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.
#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 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 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# 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 |
<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)