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Prim’s algorithm using priority_queue in STL

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Given an undirected, connected and weighted graph, find Minimum Spanning Tree (MST) of the graph using Prim’s algorithm.

Input : Adjacency List representation
of above graph
Output : Edges in MST
0 - 1
1 - 2
2 - 3
3 - 4
2 - 5
5 - 6
6 - 7
2 - 8
Note: There are two possible MSTs, the other
MST includes edge 0-7 in place of 1-2.

We have discussed below Prim’s MST implementations. 

The second implementation is time complexity wise better, but is really complex as we have implemented our own priority queue. STL provides priority_queue, but the provided priority queue doesn’t support decrease key operation. And in Prim’s algorithm, we need a priority queue and below operations on priority queue : 

  • ExtractMin : from all those vertices which have not yet been included in MST, we need to get vertex with minimum key value.
  • DecreaseKey : After extracting vertex we need to update keys of its adjacent vertices, and if new key is smaller, then update that in data structure.

The algorithm discussed here can be modified so that decrease key is never required. The idea is, not to insert all vertices in priority queue, but only those which are not MST and have been visited through a vertex that has included in MST. We keep track of vertices included in MST in a separate boolean array inMST[].

1) Initialize keys of all vertices as infinite and 
parent of every vertex as -1.
2) Create an empty priority_queue pq. Every item
of pq is a pair (weight, vertex). Weight (or
key) is used as first item of pair
as first item is by default used to compare
two pairs.
3) Initialize all vertices as not part of MST yet.
We use boolean array inMST[] for this purpose.
This array is required to make sure that an already
considered vertex is not included in pq again. This
is where Prim's implementation differs from Dijkstra.
In Dijkstra's algorithm, we didn't need this array as
distances always increase. We require this array here
because key value of a processed vertex may decrease
if not checked.
4) Insert source vertex into pq and make its key as 0.
5) While either pq doesn't become empty
a) Extract minimum key vertex from pq.
Let the extracted vertex be u.
b) Include u in MST using inMST[u] = true.
c) Loop through all adjacent of u and do
following for every vertex v.
// If weight of edge (u,v) is smaller than
// key of v and v is not already in MST
If inMST[v] = false && key[v] > weight(u, v)
(i) Update key of v, i.e., do
key[v] = weight(u, v)
(ii) Insert v into the pq
(iv) parent[v] = u

6) Print MST edges using parent array.

Below is C++ implementation of above idea. 

C++

// STL implementation of Prim's algorithm for MST
#include<bits/stdc++.h>
using namespace std;
# define INF 0x3f3f3f3f
 
// iPair ==>  Integer Pair
typedef pair<int, int> iPair;
 
// This class represents a directed graph using
// adjacency list representation
class 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);
 
    // Print MST using Prim's algorithm
    void primMST();
};
 
// Allocates memory for adjacency list
Graph::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 vertices
void Graph::primMST()
{
    // Create a priority queue to store vertices that
    // are being primMST. This is weird syntax in C++.
    // Refer below link for details of this syntax
    priority_queue< iPair, vector <iPair> , greater<iPair> > pq;
 
    int src = 0; // Taking vertex 0 as source
 
    // Create a vector for keys and initialize all
    // keys as infinite (INF)
    vector<int> key(V, INF);
 
    // To store parent array which in turn store MST
    vector<int> parent(V, -1);
 
    // To keep track of vertices included in MST
    vector<bool> inMST(V, false);
 
    // Insert source itself in priority queue and initialize
    // its key as 0.
    pq.push(make_pair(0, src));
    key[src] = 0;
 
    /* Looping till priority queue becomes empty */
    while (!pq.empty())
    {
        // The first vertex in pair is the minimum key
        // 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 key (key must be first item
        // in pair)
        int u = pq.top().second;
        pq.pop();
         
          //Different key values for same vertex may exist in the priority queue.
          //The one with the least key value is always processed first.
          //Therefore, ignore the rest.
          if(inMST[u] == true){
            continue;
        }
       
        inMST[u] = true// Include vertex in MST
 
        // '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 v is not in MST and weight of (u,v) is smaller
            // than current key of v
            if (inMST[v] == false && key[v] > weight)
            {
                // Updating key of v
                key[v] = weight;
                pq.push(make_pair(key[v], v));
                parent[v] = u;
            }
        }
    }
 
    // Print edges of MST using parent array
    for (int i = 1; i < V; ++i)
        printf("%d - %d\n", parent[i], i);
}
 
// Driver program to test methods of graph class
int 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.primMST();
 
    return 0;
}

                    

Java

import java.util.*;
 
// iPair ==>  Integer Pair
class iPair {
    int first, second;
    public iPair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}
 
// This class represents a directed graph using
// adjacency list representation
class Graph {
    int V; // No. of vertices
 
    // In a weighted graph, we need to store vertex
    // and weight pair for every edge
    List<List<iPair> > adj;
 
    public Graph(int V)
    {
        this.V = V;
        adj = new ArrayList<>();
        for (int i = 0; i < V; i++) {
            adj.add(new ArrayList<>());
        }
    }
 
    // function to add an edge to graph
    void addEdge(int u, int v, int w)
    {
        adj.get(u).add(new iPair(v, w));
        adj.get(v).add(new iPair(u, w));
    }
 
    // Print MST using Prim's algorithm
    void primMST()
    {
        // Create a priority queue to store vertices that
        // are being primMST. This is weird syntax in Java.
        // Refer below link for details of this syntax
        PriorityQueue<iPair> pq = new PriorityQueue<>(
            V, new Comparator<iPair>() {
                public int compare(iPair a, iPair b)
                {
                    return a.second - b.second;
                }
            });
 
        int src = 0; // Taking vertex 0 as source
 
        // Create a vector for keys and initialize all
        // keys as infinite (INF)
        int INF = Integer.MAX_VALUE;
        int[] key = new int[V];
        Arrays.fill(key, INF);
 
        // To store parent array which in turn store MST
        int[] parent = new int[V];
        Arrays.fill(parent, -1);
 
        // To keep track of vertices included in MST
        boolean[] inMST = new boolean[V];
 
        // Insert source itself in priority queue and
        // initialize its key as 0.
        pq.offer(new iPair(0, src));
        key[src] = 0;
 
        /* Looping till priority queue becomes empty */
        while (!pq.isEmpty()) {
            // The first vertex in pair is the minimum key
            // 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 key (key must be first item
            // in pair)
            int u = pq.peek().second;
            pq.poll();
 
            // Different key values for same vertex may
            // exist in the priority queue. The one with the
            // least key value is always processed first.
            // Therefore, ignore the rest.
            if (inMST[u]) {
                continue;
            }
 
            inMST[u] = true; // Include vertex in MST
 
            // 'i' is used to get all adjacent vertices of a
            // vertex
            for (iPair i : adj.get(u)) {
                // Get vertex label and weight of current
                // adjacent of u.
                int v = i.first;
                int weight = i.second;
 
                //  If v is not in MST and weight of (u,v)
                //  is smaller
                // than current key of v
                if (!inMST[v] && key[v] > weight) {
                    // Updating key of v
                    key[v] = weight;
                    pq.offer(new iPair(key[v], v));
                    parent[v] = u;
                }
            }
        }
 
        // Print edges of MST using parent array
        for (int i = 1; i < V; i++) {
            System.out.println(parent[i] + " - " + i);
        }
    }
}
 
// Driver class
public class Main {
    public static void main(String[] args)
    {
        // create the graph given in above figure
        int V = 9;
 
        Graph graph = new Graph(V);
 
        //  making above shown graph
        graph.addEdge(0, 1, 4);
        graph.addEdge(0, 7, 8);
        graph.addEdge(1, 2, 8);
        graph.addEdge(1, 7, 11);
        graph.addEdge(2, 3, 7);
        graph.addEdge(2, 8, 2);
        graph.addEdge(2, 5, 4);
        graph.addEdge(3, 4, 9);
        graph.addEdge(3, 5, 14);
        graph.addEdge(4, 5, 10);
        graph.addEdge(5, 6, 2);
        graph.addEdge(6, 7, 1);
        graph.addEdge(6, 8, 6);
        graph.addEdge(7, 8, 7);
 
        graph.primMST();
    }
}

                    

Python3

import heapq
 
# This class represents a directed graph using adjacency list representation
class Graph:
    def __init__(self, V):
        self.V = V
        self.adj = [[] for _ in range(V)]
 
    # Function to add an edge to the graph
    def add_edge(self, u, v, w):
        self.adj[u].append((v, w))
        self.adj[v].append((u, w))
 
    # Function to print MST using Prim's algorithm
    def prim_mst(self):
        pq = []  # Priority queue to store vertices that are being processed
        src = 0  # Taking vertex 0 as the source
 
        # Create a list for keys and initialize all keys as infinite (INF)
        key = [float('inf')] * self.V
 
        # To store the parent array which, in turn, stores MST
        parent = [-1] * self.V
 
        # To keep track of vertices included in MST
        in_mst = [False] * self.V
 
        # Insert source itself into the priority queue and initialize its key as 0
        heapq.heappush(pq, (0, src))
        key[src] = 0
 
        # Loop until the priority queue becomes empty
        while pq:
            # The first vertex in the pair is the minimum key vertex
            # Extract it from the priority queue
            # The vertex label is stored in the second of the pair
            u = heapq.heappop(pq)[1]
 
            # Different key values for the same vertex may exist in the priority queue.
            # The one with the least key value is always processed first.
            # Therefore, ignore the rest.
            if in_mst[u]:
                continue
 
            in_mst[u] = True  # Include the vertex in MST
 
            # Iterate through all adjacent vertices of a vertex
            for v, weight in self.adj[u]:
                # If v is not in MST and the weight of (u, v) is smaller than the current key of v
                if not in_mst[v] and key[v] > weight:
                    # Update the key of v
                    key[v] = weight
                    heapq.heappush(pq, (key[v], v))
                    parent[v] = u
 
        # Print edges of MST using the parent array
        for i in range(1, self.V):
            print(f"{parent[i]} - {i}")
 
# Driver program to test methods of the graph class
if __name__ == "__main__":
    # Create the graph given in the above figure
    V = 9
    g = Graph(V)
 
    # Making the above shown graph
    g.add_edge(0, 1, 4)
    g.add_edge(0, 7, 8)
    g.add_edge(1, 2, 8)
    g.add_edge(1, 7, 11)
    g.add_edge(2, 3, 7)
    g.add_edge(2, 8, 2)
    g.add_edge(2, 5, 4)
    g.add_edge(3, 4, 9)
    g.add_edge(3, 5, 14)
    g.add_edge(4, 5, 10)
    g.add_edge(5, 6, 2)
    g.add_edge(6, 7, 1)
    g.add_edge(6, 8, 6)
    g.add_edge(7, 8, 7)
 
    g.prim_mst()

                    

C#

using System;
using System.Collections.Generic;
 
class Graph
{
    private int V;
    private List<Tuple<int, int>>[] adj;
 
    public Graph(int V)
    {
        this.V = V;
        adj = new List<Tuple<int, int>>[V];
        for (int i = 0; i < V; ++i)
            adj[i] = new List<Tuple<int, int>>();
    }
 
    public void AddEdge(int u, int v, int w)
    {
        adj[u].Add(new Tuple<int, int>(v, w));
        adj[v].Add(new Tuple<int, int>(u, w));
    }
 
    public void PrimMST()
    {
        PriorityQueue<(int key, int vertex)> pq = new PriorityQueue<(int key, int vertex)>(Comparer<(int key, int vertex)>.Create((x, y) => x.key.CompareTo(y.key)));
        int src = 0;
 
        int[] key = new int[V];
        int[] parent = new int[V];
        bool[] inMST = new bool[V];
 
        for (int i = 0; i < V; ++i)
        {
            key[i] = int.MaxValue;
            parent[i] = -1;
            inMST[i] = false;
        }
 
        pq.Enqueue((0, src));
        key[src] = 0;
 
        while (pq.Count > 0)
        {
            int u = pq.Dequeue().vertex;
 
            if (inMST[u])
                continue;
 
            inMST[u] = true;
 
            foreach (var edge in adj[u])
            {
                int v = edge.Item1;
                int weight = edge.Item2;
 
                if (!inMST[v] && key[v] > weight)
                {
                    key[v] = weight;
                    pq.Enqueue((key[v], v));
                    parent[v] = u;
                }
            }
        }
 
        for (int i = 1; i < V; ++i)
        {
            Console.WriteLine($"{parent[i]} - {i}");
        }
    }
}
 
class PriorityQueue<T>
{
    private List<T> heap = new List<T>();
    private readonly Comparer<T> comparer;
 
    public int Count => heap.Count;
 
    public PriorityQueue(Comparer<T> comparer)
    {
        this.comparer = comparer;
    }
 
    // Added parameterless constructor
    public PriorityQueue() : this(Comparer<T>.Default) { }
 
    public void Enqueue(T item)
    {
        heap.Add(item);
        int i = heap.Count - 1;
 
        while (i > 0)
        {
            int parent = (i - 1) / 2;
            if (comparer.Compare(heap[parent], heap[i]) <= 0)
                break;
 
            Swap(parent, i);
            i = parent;
        }
    }
 
    public T Dequeue()
    {
        int count = heap.Count - 1;
        T front = heap[0];
        heap[0] = heap[count];
        heap.RemoveAt(count);
 
        count--;
        int i = 0;
 
        while (true)
        {
            int left = i * 2 + 1;
            int right = i * 2 + 2;
 
            if (left > count)
                break;
 
            int smallest = left;
 
            if (right <= count && comparer.Compare(heap[right], heap[left]) < 0)
                smallest = right;
 
            if (comparer.Compare(heap[i], heap[smallest]) <= 0)
                break;
 
            Swap(i, smallest);
            i = smallest;
        }
 
        return front;
    }
 
    private void Swap(int i, int j)
    {
        T temp = heap[i];
        heap[i] = heap[j];
        heap[j] = temp;
    }
}
 
class Program
{
    static void Main()
    {
        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.PrimMST();
    }
}

                    

Javascript

class Graph {
    constructor(V) {
        this.V = V;
        this.adj = Array.from({ length: V }, () => []);
    }
 
    addEdge(u, v, w) {
        this.adj[u].push([v, w]);
        this.adj[v].push([u, w]);
    }
 
    primMST() {
        const pq = [];
        const src = 0;
        const key = new Array(this.V).fill(Infinity);
        const parent = new Array(this.V).fill(-1);
        const inMST = new Array(this.V).fill(false);
 
        pq.push([0, src]);
        key[src] = 0;
 
        while (pq.length > 0) {
            const [uDist, u] = pq.shift();
 
            if (inMST[u]) {
                continue;
            }
 
            inMST[u] = true;
 
            for (const [v, weight] of this.adj[u]) {
                if (!inMST[v] && key[v] > weight) {
                    key[v] = weight;
                    pq.push([key[v], v]);
                    parent[v] = u;
                }
            }
 
            // Sort the priority queue based on the second element (vertex)
            pq.sort((a, b) => a[1] - b[1]);
        }
 
        // Print edges of MST using the parent array in the desired order
        console.log("Edge   Weight");
        for (let i = 1; i < this.V; ++i) {
            console.log(`${parent[i]} - ${i}    ${key[i]}`);
        }
    }
}
 
// Driver program to test methods of the graph class
function main() {
    const V = 9;
    const 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.primMST();
}
 
// Call the main function
main();

                    

Output
0 - 1
1 - 2
2 - 3
3 - 4
2 - 5
5 - 6
6 - 7
2 - 8


Time complexity : O(E Log V))

Auxiliary Space :O(V)

A Quicker Implementation using array of vectors representation of a weighted graph

C++

// STL implementation of Prim's algorithm for MST
#include<bits/stdc++.h>
using namespace std;
# define INF 0x3f3f3f3f
 
// iPair ==> Integer Pair
typedef pair<int, int> iPair;
 
// To add an edge
void 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 vertices
void primMST(vector<pair<int,int> > adj[], int V)
{
    // Create a priority queue to store vertices that
    // are being primMST. This is weird syntax in C++.
    // Refer below link for details of this syntax
    priority_queue< iPair, vector <iPair> , greater<iPair> > pq;
 
    int src = 0; // Taking vertex 0 as source
 
    // Create a vector for keys and initialize all
    // keys as infinite (INF)
    vector<int> key(V, INF);
 
    // To store parent array which in turn store MST
    vector<int> parent(V, -1);
 
    // To keep track of vertices included in MST
    vector<bool> inMST(V, false);
 
    // Insert source itself in priority queue and initialize
    // its key as 0.
    pq.push(make_pair(0, src));
    key[src] = 0;
 
    /* Looping till priority queue becomes empty */
    while (!pq.empty())
    {
        // The first vertex in pair is the minimum key
        // 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 key (key must be first item
        // in pair)
        int u = pq.top().second;
        pq.pop();
 
          //Different key values for same vertex may exist in the priority queue.
          //The one with the least key value is always processed first.
          //Therefore, ignore the rest.
          if(inMST[u] == true){
            continue;
        }
       
        inMST[u] = true; // Include vertex in MST
 
        // Traverse 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 v is not in MST and weight of (u,v) is smaller
            // than current key of v
            if (inMST[v] == false && key[v] > weight)
            {
                // Updating key of v
                key[v] = weight;
                pq.push(make_pair(key[v], v));
                parent[v] = u;
            }
        }
    }
 
    // Print edges of MST using parent array
    for (int i = 1; i < V; ++i)
        printf("%d - %d\n", parent[i], i);
}
 
// Driver program to test methods of graph class
int 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);
 
    primMST(adj, V);
 
    return 0;
}

                    

Python3

import heapq
 
# To add an edge
def addEdge(adj, u, v, wt):
    adj[u].append((v, wt))
    adj[v].append((u, wt))
 
# Prints shortest paths from src to all other vertices
def primMST(adj, V):
    # Create a priority queue to store vertices that
    # are being primMST. This is implemented using heapq in Python.
    # It is a min heap, and we use tuples for pairs (key, vertex).
    pq = [(0, 0)]  # Start from vertex 0 with key 0
 
    # Create a list for keys and initialize all
    # keys as infinite (float('inf') is used for infinity)
    key = [float('inf')] * V
 
    # To store parent array which in turn store MST
    parent = [-1] * V
 
    # To keep track of vertices included in MST
    inMST = [False] * V
 
    # Insert source itself in priority queue and initialize
    # its key as 0.
    key[0] = 0
 
    # Looping till priority queue becomes empty
    while pq:
        # The first vertex in tuple is the minimum key
        # vertex, extract it from priority queue.
        # vertex label is stored in second of tuple (it
        # has to be done this way to keep the vertices
        # sorted by key, key must be first item
        # in tuple)
        u = heapq.heappop(pq)[1]
 
        # Different key values for the same vertex may exist in the priority queue.
        # The one with the least key value is always processed first.
        # Therefore, ignore the rest.
        if inMST[u]:
            continue
 
        inMST[u] = True  # Include vertex in MST
 
        # Traverse all adjacent vertices of u
        for x in adj[u]:
            # Get vertex label and weight of the current adjacent
            # of u.
            v, weight = x
 
            # If v is not in MST and weight of (u, v) is smaller
            # than the current key of v
            if not inMST[v] and key[v] > weight:
                # Updating key of v
                key[v] = weight
                heapq.heappush(pq, (key[v], v))
                parent[v] = u
 
    # Print edges of MST using parent array
    for i in range(1, V):
        print(f"{parent[i]} - {i}")
 
# Driver program to test methods of graph class
def main():
    V = 9
    adj = [[] for _ in range(V)]
 
    # Making the graph as shown in the C++ code
    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)
 
    primMST(adj, V)
 
if __name__ == "__main__":
    main()

                    

C#

using System;
using System.Collections.Generic;
 
public class PrimMST
{
    // iPair ==> Integer Pair
    private class iPair : IComparable<iPair>
    {
        public int First { get; set; }
        public int Second { get; set; }
 
        public iPair(int first, int second)
        {
            First = first;
            Second = second;
        }
 
        public int CompareTo(iPair other)
        {
            return First.CompareTo(other.First);
        }
    }
 
    // To add an edge
    private static void AddEdge(List<List<iPair>> adj, int u, int v, int wt)
    {
        adj[u].Add(new iPair(v, wt));
        adj[v].Add(new iPair(u, wt));
    }
 
    // Prints shortest paths from src to all other vertices
    private static void PrimMSTAlgorithm(List<List<iPair>> adj, int V)
    {
        // Create a priority queue to store vertices that are being primMST.
        // It is a min heap, and we use tuples for pairs (key, vertex).
        PriorityQueue<iPair> pq = new PriorityQueue<iPair>();
 
        int src = 0; // Taking vertex 0 as the source
 
        // Create a list for keys and initialize all keys as infinite
        List<int> key = new List<int>(new int[V]);
        for (int i = 0; i < V; i++)
        {
            key[i] = int.MaxValue;
        }
 
        // To store the parent array which, in turn, stores MST
        List<int> parent = new List<int>(new int[V]);
        for (int i = 0; i < V; i++)
        {
            parent[i] = -1;
        }
 
        // To keep track of vertices included in MST
        List<bool> inMST = new List<bool>(new bool[V]);
 
        // Insert the source itself into the priority queue and initialize its key as 0.
        pq.Enqueue(new iPair(0, src));
        key[src] = 0;
 
        // Looping until the priority queue becomes empty
        while (pq.Count > 0)
        {
            // The first vertex in the pair is the minimum key vertex, extract it from the priority queue.
            // The vertex label is stored in the second of the pair (it has to be done this way to keep the vertices
            // sorted by key; key must be the first item in the pair)
            int u = pq.Dequeue().Second;
 
            // Different key values for the same vertex may exist in the priority queue.
            // The one with the least key value is always processed first.
            // Therefore, ignore the rest.
            if (inMST[u])
            {
                continue;
            }
 
            inMST[u] = true; // Include the vertex in MST
 
            // Traverse all adjacent vertices of u
            foreach (var x in adj[u])
            {
                // Get the vertex label and weight of the current adjacent of u.
                int v = x.First;
                int weight = x.Second;
 
                // If v is not in MST and the weight of (u, v) is smaller than the current key of v
                if (!inMST[v] && key[v] > weight)
                {
                    // Updating the key of v
                    key[v] = weight;
                    pq.Enqueue(new iPair(key[v], v));
                    parent[v] = u;
                }
            }
        }
 
        // Print edges of MST using the parent array
        for (int i = 1; i < V; ++i)
        {
            Console.WriteLine($"{parent[i]} - {i}");
        }
    }
 
    // Driver program to test methods of the PrimMST class
    public static void Main()
    {
        int V = 9;
        List<List<iPair>> adj = new List<List<iPair>>(new List<iPair>[V]);
 
        // Making the graph as shown in the C++ code
        for (int i = 0; i < V; ++i)
        {
            adj[i] = new List<iPair>();
        }
 
        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);
 
        PrimMSTAlgorithm(adj, V);
    }
}
 
// A simple priority queue implementation
public class PriorityQueue<T> where T : IComparable<T>
{
    private List<T> heap;
 
    public int Count => heap.Count;
 
    public PriorityQueue()
    {
        heap = new List<T>();
    }
 
    public void Enqueue(T item)
    {
        heap.Add(item);
        int i = heap.Count - 1;
        while (i > 0)
        {
            int parent = (i - 1) / 2;
            if (heap[parent].CompareTo(heap[i]) <= 0)
                break;
 
            Swap(parent, i);
            i = parent;
        }
    }
 
    public T Dequeue()
    {
        if (heap.Count == 0)
            throw new InvalidOperationException("Priority queue is empty.");
 
        T item = heap[0];
        int lastIndex = heap.Count - 1;
        heap[0] = heap[lastIndex];
        heap.RemoveAt(lastIndex);
 
        int i = 0;
        while (true)
        {
            int leftChild = 2 * i + 1;
            if (leftChild >= heap.Count)
                break;
 
            int rightChild = leftChild + 1;
            int minChild = (rightChild < heap.Count && heap[rightChild].CompareTo(heap[leftChild]) < 0) ? rightChild : leftChild;
 
            if (heap[i].CompareTo(heap[minChild]) <= 0)
                break;
 
            Swap(i, minChild);
            i = minChild;
        }
 
        return item;
    }
 
    private void Swap(int i, int j)
    {
        T temp = heap[i];
        heap[i] = heap[j];
        heap[j] = temp;
    }
}

                    

Javascript

class Graph {
    constructor(V) {
        this.V = V;
        this.adj = Array.from({ length: V }, () => []); // Adjacency list to store the graph
    }
 
    addEdge(u, v, weight) {
        this.adj[u].push({ v, weight }); // Add edge to the graph
        this.adj[v].push({ v: u, weight }); // Undirected graph, so add edge in both directions
    }
 
    primMST() {
        const pq = new PriorityQueue((a, b) => a.weight - b.weight);
        // Priority queue to store vertices
 
        const src = 0;
        const key = new Array(this.V).fill(Number.MAX_SAFE_INTEGER);
        // Key values to track the minimum edge weight to each vertex
        const parent = new Array(this.V).fill(-1);
        // Parent array to store the resulting MST
        const inMST = new Array(this.V).fill(false);
        // To keep track of vertices included in MST
 
        pq.enqueue({ v: src, weight: 0 }); // Start with the source vertex
        key[src] = 0;
 
        while (!pq.isEmpty()) {
            const { v: u, weight } = pq.dequeue(); // Extract the vertex with the minimum key
 
            if (inMST[u]) {
                continue; // Skip if the vertex is already in the MST
            }
 
            inMST[u] = true;
 
            for (const edge of this.adj[u]) {
                const { v, weight } = edge;
 
                if (!inMST[v] && key[v] > weight) {
                    // If vertex v is not in MST and the weight of edge (u, v) is smaller than the current key of v
                    key[v] = weight;
                    pq.enqueue({ v, weight: key[v] });
                    parent[v] = u;
                }
            }
        }
 
        // Print edges of MST using parent array
        for (let i = 1; i < this.V; ++i) {
            console.log(parent[i] + " - " + i);
        }
    }
}
 
class PriorityQueue {
    constructor(compare) {
        this.heap = [];
        this.compare = compare;
    }
 
    enqueue(element) {
        this.heap.push(element);
        this.bubbleUp();
    }
 
    dequeue() {
        const root = this.heap[0];
        const last = this.heap.pop();
        if (this.heap.length > 0) {
            this.heap[0] = last;
            this.bubbleDown();
        }
        return root;
    }
 
    isEmpty() {
        return this.heap.length === 0;
    }
 
    bubbleUp() {
        let index = this.heap.length - 1;
        while (index > 0) {
            const element = this.heap[index];
            const parentIndex = Math.floor((index - 1) / 2);
            const parent = this.heap[parentIndex];
 
            if (this.compare(parent, element) < 0) {
                break;
            }
 
            this.heap[index] = parent;
            this.heap[parentIndex] = element;
            index = parentIndex;
        }
    }
 
    bubbleDown() {
        let index = 0;
        const length = this.heap.length;
        const element = this.heap[0];
 
        while (true) {
            let leftChildIndex = 2 * index + 1;
            let rightChildIndex = 2 * index + 2;
            let leftChild, rightChild;
            let swap = null;
 
            if (leftChildIndex < length) {
                leftChild = this.heap[leftChildIndex];
                if (this.compare(leftChild, element) < 0) {
                    swap = leftChildIndex;
                }
            }
 
            if (rightChildIndex < length) {
                rightChild = this.heap[rightChildIndex];
                if (
                    (swap === null && this.compare(rightChild, element) < 0) ||
                    (swap !== null && this.compare(rightChild, leftChild) < 0)
                ) {
                    swap = rightChildIndex;
                }
            }
 
            if (swap === null) {
                break;
            }
 
            this.heap[index] = this.heap[swap];
            this.heap[swap] = element;
            index = swap;
        }
    }
}
 
// Driver program to test methods of graph class
const V = 9;
const graph = new Graph(V);
 
graph.addEdge(0, 1, 4);
graph.addEdge(0, 7, 8);
graph.addEdge(1, 2, 8);
graph.addEdge(1, 7, 11);
graph.addEdge(2, 3, 7);
graph.addEdge(2, 8, 2);
graph.addEdge(2, 5, 4);
graph.addEdge(3, 4, 9);
graph.addEdge(3, 5, 14);
graph.addEdge(4, 5, 10);
graph.addEdge(5, 6, 2);
graph.addEdge(6, 7, 1);
graph.addEdge(6, 8, 6);
graph.addEdge(7, 8, 7);
 
graph.primMST();

                    

Output
0 - 1
1 - 2
2 - 3
3 - 4
2 - 5
5 - 6
6 - 7
2 - 8

Note: Like Dijkstra’s priority_queue implementation, we may have multiple entries for same vertex as we do not (and we can not) make isMST[v] = true in if condition. 

C++

// If v is not in MST and weight of (u,v) is smaller
 // than current key of v
 if (inMST[v] == false && key[v] > weight)
 {
      // Updating key of v
      key[v] = weight;
      pq.push(make_pair(key[v], v));
      parent[v] = u;
}

                    

Java

// If v is not in MST and weight of (u,v) is smaller
// than current key of v
  
if (inMST[v] == false && key[v] > weight)
{
   
  // Updating key of v
  key[v] = weight;
  pq.add(new Pair<Integer, Integer>(key[v], v));
  parent[v] = u;
}
  
// This code is contributed by avanitrachhadiya2155

                    

Python3

# If v is not in MST and weight of (u,v) is smaller
# than current key of v
 
if (inMST[v] == False and key[v] > weight) :
 
  # Updating key of v
  key[v] = weight
  pq.append([key[v], v])
  parent[v] = u
   
  # This code is contributed by divyeshrabadiya07.

                    

C#

// If v is not in MST and weight of (u,v) is smaller
// than current key of v
 
if (inMST[v] == false && key[v] > weight)
{
  // Updating key of v
  key[v] = weight;
  pq.Add(new Tuple<int,int>(key[v], v));
  parent[v] = u;
}
 
// This code is contributed by divyesh072019.

                    

Javascript

<script>
 
// If v is not in MST and weight of (u,v)
// is smaller than current key of v
if (inMST[v] == false && key[v] > weight)
{
     
    // Updating key of v
    key[v] = weight;
    value = [key[v], v];
    pq.push(value);
    parent[v] = u;
}
 
// This code is contributed by suresh07
 
</script>

                    

But as explained in Dijkstra’s algorithm, time complexity remains O(E Log V) as there will be at most O(E) vertices in priority queue and O(Log E) is same as O(Log V).

Unlike Dijkstra’s implementation, a boolean array inMST[] is mandatory here because the key values of newly inserted items can be less than the key values of extracted items. So we must not consider extracted items.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



Last Updated : 15 Feb, 2024
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