Spanning Tree With Maximum Degree (Using Kruskal’s Algorithm)
Given an undirected unweighted connected graph consisting of n vertices and m edges. The task is to find any spanning tree of this graph such that the maximum degree over all vertices is maximum possible. The order in which you print the output edges does not matter and an edge can be printed in reverse also i.e. (u, v) can also be printed as (v, u).
Examples:
Input: 1 / \ 2 5 \ / 3 | 4 Output: 3 2 3 5 3 4 1 2 The maximum degree over all vertices is of vertex 3 which is 3 and is maximum possible. Input: 1 / 2 / \ 5 3 | 4 Output: 2 1 2 5 2 3 3 4
Prerequisite: Kruskal Algorithm to find Minimum Spanning Tree
Approach: The given problem can be solved using Kruskal’s algorithm to find the Minimum Spanning tree.
We find the vertex which has maximum degree in the graph. At first we will perform the union of all the edges which are incident to this vertex and then carry out normal Kruskal’s algorithm. This gives us optimal spanning tree.
C++
#include<bits/stdc++.h> using namespace std; // par and sz will store the parent // and rank of particular node // in the Union Find Algorithm vector< int > par,sz; // Find function of Union Find Algorithm int find( int x) { if (par[x]!=x) par[x]=find(par[x]); return par[x]; } // Union function of Union Find Algorithm void Union( int u, int v) { int x = find(u); int y = find(v); if (x == y) return ; if (sz[x] > sz[y]) par[y] = x; else if (sz[x] < sz[y]) par[x] = y; else { par[x] = y; sz[y]++; } } // Function to find the required spanning tree void findSpanningTree(vector< int > deg, int n, int m,vector<vector< int >> g) { par.resize(n+1); sz.resize(n+1); // Initialising parent of a node // by itself for ( int i = 1; i <= n; i++) par[i] = i; // Variable to store the node // with maximum degree int max = 1; // Finding the node with maximum degree for ( int i = 2; i <= n; i++) if (deg[i] > deg[max]) max = i; // Union of all edges incident // on vertex with maximum degree for ( int v : g[max]) { cout << max << " " << v << '\n' ; Union(max, v); } // Carrying out normal Kruskal Algorithm for ( int u = 1; u <= n; u++) { for ( int v : g[u]) { int x = find(u); int y = find(v); if (x == y) continue ; Union(x, y); cout << u << " " << v << '\n' ; } } } int main() { // Number of nodes int n = 5; // Number of edges int m = 5; // store the graph vector<vector< int >> g(n+1); // store the degree // of each node in the graph vector< int > deg(n+1); // add edges and update degrees g[1].push_back(2); g[2].push_back(1); deg[1]++; deg[2]++; g[1].push_back(5); g[5].push_back(1); deg[1]++; deg[5]++; g[2].push_back(3); g[3].push_back(2); deg[2]++; deg[3]++; g[5].push_back(3); g[3].push_back(5); deg[3]++; deg[5]++; g[3].push_back(4); g[4].push_back(3); deg[3]++; deg[4]++; findSpanningTree(deg, n, m, g); return 0; } |
Java
// Java implementation of the approach import java.util.*; public class GFG { // par and rank will store the parent // and rank of particular node // in the Union Find Algorithm static int par[], rank[]; // Find function of Union Find Algorithm static int find( int x) { if (par[x] != x) par[x] = find(par[x]); return par[x]; } // Union function of Union Find Algorithm static void union( int u, int v) { int x = find(u); int y = find(v); if (x == y) return ; if (rank[x] > rank[y]) par[y] = x; else if (rank[x] < rank[y]) par[x] = y; else { par[x] = y; rank[y]++; } } // Function to find the required spanning tree static void findSpanningTree( int deg[], int n, int m, ArrayList<Integer> g[]) { par = new int [n + 1 ]; rank = new int [n + 1 ]; // Initialising parent of a node // by itself for ( int i = 1 ; i <= n; i++) par[i] = i; // Variable to store the node // with maximum degree int max = 1 ; // Finding the node with maximum degree for ( int i = 2 ; i <= n; i++) if (deg[i] > deg[max]) max = i; // Union of all edges incident // on vertex with maximum degree for ( int v : g[max]) { System.out.println(max + " " + v); union(max, v); } // Carrying out normal Kruskal Algorithm for ( int u = 1 ; u <= n; u++) { for ( int v : g[u]) { int x = find(u); int y = find(v); if (x == y) continue ; union(x, y); System.out.println(u + " " + v); } } } // Driver code public static void main(String args[]) { // Number of nodes int n = 5 ; // Number of edges int m = 5 ; // ArrayList to store the graph ArrayList<Integer> g[] = new ArrayList[n + 1 ]; for ( int i = 1 ; i <= n; i++) g[i] = new ArrayList<>(); // Array to store the degree // of each node in the graph int deg[] = new int [n + 1 ]; // Add edges and update degrees g[ 1 ].add( 2 ); g[ 2 ].add( 1 ); deg[ 1 ]++; deg[ 2 ]++; g[ 1 ].add( 5 ); g[ 5 ].add( 1 ); deg[ 1 ]++; deg[ 5 ]++; g[ 2 ].add( 3 ); g[ 3 ].add( 2 ); deg[ 2 ]++; deg[ 3 ]++; g[ 5 ].add( 3 ); g[ 3 ].add( 5 ); deg[ 3 ]++; deg[ 5 ]++; g[ 3 ].add( 4 ); g[ 4 ].add( 3 ); deg[ 3 ]++; deg[ 4 ]++; findSpanningTree(deg, n, m, g); } } |
Python3
# Python3 implementation of the approach from typing import List # par and rank will store the parent # and rank of particular node # in the Union Find Algorithm par = [] rnk = [] # Find function of Union Find Algorithm def find(x: int ) - > int : global par if (par[x] ! = x): par[x] = find(par[x]) return par[x] # Union function of Union Find Algorithm def Union(u: int , v: int ) - > None : global par, rnk x = find(u) y = find(v) if (x = = y): return if (rnk[x] > rnk[y]): par[y] = x elif (rnk[x] < rnk[y]): par[x] = y else : par[x] = y rnk[y] + = 1 # Function to find the required spanning tree def findSpanningTree(deg: List [ int ], n: int , m: int , g: List [ List [ int ]]) - > None : global rnk, par # Initialising parent of a node # by itself par = [i for i in range (n + 1 )] rnk = [ 0 ] * (n + 1 ) # Variable to store the node # with maximum degree max = 1 # Finding the node with maximum degree for i in range ( 2 , n + 1 ): if (deg[i] > deg[ max ]): max = i # Union of all edges incident # on vertex with maximum degree for v in g[ max ]: print ( "{} {}" . format ( max , v)) Union( max , v) # Carrying out normal Kruskal Algorithm for u in range ( 1 , n + 1 ): for v in g[u]: x = find(u) y = find(v) if (x = = y): continue Union(x, y) print ( "{} {}" . format (u, v)) # Driver code if __name__ = = "__main__" : # Number of nodes n = 5 # Number of edges m = 5 # ArrayList to store the graph g = [[] for _ in range (n + 1 )] # Array to store the degree # of each node in the graph deg = [ 0 ] * (n + 1 ) # Add edges and update degrees g[ 1 ].append( 2 ) g[ 2 ].append( 1 ) deg[ 1 ] + = 1 deg[ 2 ] + = 1 g[ 1 ].append( 5 ) g[ 5 ].append( 1 ) deg[ 1 ] + = 1 deg[ 5 ] + = 1 g[ 2 ].append( 3 ) g[ 3 ].append( 2 ) deg[ 2 ] + = 1 deg[ 3 ] + = 1 g[ 5 ].append( 3 ) g[ 3 ].append( 5 ) deg[ 3 ] + = 1 deg[ 5 ] + = 1 g[ 3 ].append( 4 ) g[ 4 ].append( 3 ) deg[ 3 ] + = 1 deg[ 4 ] + = 1 findSpanningTree(deg, n, m, g) # This code is contributed by sanjeev2552 |
C#
// C# implementation of the approach using System; using System.Collections.Generic; class GFG { // par and rank will store the parent // and rank of particular node // in the Union Find Algorithm static int []par; static int []rank; // Find function of Union Find Algorithm static int find( int x) { if (par[x] != x) par[x] = find(par[x]); return par[x]; } // Union function of Union Find Algorithm static void union( int u, int v) { int x = find(u); int y = find(v); if (x == y) return ; if (rank[x] > rank[y]) par[y] = x; else if (rank[x] < rank[y]) par[x] = y; else { par[x] = y; rank[y]++; } } // Function to find the required spanning tree static void findSpanningTree( int []deg, int n, int m, List< int > []g) { par = new int [n + 1]; rank = new int [n + 1]; // Initialising parent of a node // by itself for ( int i = 1; i <= n; i++) par[i] = i; // Variable to store the node // with maximum degree int max = 1; // Finding the node with maximum degree for ( int i = 2; i <= n; i++) if (deg[i] > deg[max]) max = i; // Union of all edges incident // on vertex with maximum degree foreach ( int v in g[max]) { Console.WriteLine(max + " " + v); union(max, v); } // Carrying out normal Kruskal Algorithm for ( int u = 1; u <= n; u++) { foreach ( int v in g[u]) { int x = find(u); int y = find(v); if (x == y) continue ; union(x, y); Console.WriteLine(u + " " + v); } } } // Driver code public static void Main(String []args) { // Number of nodes int n = 5; // Number of edges int m = 5; // ArrayList to store the graph List< int > []g = new List< int >[n + 1]; for ( int i = 1; i <= n; i++) g[i] = new List< int >(); // Array to store the degree // of each node in the graph int []deg = new int [n + 1]; // Add edges and update degrees g[1].Add(2); g[2].Add(1); deg[1]++; deg[2]++; g[1].Add(5); g[5].Add(1); deg[1]++; deg[5]++; g[2].Add(3); g[3].Add(2); deg[2]++; deg[3]++; g[5].Add(3); g[3].Add(5); deg[3]++; deg[5]++; g[3].Add(4); g[4].Add(3); deg[3]++; deg[4]++; findSpanningTree(deg, n, m, g); } } // This code has been contributed by 29AjayKumar |
Javascript
<script> // JavaScript implementation of the approach // par and rank will store the parent // and rank of particular node // in the Union Find Algorithm let par, rank; // Find function of Union Find Algorithm function find(x) { if (par[x] != x) par[x] = find(par[x]); return par[x]; } // Union function of Union Find Algorithm function union(u, v) { let x = find(u); let y = find(v); if (x == y) return ; if (rank[x] > rank[y]) par[y] = x; else if (rank[x] < rank[y]) par[x] = y; else { par[x] = y; rank[y]++; } } // Function to find the required spanning tree function findSpanningTree(deg, n, m, g) { par = new Array(n + 1); rank = new Array(n + 1); // Initialising parent of a node // by itself for (let i = 1; i <= n; i++) par[i] = i; // Variable to store the node // with maximum degree let max = 1; // Finding the node with maximum degree for (let i = 2; i <= n; i++) if (deg[i] > deg[max]) max = i; // Union of all edges incident // on vertex with maximum degree for (let v = 0; v < g[max].length; v++) { document.write(max + " " + g[max][v] + "</br>" ); union(max, g[max][v]); } // Carrying out normal Kruskal Algorithm for (let u = 1; u <= n; u++) { for (let v = 0; v < g[u].length; v++) { let x = find(u); let y = find(g[u][v]); if (x == y) continue ; union(x, y); document.write(u + " " + g[u][v] + "</br>" ); } } } // Number of nodes let n = 5; // Number of edges let m = 5; // ArrayList to store the graph let g = new Array(n + 1); for (let i = 1; i <= n; i++) g[i] = []; // Array to store the degree // of each node in the graph let deg = new Array(n + 1); deg.fill(0); // Add edges and update degrees g[1].push(2); g[2].push(1); deg[1]++; deg[2]++; g[1].push(5); g[5].push(1); deg[1]++; deg[5]++; g[2].push(3); g[3].push(2); deg[2]++; deg[3]++; g[5].push(3); g[3].push(5); deg[3]++; deg[5]++; g[3].push(4); g[4].push(3); deg[3]++; deg[4]++; findSpanningTree(deg, n, m, g); </script> |
3 2 3 5 3 4 1 2
Time Complexity: O(N * logN), where N is the total number of nodes in the graph.
Auxiliary Space: O(N)
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