Given an undirected, connected and weighted graph, find **M**inimum **S**panning **T**ree (MST) of the graph using Kruskal’s algorithm.

Input : Graph as an array of edges Output : Edges of MST are 6 - 7 2 - 8 5 - 6 0 - 1 2 - 5 2 - 3 0 - 7 3 - 4 Weight of MST is 37Note :There are two possible MSTs, the other MST includes edge 1-2 in place of 0-7.

We have discussed below Kruskal’s MST implementations.

Greedy Algorithms | Set 2 (Kruskal’s Minimum Spanning Tree Algorithm)

Below are the steps for finding MST using Kruskal’s algorithm

- Sort all the edges in non-decreasing order of their weight.
- Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it.
- Repeat step#2 until there are (V-1) edges in the spanning tree.

Here are some key points which will be useful for us in implementing the Kruskal’s algorithm using STL.

- Use a vector of edges which consist of all the edges in the graph and each item of a vector will contain 3 parameters: source, destination and the cost of an edge between the source and destination.
vector<pair<int, pair<int, int> > > edges;

Here in the outer pair (i.e pair<int,pair<int,int> > ) the first element corresponds to the cost of a edge while the second element is itself a pair, and it contains two vertices of edge.

- Use the inbuilt std::sort to sort the edges in the non-decreasing order; by default the sort function sort in non-decreasing order.
- We use the Union Find Algorithm to check if it the current edge forms a cycle if it is added in the current MST. If yes discard it, else include it (union).

**Pseudo Code: **

// Initialize result mst_weight = 0 // Create V single item setsforeachvertex v parent[v] = v; rank[v] = 0;Sortall edges into non decreasing order by weight wforeach (u, v) taken from the sorted list Edo ifFIND-SET(u) !=FIND-SET(v)UNION(u, v)

Below is C++ implementation of above algorithm.

// C++ program for Kruskal's algorithm to find Minimum // Spanning Tree of a given connected, undirected and // weighted graph #include<bits/stdc++.h> using namespace std; // Creating shortcut for an integer pair typedef pair<int, int> iPair; // Structure to represent a graph struct Graph { int V, E; vector< pair<int, iPair> > edges; // Constructor Graph(int V, int E) { this->V = V; this->E = E; } // Utility function to add an edge void addEdge(int u, int v, int w) { edges.push_back({w, {u, v}}); } // Function to find MST using Kruskal's // MST algorithm int kruskalMST(); }; // To represent Disjoint Sets struct DisjointSets { int *parent, *rnk; int n; // Constructor. DisjointSets(int n) { // Allocate memory this->n = n; parent = new int[n+1]; rnk = new int[n+1]; // Initially, all vertices are in // different sets and have rank 0. for (int i = 0; i <= n; i++) { rnk[i] = 0; //every element is parent of itself parent[i] = i; } } // Find the parent of a node 'u' // Path Compression int find(int u) { /* Make the parent of the nodes in the path from u--> parent[u] point to parent[u] */ if (u != parent[u]) parent[u] = find(parent[u]); return parent[u]; } // Union by rank void merge(int x, int y) { x = find(x), y = find(y); /* Make tree with smaller height a subtree of the other tree */ if (rnk[x] > rnk[y]) parent[y] = x; else // If rnk[x] <= rnk[y] parent[x] = y; if (rnk[x] == rnk[y]) rnk[y]++; } }; /* Functions returns weight of the MST*/ int Graph::kruskalMST() { int mst_wt = 0; // Initialize result // Sort edges in increasing order on basis of cost sort(edges.begin(), edges.end()); // Create disjoint sets DisjointSets ds(V); // Iterate through all sorted edges vector< pair<int, iPair> >::iterator it; for (it=edges.begin(); it!=edges.end(); it++) { int u = it->second.first; int v = it->second.second; int set_u = ds.find(u); int set_v = ds.find(v); // Check if the selected edge is creating // a cycle or not (Cycle is created if u // and v belong to same set) if (set_u != set_v) { // Current edge will be in the MST // so print it cout << u << " - " << v << endl; // Update MST weight mst_wt += it->first; // Merge two sets ds.merge(set_u, set_v); } } return mst_wt; } // Driver program to test above functions int main() { /* Let us create above shown weighted and unidrected graph */ int V = 9, E = 14; Graph g(V, E); // 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); cout << "Edges of MST are \n"; int mst_wt = g.kruskalMST(); cout << "\nWeight of MST is " << mst_wt; return 0; }

Output :

Edges of MST are 6 - 7 2 - 8 5 - 6 0 - 1 2 - 5 2 - 3 0 - 7 3 - 4 Weight of MST is 37

**Optimization:**

The above code can be optimized to stop the main loop of Kruskal when number of selected edges become V-1. We know that MST has V-1 edges and there is no point iterating after V-1 edges are selected. We have not added this optimization to keep code simple.

**References:**

Introduction to Algorithms by Cormen Leiserson Rivest and Stein(CLRS) 3

Time complexity and step by step illustration are discussed in previous post on Kruskal’s algorithm.

This article is contributed by **Chirag Agrawal**. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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