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Dijkstra's Algorithm to find Shortest Paths from a Source to all
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Dijkstra's Algorithm to find Shortest Paths from a Source to all

Last Updated : 21 May, 2025
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Given a weighted undirected graph represented as an edge list and a source vertex src, find the shortest path distances from the source vertex to all other vertices in the graph. The graph contains V vertices, numbered from 0 to V - 1.

Note: The given graph does not contain any negative edge.

Examples:

Input: src = 0, V = 5, edges[][] = [[0, 1, 4], [0, 2, 8], [1, 4, 6], [2, 3, 2], [3, 4, 10]]

1
Graph with 5 node

Output: 0 4 8 10 10
Explanation: Shortest Paths:
0 to 1 = 4. 0 → 1
0 to 2 = 8. 0 → 2
0 to 3 = 10. 0 → 2 → 3
0 to 4 = 10. 0 → 1 → 4

Dijkstra’s Algorithm using Min Heap - O(E*logV) Time and O(V) Space

In Dijkstra's Algorithm, the goal is to find the shortest distance from a given source node to all other nodes in the graph. As the source node is the starting point, its distance is initialized to zero. From there, we iteratively pick the unprocessed node with the minimum distance from the source, this is where a min-heap (priority queue) or a set is typically used for efficiency. For each picked node u, we update the distance to its neighbors v using the formula: dist[v] = dist[u] + weight[u][v], but only if this new path offers a shorter distance than the current known one. This process continues until all nodes have been processed.

Step-by-Step Implementation

  1. Set dist[source]=0 and all other distances as infinity.
  2. Push the source node into the min heap as a pair <distance, node> → i.e., <0, source>.
  3. Pop the top element (node with the smallest distance) from the min heap.
    1. For each adjacent neighbor of the current node:
    2. Calculate the distance using the formula:
      dist[v] = dist[u] + weight[u][v]
          If this new distance is shorter than the current dist[v], update it.
          Push the updated pair <dist[v], v> into the min heap
  4. Repeat step 3 until the min heap is empty.
  5. Return the distance array, which holds the shortest distance from the source to all nodes.

Illustration:


C++ 
//Driver Code Starts #include <iostream> #include <vector> #include <queue> #include <climits> using namespace std;  // Function to construct adjacency  vector<vector<vector<int>>> constructAdj(vector<vector<int>>                               &edges, int V) {                                       // adj[u] = list of {v, wt}     vector<vector<vector<int>>> adj(V);       for (const auto &edge : edges) {         int u = edge[0];         int v = edge[1];         int wt = edge[2];         adj[u].push_back({v, wt});         adj[v].push_back({u, wt});      }          return adj; }  //Driver Code Ends  // Returns shortest distances from src to all other vertices vector<int> dijkstra(int V, vector<vector<int>> &edges, int src){          // Create adjacency list     vector<vector<vector<int>>> adj = constructAdj(edges, V);      // Create a priority queue to store vertices that     // are being preprocessed.     priority_queue<vector<int>, vector<vector<int>>,                     greater<vector<int>>> pq;      // Create a vector for distances and initialize all     // distances as infinite     vector<int> dist(V, INT_MAX);      // 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.empty()){                  // The first vertex in pair is the minimum distance         // vertex, extract it from priority queue.         int u = pq.top()[1];         pq.pop();          // Get all adjacent of u.         for (auto x : adj[u]){                          // Get vertex label and weight of current             // adjacent of u.             int v = x[0];             int weight = x[1];              // If there is shorter 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});             }         }     }      return dist; }  //Driver Code Starts  // Driver program to test methods of graph class int main(){     int V = 5;     int src = 0;      // edge list format: {u, v, weight}     vector<vector<int>> edges = {{0, 1, 4}, {0, 2, 8}, {1, 4, 6},                                   {2, 3, 2}, {3, 4, 10}};      vector<int> result = dijkstra(V, edges, src);      // Print shortest distances in one line     for (int dist : result)         cout << dist << " ";       return 0; }  //Driver Code Ends
Java 
//Driver Code Starts import java.util.*;  class GfG {      // Construct adjacency list using ArrayList of ArrayList     static ArrayList<ArrayList<ArrayList<Integer>>>                               constructAdj(int[][] edges, int V) {         // Initialize the adjacency list         ArrayList<ArrayList<ArrayList<Integer>>>                                     adj = new ArrayList<>();          for (int i = 0; i < V; i++) {             adj.add(new ArrayList<>());         }          // Fill the adjacency list from edges         for (int[] edge : edges) {             int u = edge[0];             int v = edge[1];             int wt = edge[2];              // Add edge from u to v             ArrayList<Integer> e1 = new ArrayList<>();             e1.add(v);             e1.add(wt);             adj.get(u).add(e1);              // Since the graph is undirected, add edge from v to u             ArrayList<Integer> e2 = new ArrayList<>();             e2.add(u);             e2.add(wt);             adj.get(v).add(e2);         }          return adj;     }  //Driver Code Ends      // Returns shortest distances from src to all other vertices     static int[] dijkstra(int V, int[][] edges, int src) {                  // Create adjacency list         ArrayList<ArrayList<ArrayList<Integer>>> adj =                                 constructAdj(edges, V);          // PriorityQueue to store vertices to be processed         // Each element is a pair: [distance, node]         PriorityQueue<ArrayList<Integer>> pq =            new PriorityQueue<>(Comparator.comparingInt(a -> a.get(0)));          // Create a distance array and initialize all distances as infinite         int[] dist = new int[V];         Arrays.fill(dist, Integer.MAX_VALUE);          // Insert source with distance 0         dist[src] = 0;         ArrayList<Integer> start = new ArrayList<>();                  start.add(0);         start.add(src);         pq.offer(start);          // Loop until the priority queue is empty         while (!pq.isEmpty()) {                          // Get the node with the minimum distance             ArrayList<Integer> curr = pq.poll();             int d = curr.get(0);             int u = curr.get(1);              // Traverse all adjacent vertices of the current node             for (ArrayList<Integer> neighbor : adj.get(u)) {                 int v = neighbor.get(0);                 int weight = neighbor.get(1);                  // If there is a shorter path to v through u                 if (dist[v] > dist[u] + weight) {                     // Update distance of v                     dist[v] = dist[u] + weight;                      // Add updated pair to the queue                     ArrayList<Integer> temp = new ArrayList<>();                     temp.add(dist[v]);                     temp.add(v);                     pq.offer(temp);                 }             }         }          // Return the shortest distance array         return dist;     }  //Driver Code Starts      // Driver program to test methods of graph class     public static void main(String[] args) {         int V = 5;               int src = 0;              // Edge list format: {u, v, weight}         int[][] edges = {             {0, 1, 4}, {0, 2, 8}, {1, 4, 6},             {2, 3, 2}, {3, 4, 10}         };          // Get shortest path distances         int[] result = dijkstra(V, edges, src);          // Print shortest distances in one line         for (int d : result)             System.out.print(d + " ");     } }  //Driver Code Ends
Python 
#Driver Code Starts import heapq import sys  # Function to construct adjacency  def constructAdj(edges, V):          # adj[u] = list of [v, wt]     adj = [[] for _ in range(V)]      for edge in edges:         u, v, wt = edge         adj[u].append([v, wt])         adj[v].append([u, wt])      return adj  #Driver Code Ends  # Returns shortest distances from src to all other vertices def dijkstra(V, edges, src):     # Create adjacency list     adj = constructAdj(edges, V)      # Create a priority queue to store vertices that     # are being preprocessed.     pq = []          # Create a list for distances and initialize all     # distances as infinite     dist = [sys.maxsize] * V      # Insert source itself in priority queue and initialize     # its distance as 0.     heapq.heappush(pq, [0, src])     dist[src] = 0      # Looping till priority queue becomes empty (or all     # distances are not finalized)      while pq:         # The first vertex in pair is the minimum distance         # vertex, extract it from priority queue.         u = heapq.heappop(pq)[1]          # Get all adjacent of u.         for x in adj[u]:             # Get vertex label and weight of current             # adjacent of u.             v, weight = x[0], x[1]              # If there is shorter 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])      # Return the shortest distance array     return dist  #Driver Code Starts  # Driver program to test methods of graph class if __name__ == "__main__":     V = 5     src = 0      # edge list format: {u, v, weight}     edges =[[0, 1, 4], [0, 2, 8], [1, 4, 6], [2, 3, 2], [3, 4, 10]];      result = dijkstra(V, edges, src)      # Print shortest distances in one line     print(' '.join(map(str, result)))  #Driver Code Ends
C# 
//Driver Code Starts using System; using System.Collections.Generic;  class GfG {      // MinHeap Node (stores vertex and its distance)     class HeapNode {         public int Vertex;         public int Distance;          public HeapNode(int v, int d) {             Vertex = v;             Distance = d;         }     }      // Custom MinHeap class     class MinHeap {         private List<HeapNode> heap = new List<HeapNode>();          public int Count => heap.Count;          private void Swap(int i, int j) {             var temp = heap[i];             heap[i] = heap[j];             heap[j] = temp;         }          public void Push(HeapNode node) {             heap.Add(node);             int i = heap.Count - 1;              while (i > 0) {                 int parent = (i - 1) / 2;                 if (heap[parent].Distance <= heap[i].Distance)                     break;                 Swap(i, parent);                 i = parent;             }         }          public HeapNode Pop() {             if (heap.Count == 0) return null;              var root = heap[0];             heap[0] = heap[heap.Count - 1];             heap.RemoveAt(heap.Count - 1);             Heapify(0);              return root;         }          private void Heapify(int i) {             int smallest = i;             int left = 2 * i + 1;             int right = 2 * i + 2;              if (left < heap.Count && heap[left].Distance < heap[smallest].Distance)                 smallest = left;              if (right < heap.Count && heap[right].Distance < heap[smallest].Distance)                 smallest = right;              if (smallest != i) {                 Swap(i, smallest);                 Heapify(smallest);             }         }     }      // Build adjacency list from edge list     static List<int[]>[] constructAdj(int[,] edges, int V) {         List<int[]>[] adj = new List<int[]>[V];         for (int i = 0; i < V; i++)             adj[i] = new List<int[]>();          int E = edges.GetLength(0);         for (int i = 0; i < E; i++) {             int u = edges[i, 0];             int v = edges[i, 1];             int wt = edges[i, 2];              adj[u].Add(new int[] { v, wt });             adj[v].Add(new int[] { u, wt }); // Undirected graph         }          return adj;     }  //Driver Code Ends      // Dijkstra's algorithm using custom MinHeap     static int[] dijkstra(int V, int[,] edges, int src) {         var adj = constructAdj(edges, V);         int[] dist = new int[V];         bool[] visited = new bool[V];          for (int i = 0; i < V; i++)             dist[i] = int.MaxValue;          dist[src] = 0;         MinHeap pq = new MinHeap();         pq.Push(new HeapNode(src, 0));          while (pq.Count > 0) {             HeapNode node = pq.Pop();             int u = node.Vertex;              if (visited[u]) continue;             visited[u] = true;              foreach (var neighbor in adj[u]) {                 int v = neighbor[0];                 int weight = neighbor[1];                  if (!visited[v] && dist[u] + weight < dist[v]) {                     dist[v] = dist[u] + weight;                     pq.Push(new HeapNode(v, dist[v]));                 }             }         }          return dist;     }  //Driver Code Starts      // Main method     static void Main(string[] args) {         int V = 5;         int src = 0;          int[,] edges = {             {0, 1, 4},             {0, 2, 8},             {1, 4, 6},             {2, 3, 2},             {3, 4, 10}         };          int[] result = dijkstra(V, edges, src);          foreach (int d in result)             Console.Write(d + " ");     } }  //Driver Code Ends
JavaScript 
//Driver Code Starts  class MinHeap {     constructor() {         this.heap = [];     }      push(val) {         this.heap.push(val);         this._heapifyUp(this.heap.length - 1);     }      pop() {         if (this.size() === 0) return null;         if (this.size() === 1) return this.heap.pop();          const min = this.heap[0];         this.heap[0] = this.heap.pop();         this._heapifyDown(0);         return min;     }      size() {         return this.heap.length;     }      _heapifyUp(index) {         while (index > 0) {             const parent = Math.floor((index - 1) / 2);             if (this.heap[parent][0] <= this.heap[index][0]) break;             [this.heap[parent], this.heap[index]] = [this.heap[index],                                  this.heap[parent]];             index = parent;         }     }      _heapifyDown(index) {         const n = this.heap.length;         while (true) {             let smallest = index;             const left = 2 * index + 1;             const right = 2 * index + 2;              if (left < n && this.heap[left][0] < this.heap[smallest][0]){                 smallest = left;             }                          if (right < n && this.heap[right][0] < this.heap[smallest][0]){                 smallest = right;             }              if (smallest === index) break;             [this.heap[smallest], this.heap[index]] =                                [this.heap[index], this.heap[smallest]];             index = smallest;         }     } }  // Function to construct adjacency  function constructAdj(edges, V) {     // adj[u] = list of [v, wt]     const adj = Array.from({ length: V }, () => []);      for (const edge of edges) {         const [u, v, wt] = edge;         adj[u].push([v, wt]);         adj[v].push([u, wt]);      }      return adj; }  //Driver Code Ends  // Returns shortest distances from src to all other vertices function dijkstra(V, edges, src) {          // Create adjacency list     const adj = constructAdj(edges, V);      // Create a min heap to store <distance, node>     const minHeap = new MinHeap();      // Create an array for distances and initialize all distances as infinity     const dist = Array(V).fill(Number.MAX_SAFE_INTEGER);      // Push the source node with distance 0     minHeap.push([0, src]);     dist[src] = 0;      // Process the heap     while (minHeap.size() > 0) {         const [d, u] = minHeap.pop();          // Traverse all adjacent of u         for (const [v, weight] of adj[u]) {             if (dist[v] > dist[u] + weight) {                 dist[v] = dist[u] + weight;                 minHeap.push([dist[v], v]);             }         }     }      return dist; }  //Driver Code Starts  // Driver code const V = 5; const src = 0;  // edge list format: [u, v, weight] const edges = [[0, 1, 4], [0, 2, 8], [1, 4, 6], [2, 3, 2], [3, 4, 10]];  const result = dijkstra(V, edges, src);  // Print shortest distances in one line console.log(result.join(' '));  //Driver Code Ends

Output
0 4 8 10 10

Time Complexity: O(E*logV), Where E is the number of edges and V is the number of vertices.
Auxiliary Space: O(V), Where V is the number of vertices, We do not count the adjacency list in auxiliary space as it is necessary for representing the input graph.

Problems based on Shortest Path

  • Shortest Path in Directed Acyclic Graph
  • Shortest path with one curved edge in an undirected Graph
  • Minimum Cost Path
  • Path with smallest difference between consecutive cells
  • Print negative weight cycle in a Directed Graph
  • 1st to Kth shortest path lengths in given Graph
  • Shortest path in a Binary Maze
  • Minimum steps to reach target by a Knight
  • Number of ways to reach at destination in shortest time
  • Snake and Ladder Problem
  • Word Ladder

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