A minimum spanning tree (MST) or minimum weight spanning tree for a weighted, connected, undirected graph is a spanning tree with a weight less than or equal to the weight of every other spanning tree. To learn more about Minimum Spanning Tree, refer to this article.
Introduction to Kruskal’s Algorithm:
Here we will discuss Kruskal’s algorithm to find the MST of a given weighted graph.
In Kruskal’s algorithm, sort all edges of the given graph in increasing order. Then it keeps on adding new edges and nodes in the MST if the newly added edge does not form a cycle. It picks the minimum weighted edge at first and the maximum weighted edge at last. Thus we can say that it makes a locally optimal choice in each step in order to find the optimal solution. Hence this is a Greedy Algorithm.
How to find MST using Kruskal’s 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 the cycle is not formed, include this edge. Else, discard it.
- Repeat step#2 until there are (V-1) edges in the spanning tree.
Step 2 uses the Union-Find algorithm to detect cycles.
So we recommend reading the following post as a prerequisite.
Kruskal’s algorithm to find the minimum cost spanning tree uses the greedy approach. The Greedy Choice is to pick the smallest weight edge that does not cause a cycle in the MST constructed so far. Let us understand it with an example:
Illustration:
Below is the illustration of the above approach:
Input Graph:
The graph contains 9 vertices and 14 edges. So, the minimum spanning tree formed will be having (9 – 1) = 8 edges.
After sorting:
Weight Source Destination 1 7 6 2 8 2 2 6 5 4 0 1 4 2 5 6 8 6 7 2 3 7 7 8 8 0 7 8 1 2 9 3 4 10 5 4 11 1 7 14 3 5 Now pick all edges one by one from the sorted list of edges
Step 1: Pick edge 7-6. No cycle is formed, include it.
Add edge 7-6 in the MST
Step 2: Pick edge 8-2. No cycle is formed, include it.
Add edge 8-2 in the MST
Step 3: Pick edge 6-5. No cycle is formed, include it.
Add edge 6-5 in the MST
Step 4: Pick edge 0-1. No cycle is formed, include it.
Add edge 0-1 in the MST
Step 5: Pick edge 2-5. No cycle is formed, include it.
Add edge 2-5 in the MST
Step 6: Pick edge 8-6. Since including this edge results in the cycle, discard it. Pick edge 2-3: No cycle is formed, include it.
Add edge 2-3 in the MST
Step 7: Pick edge 7-8. Since including this edge results in the cycle, discard it. Pick edge 0-7. No cycle is formed, include it.
Add edge 0-7 in MST
Step 8: Pick edge 1-2. Since including this edge results in the cycle, discard it. Pick edge 3-4. No cycle is formed, include it.
Add edge 3-4 in the MST
Note: Since the number of edges included in the MST equals to (V – 1), so the algorithm stops here
Below is the implementation of the above approach:
C++
// C++ program for the above approach #include <bits/stdc++.h> using namespace std; // DSU data structure // path compression + rank by union class DSU { int* parent; int* rank; public: DSU(int n) { parent = new int[n]; rank = new int[n]; for (int i = 0; i < n; i++) { parent[i] = -1; rank[i] = 1; } } // Find function int find(int i) { if (parent[i] == -1) return i; return parent[i] = find(parent[i]); } // Union function void unite(int x, int y) { int s1 = find(x); int s2 = find(y); if (s1 != s2) { if (rank[s1] < rank[s2]) { parent[s1] = s2; } else if (rank[s1] > rank[s2]) { parent[s2] = s1; } else { parent[s2] = s1; rank[s1] += 1; } } } }; class Graph { vector<vector<int> > edgelist; int V; public: Graph(int V) { this->V = V; } // Function to add edge in a graph void addEdge(int x, int y, int w) { edgelist.push_back({ w, x, y }); } void kruskals_mst() { // Sort all edges sort(edgelist.begin(), edgelist.end()); // Initialize the DSU DSU s(V); int ans = 0; cout << "Following are the edges in the " "constructed MST" << endl; for (auto edge : edgelist) { int w = edge[0]; int x = edge[1]; int y = edge[2]; // Take this edge in MST if it does // not forms a cycle if (s.find(x) != s.find(y)) { s.unite(x, y); ans += w; cout << x << " -- " << y << " == " << w << endl; } } cout << "Minimum Cost Spanning Tree: " << ans; } }; // Driver code int main() { Graph g(4); g.addEdge(0, 1, 10); g.addEdge(1, 3, 15); g.addEdge(2, 3, 4); g.addEdge(2, 0, 6); g.addEdge(0, 3, 5); // Function call g.kruskals_mst(); return 0; } |
C
// C code to implement Kruskal's algorithm #include <stdio.h> #include <stdlib.h> // Comparator function to use in sorting int comparator(const void* p1, const void* p2) { const int(*x)[3] = p1; const int(*y)[3] = p2; return (*x)[2] - (*y)[2]; } // Initialization of parent[] and rank[] arrays void makeSet(int parent[], int rank[], int n) { for (int i = 0; i < n; i++) { parent[i] = i; rank[i] = 0; } } // Function to find the parent of a node int findParent(int parent[], int component) { if (parent[component] == component) return component; return parent[component] = findParent(parent, parent[component]); } // Function to unite two sets void unionSet(int u, int v, int parent[], int rank[], int n) { // Finding the parents u = findParent(parent, u); v = findParent(parent, v); if (rank[u] < rank[v]) { parent[u] = v; } else if (rank[u] > rank[v]) { parent[v] = u; } else { parent[v] = u; // Since the rank increases if // the ranks of two sets are same rank[u]++; } } // Function to find the MST void kruskalAlgo(int n, int edge[n][3]) { // First we sort the edge array in ascending order // so that we can access minimum distances/cost qsort(edge, n, sizeof(edge[0]), comparator); int parent[n]; int rank[n]; // Function to initialize parent[] and rank[] makeSet(parent, rank, n); // To store the minimun cost int minCost = 0; printf( "Following are the edges in the constructed MST\n"); for (int i = 0; i < n; i++) { int v1 = findParent(parent, edge[i][0]); int v2 = findParent(parent, edge[i][1]); int wt = edge[i][2]; // If the parents are different that // means they are in different sets so // union them if (v1 != v2) { unionSet(v1, v2, parent, rank, n); minCost += wt; printf("%d -- %d == %d\n", edge[i][0], edge[i][1], wt); } } printf("Minimum Cost Spanning Tree: %d\n", minCost); } // Driver code int main() { int edge[5][3] = { { 0, 1, 10 }, { 0, 2, 6 }, { 0, 3, 5 }, { 1, 3, 15 }, { 2, 3, 4 } }; kruskalAlgo(5, edge); return 0; } |
Java
// Java program for Kruskal's algorithm import java.util.ArrayList; import java.util.Comparator; import java.util.List; public class KruskalsMST { // Defines edge structure static class Edge { int src, dest, weight; public Edge(int src, int dest, int weight) { this.src = src; this.dest = dest; this.weight = weight; } } // Defines subset element structure static class Subset { int parent, rank; public Subset(int parent, int rank) { this.parent = parent; this.rank = rank; } } // Starting point of program execution public static void main(String[] args) { int V = 4; List<Edge> graphEdges = new ArrayList<Edge>( List.of(new Edge(0, 1, 10), new Edge(0, 2, 6), new Edge(0, 3, 5), new Edge(1, 3, 15), new Edge(2, 3, 4))); // Sort the edges in non-decreasing order // (increasing with repetition allowed) graphEdges.sort(new Comparator<Edge>() { @Override public int compare(Edge o1, Edge o2) { return o1.weight - o2.weight; } }); kruskals(V, graphEdges); } // Function to find the MST private static void kruskals(int V, List<Edge> edges) { int j = 0; int noOfEdges = 0; // Allocate memory for creating V subsets Subset subsets[] = new Subset[V]; // Allocate memory for results Edge results[] = new Edge[V]; // Create V subsets with single elements for (int i = 0; i < V; i++) { subsets[i] = new Subset(i, 0); } // Number of edges to be taken is equal to V-1 while (noOfEdges < V - 1) { // Pick the smallest edge. And increment // the index for next iteration Edge nextEdge = edges.get(j); int x = findRoot(subsets, nextEdge.src); int y = findRoot(subsets, nextEdge.dest); // If including this edge doesn't cause cycle, // include it in result and increment the index // of result for next edge if (x != y) { results[noOfEdges] = nextEdge; union(subsets, x, y); noOfEdges++; } j++; } // Print the contents of result[] to display the // built MST System.out.println( "Following are the edges of the constructed MST:"); int minCost = 0; for (int i = 0; i < noOfEdges; i++) { System.out.println(results[i].src + " -- " + results[i].dest + " == " + results[i].weight); minCost += results[i].weight; } System.out.println("Total cost of MST: " + minCost); } // Function to unite two disjoint sets private static void union(Subset[] subsets, int x, int y) { int rootX = findRoot(subsets, x); int rootY = findRoot(subsets, y); if (subsets[rootY].rank < subsets[rootX].rank) { subsets[rootY].parent = rootX; } else if (subsets[rootX].rank < subsets[rootY].rank) { subsets[rootX].parent = rootY; } else { subsets[rootY].parent = rootX; subsets[rootX].rank++; } } // Function to find parent of a set private static int findRoot(Subset[] subsets, int i) { if (subsets[i].parent == i) return subsets[i].parent; subsets[i].parent = findRoot(subsets, subsets[i].parent); return subsets[i].parent; } } // This code is contributed by Salvino D'sa |
Python3
# Python program for Kruskal's algorithm to find # Minimum Spanning Tree of a given connected, # undirected and weighted graph # Class to represent a graph class Graph: def __init__(self, vertices): self.V = vertices self.graph = [] # Function to add an edge to graph def addEdge(self, u, v, w): self.graph.append([u, v, w]) # A utility function to find set of an element i # (truly uses path compression technique) def find(self, parent, i): if parent[i] != i: # Reassignment of node's parent # to root node as # path compression requires parent[i] = self.find(parent, parent[i]) return parent[i] # A function that does union of two sets of x and y # (uses union by rank) def union(self, parent, rank, x, y): # Attach smaller rank tree under root of # high rank tree (Union by Rank) if rank[x] < rank[y]: parent[x] = y elif rank[x] > rank[y]: parent[y] = x # If ranks are same, then make one as root # and increment its rank by one else: parent[y] = x rank[x] += 1 # The main function to construct MST # using Kruskal's algorithm def KruskalMST(self): # This will store the resultant MST result = [] # An index variable, used for sorted edges i = 0 # An index variable, used for result[] e = 0 # Sort all the edges in # non-decreasing order of their # weight self.graph = sorted(self.graph, key=lambda item: item[2]) parent = [] rank = [] # Create V subsets with single elements for node in range(self.V): parent.append(node) rank.append(0) # Number of edges to be taken is less than to V-1 while e < self.V - 1: # Pick the smallest edge and increment # the index for next iteration u, v, w = self.graph[i] i = i + 1 x = self.find(parent, u) y = self.find(parent, v) # If including this edge doesn't # cause cycle, then include it in result # and increment the index of result # for next edge if x != y: e = e + 1 result.append([u, v, w]) self.union(parent, rank, x, y) # Else discard the edge minimumCost = 0 print("Edges in the constructed MST") for u, v, weight in result: minimumCost += weight print("%d -- %d == %d" % (u, v, weight)) print("Minimum Spanning Tree", minimumCost) # Driver code if __name__ == '__main__': g = Graph(4) g.addEdge(0, 1, 10) g.addEdge(0, 2, 6) g.addEdge(0, 3, 5) g.addEdge(1, 3, 15) g.addEdge(2, 3, 4) # Function call g.KruskalMST() # This code is contributed by Neelam Yadav # Improved by James Graça-Jones |
C#
// C# Code for the above approach using System; class Graph { // A class to represent a graph edge class Edge : IComparable<Edge> { public int src, dest, weight; // Comparator function used for sorting edges // based on their weight public int CompareTo(Edge compareEdge) { return this.weight - compareEdge.weight; } } // A class to represent // a subset for union-find public class subset { public int parent, rank; }; // V-> no. of vertices & E->no.of edges int V, E; // Collection of all edges Edge[] edge; // Creates a graph with V vertices and E edges Graph(int v, int e) { V = v; E = e; edge = new Edge[E]; for (int i = 0; i < e; ++i) edge[i] = new Edge(); } // A utility function to find set of an element i // (uses path compression technique) int find(subset[] subsets, int i) { // Find root and make root as // parent of i (path compression) if (subsets[i].parent != i) subsets[i].parent = find(subsets, subsets[i].parent); return subsets[i].parent; } // A function that does union of // two sets of x and y (uses union by rank) void Union(subset[] subsets, int x, int y) { int xroot = find(subsets, x); int yroot = find(subsets, y); // Attach smaller rank tree under root of // high rank tree (Union by Rank) if (subsets[xroot].rank < subsets[yroot].rank) subsets[xroot].parent = yroot; else if (subsets[xroot].rank > subsets[yroot].rank) subsets[yroot].parent = xroot; // If ranks are same, then make one as root // and increment its rank by one else { subsets[yroot].parent = xroot; subsets[xroot].rank++; } } // The main function to construct MST // using Kruskal's algorithm void KruskalMST() { // This will store the // resultant MST Edge[] result = new Edge[V]; // An index variable, used for result[] int e = 0; // An index variable, used for sorted edges int i = 0; for (i = 0; i < V; ++i) result[i] = new Edge(); // Sort all the edges in non-decreasing // order of their weight. If we are not allowed // to change the given graph, we can create // a copy of array of edges Array.Sort(edge); // Allocate memory for creating V subsets subset[] subsets = new subset[V]; for (i = 0; i < V; ++i) subsets[i] = new subset(); // Create V subsets with single elements for (int v = 0; v < V; ++v) { subsets[v].parent = v; subsets[v].rank = 0; } i = 0; // Number of edges to be taken is equal to V-1 while (e < V - 1) { // Pick the smallest edge. And increment // the index for next iteration Edge next_edge = new Edge(); next_edge = edge[i++]; int x = find(subsets, next_edge.src); int y = find(subsets, next_edge.dest); // If including this edge doesn't cause cycle, // include it in result and increment the index // of result for next edge if (x != y) { result[e++] = next_edge; Union(subsets, x, y); } } // Print the contents of result[] to display // the built MST Console.WriteLine("Following are the edges in " + "the constructed MST"); int minimumCost = 0; for (i = 0; i < e; ++i) { Console.WriteLine(result[i].src + " -- " + result[i].dest + " == " + result[i].weight); minimumCost += result[i].weight; } Console.WriteLine("Minimum Cost Spanning Tree: " + minimumCost); Console.ReadLine(); } // Driver's Code public static void Main(String[] args) { int V = 4; int E = 5; Graph graph = new Graph(V, E); // add edge 0-1 graph.edge[0].src = 0; graph.edge[0].dest = 1; graph.edge[0].weight = 10; // add edge 0-2 graph.edge[1].src = 0; graph.edge[1].dest = 2; graph.edge[1].weight = 6; // add edge 0-3 graph.edge[2].src = 0; graph.edge[2].dest = 3; graph.edge[2].weight = 5; // add edge 1-3 graph.edge[3].src = 1; graph.edge[3].dest = 3; graph.edge[3].weight = 15; // add edge 2-3 graph.edge[4].src = 2; graph.edge[4].dest = 3; graph.edge[4].weight = 4; // Function call graph.KruskalMST(); } } // This code is contributed by Aakash Hasija |
Javascript
<script> // JavaScript implementation of the krushkal's algorithm. function makeSet(parent,rank,n) { for(let i=0;i<n;i++) { parent[i]=i; rank[i]=0; } } function findParent(parent,component) { if(parent[component]==component) return component; return parent[component] = findParent(parent,parent[component]); } function unionSet(u, v, parent, rank,n) { //this function unions two set on the basis of rank //as shown below u=findParent(parent,u); v=findParent(parent,v); if(rank[u]<rank[v]) { parent[u]=v; } else if(rank[u]<rank[v]) { parent[v]=u; } else { parent[v]=u; rank[u]++;//since the rank increases if the ranks of two sets are same } } function kruskalAlgo(n, edge) { //First we sort the edge array in ascending order //so that we can access minimum distances/cost edge.sort((a, b)=>{ return a[2] - b[2]; }) //inbuilt quick sort function comes with stdlib.h //if there is any doubt regarding the function let parent = new Array(n); let rank = new Array(n); makeSet(parent,rank,n);//function to initialize parent[] and rank[] let minCost=0;//to store the minimun cost document.write("Following are the edges in the constructed MST"); for(let i=0;i<n;i++) { let v1=findParent(parent,edge[i][0]); let v2=findParent(parent,edge[i][1]); let wt=edge[i][2]; if(v1!=v2)//if the parents are different that means they are in //different sets so union them { unionSet(v1,v2,parent,rank,n); minCost+=wt; document.write(edge[i][0] + " -- " + edge[i][1] + " == " + wt); } } document.write("Minimum Cost Spanning Tree:",minCost); } //Here 5 is the number of edges, can be asked from the user //when making the graph through user input //3 represents the no of index positions for storing u --> v(adjacent vertices) //and its cost/distance; let edge = [ [0,1,10], [0,2,6], [0,3,5], [1,3,15], [2,3,4] ]; kruskalAlgo(5,edge); // The code is contributed by Arushi Jindal. </script> |
Output
Following are the edges in the constructed MST 2 -- 3 == 4 0 -- 3 == 5 0 -- 1 == 10 Minimum Cost Spanning Tree: 19
Time Complexity: O(E * logE) or O(E * logV)
- Sorting of edges takes O(E * logE) time.
- After sorting, we iterate through all edges and apply the find-union algorithm. The find and union operations can take at most O(logV) time.
- So overall complexity is O(E * logE + E * logV) time.
- The value of E can be at most O(V2), so O(logV) and O(logE) are the same. Therefore, the overall time complexity is O(E * logE) or O(E*logV)
Auxiliary Space: O(V + E), where V is the number of vertices and E is the number of edges in the graph.
This article is compiled by Aashish Barnwal and reviewed by the neveropen team. Please write comments if you find anything incorrect, or if you want to share more information about the topic discussed above.
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