Kruskal’s Minimum Spanning Tree Algorithm | Greedy Algo-2
What is Minimum Spanning Tree?
Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. 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. The weight of a spanning tree is the sum of weights given to each edge of the spanning tree.
How many edges does a minimum spanning tree has?
A minimum spanning tree has (V – 1) edges where V is the number of vertices in the given graph.
What are the applications of the Minimum Spanning Tree?
See this for applications of MST.
Below are the steps for finding MST using Kruskal’s algorithm
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1. Sort all the edges in non-decreasing order of their weight.
2. 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.
3. 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.
Union-Find Algorithm | Set 1 (Detect Cycle in a Graph)
Union-Find Algorithm | Set 2 (Union By Rank and Path Compression)
The algorithm is a Greedy Algorithm. 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: Consider the below 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 Src Dest 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
1. Pick edge 7-6: No cycle is formed, include it.
2. Pick edge 8-2: No cycle is formed, include it.
3. Pick edge 6-5: No cycle is formed, include it.
4. Pick edge 0-1: No cycle is formed, include it.
5. Pick edge 2-5: No cycle is formed, include it.
6. Pick edge 8-6: Since including this edge results in the cycle, discard it.
7. Pick edge 2-3: No cycle is formed, include it.
8. Pick edge 7-8: Since including this edge results in the cycle, discard it.
9. Pick edge 0-7: No cycle is formed, include it.
10. Pick edge 1-2: Since including this edge results in the cycle, discard it.
11. Pick edge 3-4: No cycle is formed, include it.
Since the number of edges included equals (V – 1), the algorithm stops here.
Below is the implementation of the above idea:
Java
// Java program for Kruskal's algorithm to // find Minimum Spanning Tree of a given //connected, undirected and weighted graphimport java.util.*;import java.lang.*;import java.io.*;class Graph { // A class to represent a graph edge class Edge implements Comparable<Edge> { int src, dest, weight; // Comparator function used for // sorting edgesbased on their weight public int compareTo(Edge compareEdge) { return this.weight - compareEdge.weight; } }; // A class to represent a subset for // union-find class subset { int parent, rank; }; int V, E; // V-> no. of vertices & E->no.of edges Edge edge[]; // collection of all edges // 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() { // Tnis 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(); // Step 1: 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 Arrays.sort(edge); // Allocate memory for creating V ssubsets 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; // Index used to pick next edge // Number of edges to be taken is equal to V-1 while (e < V - 1) { // Step 2: Pick the smallest edge. And increment // the index for next iteration Edge next_edge = edge[i++]; int x = find(subsets, next_edge.src); int y = find(subsets, next_edge.dest); // If including this edge does'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); } // Else discard the next_edge } // print the contents of result[] to display // the built MST System.out.println("Following are the edges in " + "the constructed MST"); int minimumCost = 0; for (i = 0; i < e; ++i) { System.out.println(result[i].src + " -- " + result[i].dest + " == " + result[i].weight); minimumCost += result[i].weight; } System.out.println("Minimum Cost Spanning Tree " + minimumCost); } // Driver Code public static void main(String[] args) { /* Let us create following weighted graph 10 0--------1 | \ | 6| 5\ |15 | \ | 2--------3 4 */ int V = 4; // Number of vertices in graph int E = 5; // Number of edges in graph 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 |
Python
# Python program for Kruskal's algorithm to find# Minimum Spanning Tree of a given connected,# undirected and weighted graphfrom collections import defaultdict# Class to represent a graphclass Graph: def __init__(self, vertices): self.V = vertices # No. of vertices self.graph = [] # default dictionary # to store 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 # (uses path compression technique) def find(self, parent, i): if parent[i] == i: return i return self.find(parent, 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): xroot = self.find(parent, x) yroot = self.find(parent, y) # Attach smaller rank tree under root of # high rank tree (Union by Rank) if rank[xroot] < rank[yroot]: parent[xroot] = yroot elif rank[xroot] > rank[yroot]: parent[yroot] = xroot # If ranks are same, then make one as root # and increment its rank by one else: parent[yroot] = xroot rank[xroot] += 1 # The main function to construct MST using Kruskal's # algorithm def KruskalMST(self): result = [] # This will store the resultant MST # An index variable, used for sorted edges i = 0 # An index variable, used for result[] e = 0 # Step 1: 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 graph 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 equal to V-1 while e < self.V - 1: # Step 2: 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 does't # cause cycle, include it in result # and increment the indexof 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 codeg = 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 callg.KruskalMST()# This code is contributed by Neelam Yadav |
C#
// C# Code for above approachusing 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; }; int V, E; // V-> no. of vertices & E->no.of edges Edge[] edge; // collection of all edges // 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]; int e = 0; // An index variable, used for result[] int i = 0; // An index variable, used for sorted edges for (i = 0; i < V; ++i) result[i] = new Edge(); // Step 1: 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 ssubsets 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; // Index used to pick next edge // Number of edges to be taken is equal to V-1 while (e < V - 1) { // Step 2: 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 does'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); } // Else discard the next_edge } // 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 Code public static void Main(String[] args) { /* Let us create following weighted graph 10 0--------1 | \ | 6| 5\ |15 | \ | 2--------3 4 */ int V = 4; // Number of vertices in graph int E = 5; // Number of edges in graph 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 |
C++
#include <bits/stdc++.h>using namespace std;// DSU data structure// path compression + rank by unionclass 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; rank[s2] += rank[s1]; } else { parent[s2] = s1; rank[s1] += rank[s2]; } } }};class Graph{ vector<vector<int>> edgelist; int V;public: Graph(int V) { this->V = V; } void addEdge(int x, int y, int w) { edgelist.push_back({w, x, y}); } int kruskals_mst() { // 1. Sort all edges sort(edgelist.begin(), edgelist.end()); // Initialize the DSU DSU s(V); int ans = 0; for (auto edge : edgelist) { int w = edge[0]; int x = edge[1]; int y = edge[2]; // take that edge in MST if it does form a cycle if (s.find(x) != s.find(y)) { s.unite(x, y); ans += w; } } return ans; }};int main(){ Graph g(4); g.addEdge(0, 1, 1); g.addEdge(1, 3, 3); g.addEdge(3, 2, 4); g.addEdge(2, 0, 2); g.addEdge(0, 3, 2); g.addEdge(1, 2, 2); // int n, m; // cin >> n >> m; // Graph g(n); // for (int i = 0; i < m; i++) // { // int x, y, w; // cin >> x >> y >> w; // g.addEdge(x, y, w); // } cout << g.kruskals_mst(); return 0;} |
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(ElogE) or O(ElogV). Sorting of edges takes O(ELogE) 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(ELogE + ELogV) time. The value of E can be at most O(V2), so O(LogV) is O(LogE) the same. Therefore, the overall time complexity is O(ElogE) or O(ElogV)
References:
http://www.ics.uci.edu/~eppstein/161/960206.html
http://en.wikipedia.org/wiki/Minimum_spanning_tree
This article is compiled by Aashish Barnwal and reviewed by the GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
