Given a graph and a source vertex in graph, find shortest paths from source to all vertices in the given graph.
Input : Source = 0
Output :
Vertex Distance from Source
0 0
1 4
2 12
3 19
4 21
5 11
6 9
7 8
8 14
We have discussed Dijkstra’s shortest Path implementations.
- Dijkstra’s Algorithm for Adjacency Matrix Representation (In C/C++ with time complexity O(v2)
- Dijkstra’s Algorithm for Adjacency List Representation (In C with Time Complexity O(ELogV))
- Dijkstra’s shortest path algorithm using set in STL (In C++ with Time Complexity O(ELogV))
The second implementation is time complexity wise better but is really complex as we have implemented our own priority queue. The Third implementation is simpler as it uses STL. The issue with third implementation is, it uses set which in turn uses Self-Balancing Binary Search Trees. 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 speciality of heap (or priority queue). However, the problem is, priority_queue doesn’t support decrease key. To resolve this problem, do not update a key, but insert one more copy of it. So we allow multiple instances of same vertex in priority queue. This approach doesn’t require decrease key operation and has below important properties.
- Whenever distance of a vertex is reduced, we add one more instance of 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(ELogV)) as there will be at most O(E) vertices in priority queue and O(Log E) is same as O(Log V)
Below is algorithm based on above idea.
1) Initialize distances of all vertices as infinite.
2) Create an empty priority_queue pq. Every item
of pq is a pair (weight, vertex). Weight (or
distance) is used as first item of pair
as first item is by default used to compare
two pairs
3) Insert source vertex into pq and make its
distance as 0.
4) While either pq doesn't become empty
a) Extract minimum distance vertex from pq.
Let the extracted vertex be u.
b) Loop through all adjacent of u and do
following for every vertex v.
// If there is a shorter path to v
// through u.
If dist[v] > dist[u] + weight(u, v)
(i) Update distance of v, i.e., do
dist[v] = dist[u] + weight(u, v)
(ii) Insert v into the pq (Even if v is
already there)
5) Print distance array dist[] to print all shortest
paths.
Below is C++ implementation of above idea.
CPP
// 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 program to test methods of graph classint 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); g.shortestPath(0); return 0;} |
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 log V)
Space Complexity:O(V2) , here V is number of Vertices.
A Quicker Implementation using vector of pairs representation of weighted graph :
CPP
// 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;// To add an edgevoid addEdge(vector<pair<int, int> > adj[], int u, int v, int wt){ adj[u].push_back(make_pair(v, wt)); adj[v].push_back(make_pair(u, wt));}// Prints shortest paths from src to all other verticesvoid shortestPath(vector<pair<int, int> > adj[], int V, 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(); // Get all adjacent of u. for (auto x : adj[u]) { // Get vertex label and weight of current // adjacent of u. int v = x.first; int weight = x.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 program to test methods of graph classint main(){ int V = 9; vector<iPair> adj[V]; // making above shown graph addEdge(adj, 0, 1, 4); addEdge(adj, 0, 7, 8); addEdge(adj, 1, 2, 8); addEdge(adj, 1, 7, 11); addEdge(adj, 2, 3, 7); addEdge(adj, 2, 8, 2); addEdge(adj, 2, 5, 4); addEdge(adj, 3, 4, 9); addEdge(adj, 3, 5, 14); addEdge(adj, 4, 5, 10); addEdge(adj, 5, 6, 2); addEdge(adj, 6, 7, 1); addEdge(adj, 6, 8, 6); addEdge(adj, 7, 8, 7); shortestPath(adj, V, 0); return 0;} |
Output
Vertex Distance from Source 0 0 1 4 2 12 3 19 4 21 5 11 6 9 7 8 8 14
The time complexity of Dijkstra’s algorithm using a priority queue implemented with a binary heap is O(Elog(V)), where E is the number of edges and V is the number of vertices in the graph.
The space complexity of this implementation is O(V+E), where V is the number of vertices and E is the number of edges.
Java
import java.util.ArrayList;import java.util.Arrays;import java.util.HashMap;import java.util.PriorityQueue;public class DijkstraAlgoForShortestDistance { static class Node implements Comparable<Node> { int v; int distance; public Node(int v, int distance) { this.v = v; this.distance = distance; } @Override public int compareTo(Node n) { if (this.distance <= n.distance) { return -1; } else { return 1; } } } static int[] dijkstra( int V, ArrayList<ArrayList<ArrayList<Integer> > > adj, int S) { boolean[] visited = new boolean[V]; HashMap<Integer, Node> map = new HashMap<>(); PriorityQueue<Node> q = new PriorityQueue<>(); map.put(S, new Node(S, 0)); q.add(new Node(S, 0)); while (!q.isEmpty()) { Node n = q.poll(); int v = n.v; int distance = n.distance; visited[v] = true; ArrayList<ArrayList<Integer> > adjList = adj.get(v); for (ArrayList<Integer> adjLink : adjList) { if (visited[adjLink.get(0)] == false) { if (!map.containsKey(adjLink.get(0))) { map.put( adjLink.get(0), new Node(v, distance + adjLink.get(1))); } else { Node sn = map.get(adjLink.get(0)); if (distance + adjLink.get(1) < sn.distance) { sn.v = v; sn.distance = distance + adjLink.get(1); } } q.add(new Node(adjLink.get(0), distance + adjLink.get(1))); } } } int[] result = new int[V]; for (int i = 0; i < V; i++) { result[i] = map.get(i).distance; } return result; } public static void main(String[] args) { ArrayList<ArrayList<ArrayList<Integer> > > adj = new ArrayList<>(); HashMap<Integer, ArrayList<ArrayList<Integer> > > map = new HashMap<>(); int V = 6; int E = 5; int[] u = { 0, 0, 1, 2, 4 }; int[] v = { 3, 5, 4, 5, 5 }; int[] w = { 9, 4, 4, 10, 3 }; for (int i = 0; i < E; i++) { ArrayList<Integer> edge = new ArrayList<>(); edge.add(v[i]); edge.add(w[i]); ArrayList<ArrayList<Integer> > adjList; if (!map.containsKey(u[i])) { adjList = new ArrayList<>(); } else { adjList = map.get(u[i]); } adjList.add(edge); map.put(u[i], adjList); ArrayList<Integer> edge2 = new ArrayList<>(); edge2.add(u[i]); edge2.add(w[i]); ArrayList<ArrayList<Integer> > adjList2; if (!map.containsKey(v[i])) { adjList2 = new ArrayList<>(); } else { adjList2 = map.get(v[i]); } adjList2.add(edge2); map.put(v[i], adjList2); } for (int i = 0; i < V; i++) { if (map.containsKey(i)) { adj.add(map.get(i)); } else { adj.add(null); } } int S = 1; // Input sample //[0 [[3, 9], [5, 4]], // 1 [[4, 4]], // 2 [[5, 10]], // 3 [[0, 9]], // 4 [[1, 4], [5, 3]], // 5 [[0, 4], [2, 10], [4, 3]] //] int[] result = DijkstraAlgoForShortestDistance.dijkstra( V, adj, S); System.out.println(Arrays.toString(result)); }} |
Python3
#Python#import modulesimport heapq#Define a class Node to store the valuesclass Node: def __init__(self, v, distance): self.v = v self.distance = distance #define a comparator method to compare distance of two nodes def __lt__(self, other): return self.distance < other.distance#function to implement Dijkstra Algorithm to find shortest distancedef dijkstra(V, adj, S): #initialize a visited list and map visited = [False] * V map = {} #initialize a priority queue q = [] #insert source node in map map[S] = Node(S, 0) #add source node to priority queue heapq.heappush(q, Node(S, 0)) #while the priority queue is not empty while q: #pop the node with minimum distance from priority queue n = heapq.heappop(q) #get the vertex and distance v = n.v distance = n.distance #mark the vertex as visited visited[v] = True #get the adjacent list of the vertex adjList = adj[v] #for every adjacent node of the vertex for edge in adjList: #if the node is not yet visited if visited[edge[0]] == False: #if the node is not present in map if edge[0] not in map: #put the node in map map[edge[0]] = Node(v, distance + edge[1]) else: #get the node from map sn = map[edge[0]] #check if the new distance is less than the current distance of the node if distance + edge[1] < sn.distance: #update the node's distance sn.v = v sn.distance = distance + edge[1] #push the node to priority queue heapq.heappush(q, Node(edge[0], distance + edge[1])) #initialize a result list result = [0] * V #for every key in map for key in map.keys(): #add the distance of the node to the result result[key] = map[key].distance #return the result list return result#main functionif __name__ == '__main__': #initialize adjacency list and map adj = [] map = {} #initialize number of vertices and edges V = 6 E = 5 #define u, v and w u = [0, 0, 1, 2, 4] v = [3, 5, 4, 5, 5] w = [9, 4, 4, 10, 3] #for every edge for i in range(E): #create an edge list edge = [v[i], w[i]] #if the u[i] is not present in map if u[i] not in map: #create a new adjacency list adjList = [] else: #get the existing adjacency list adjList = map[u[i]] #add the edge to the adjacency list adjList.append(edge) #put the adjacency list in map map[u[i]] = adjList #create another edge list edge2 = [u[i], w[i]] #if the v[i] is not present in map if v[i] not in map: #create a new adjacency list adjList2 = [] else: #get the existing adjacency list adjList2 = map[v[i]] #add the edge to the adjacency list adjList2.append(edge2) #put the adjacency list in map map[v[i]] = adjList2 #for every vertex for i in range(V): #if the vertex is present in map if i in map: #add the adjacency list to the main adjacency list adj.append(map[i]) else: #add null to the main adjacency list adj.append(None) #define source node S = 1 #call the dijkstra function result = dijkstra(V, adj, S) #print the result print(result) |
Output
[11, 0, 17, 20, 4, 7]
Time Complexity: O(ElogV)
The time complexity is O(ElogV) where E is the number of edges and V is the number of vertices. This is because, for each vertex, we need to traverse all its adjacent vertices and update their distances, which takes O(E) time. Also, we use a Priority Queue which takes O(logV) time for each insertion and deletion.
Space Complexity: O(V)
The space complexity is O(V) as we need to maintain a visited array of size V, a map of size V and a Priority Queue of size V.
Further Optimization We can use a flag array to store what all vertices have been extracted from priority queue. This way we can avoid updating weights of items that have already been extracted. Please see this for optimized implementation. This article is contributed by Shubham Agrawal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
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