));}// Prints shortest paths from src to all other verticesvoid Graph::shortestPath(int 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 priority_queue<iPair, vector<iPair>, greater<iPair> > pq; // Create a vector for distances and initialize all // distances as infinite (INF) vector<int> dist(V, INF); // Insert source itself in priority queue and initialize // its distance as 0. pq.push(make_pair(0, src)); dist[src] = 0; /* Looping till priority queue becomes empty (or all distances are not finalized) */ while (!pq.empty()) { // 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) int u = pq.top().second; pq.pop(); // 'i' is used to get all adjacent vertices of a // vertex list<pair<int, int> >::iterator i; for (i = adj[u].begin(); i != adj[u].end(); ++i) { // Get vertex label and weight of current // adjacent of u. int v = (*i).first; int weight = (*i).second; // 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(make_pair(dist[v], v)); } } } // Print shortest distances stored in dist[] printf("Vertex Distance from Source\n"); for (int i = 0; i < V; ++i) printf("%d \t\t %d\n", i, dist[i]);}// Driver's codeint main(){ // create the graph given in above figure int V = 9; Graph g(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); return 0;} |
Java
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.first)); 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); }} |
Python3
import heapq# iPair ==> Integer PairiPair = tuple# This class represents a directed graph using# adjacency list representationclass 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 codeif __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) |
C#
using System;using System.Collections.Generic;// This class represents a directed graph using// adjacency list representationpublic 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
<script> // javascript Program to find Dijkstra's shortest path using// priority_queue in STLconst INF = 2147483647;// This class represents a directed graph using// adjacency list representationclass 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 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[] document.write("Vertex Distance from Source"); for (let i = 0; i < V; ++i) document.write(i, " ", dist[i]); }}// Driver's code// create the graph given in above figurelet V = 9;let g = new Graph(V);// making above shown graphg.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 callg.shortestPath(0);// The code is contributed by Nidhi goel. </script> |
Output
Vertex 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)
For a more detailed explanation refer to this article Dijkstra’s Shortest Path Algorithm using priority_queue of STL.
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