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Maximal independent set from a given Graph using Backtracking

  • Difficulty Level : Hard
  • Last Updated : 25 Nov, 2021

Given an undirected graph with V vertices and E edges, the task is to print all the independent sets and also find the maximal independent set(s)
 

Independent set is a set of vertices such that any two vertices in the set do not have a direct edge between them.

 

Maximal independent set is an independent set having highest number of vertices. 
 



Note: There can be more than one independent and maximal independent sets for a given graph.
Examples:
 

Input: 
V = 3, E = 0 
Graph: 
 

Graph for example 1

Output: 
{ }{ 1 }{ 1 2 }{ 1 2 3 }{ 1 3 }{ 2 }{ 2 3 }{ 3 } 
{ 1 2 3 } 
Explanation: 
The first line represents all the possible independent sets for the given graph. The second line has the maximal independent sets possible for the given graph.
Input: 
V = 4, E = 4 
Graph: 
 

Graph for example 2

Output: 
{ }{ 1 }{ 1 3 }{ 2 }{ 2 4 }{ 3 }{ 4 } 
{ 1 3 }{ 2 4 } 
 

 

Approach: 
The idea is to use Backtracking to solve the problem. At every step, we need to check whether the current node has any direct edge with any of the nodes already present in our independent set. If not, we can add it to our independent set and recursively repeat the same process for all the nodes.
 

Illustration: 
Recursion tree for the first example: 
 

Backtracking tree for all possible sets

In the above backtracking tree we will choose only those sets which are produced after adding a safe node to maintain the set as an independent set.

Below is the implementation of the above approach:
 

CPP




// C++ Program to print the
// independent sets and
// maximal independent sets
// of the given graph
 
#include <bits/stdc++.h>
using namespace std;
 
// To store all the independent
// sets of the graph
set<set<int> > independentSets;
 
// To store all maximal independent
// sets in the graph
set<set<int> > maximalIndependentSets;
 
map<pair<int, int>, int> edges;
vector<int> vertices;
 
// Function to print all independent sets
void printAllIndependentSets()
{
    for (auto iter : independentSets) {
        cout << "{ ";
        for (auto iter2 : iter) {
            cout << iter2 << " ";
        }
        cout << "}";
    }
    cout << endl;
}
 
// Function to extract all
// maximal independent sets
void printMaximalIndependentSets()
{
    int maxCount = 0;
    int localCount = 0;
    for (auto iter : independentSets) {
 
        localCount = 0;
        for (auto iter2 : iter) {
            localCount++;
        }
        if (localCount > maxCount)
            maxCount = localCount;
    }
    for (auto iter : independentSets) {
 
        localCount = 0;
        set<int> tempMaximalSet;
 
        for (auto iter2 : iter) {
            localCount++;
            tempMaximalSet.insert(iter2);
        }
        if (localCount == maxCount)
            maximalIndependentSets
                .insert(tempMaximalSet);
    }
    for (auto iter : maximalIndependentSets) {
        cout << "{ ";
        for (auto iter2 : iter) {
            cout << iter2 << " ";
        }
        cout << "}";
    }
    cout << endl;
}
 
// Function to check if a
// node is safe node.
bool isSafeForIndependentSet(
    int vertex,
    set<int> tempSolutionSet)
{
    for (auto iter : tempSolutionSet) {
        if (edges[make_pair(iter, vertex)]) {
            return false;
        }
    }
    return true;
}
 
// Recursive function to find
// all independent sets
void findAllIndependentSets(
    int currV,
    int setSize,
    set<int> tempSolutionSet)
{
    for (int i = currV; i <= setSize; i++) {
        if (isSafeForIndependentSet(
                vertices[i - 1],
                tempSolutionSet)) {
            tempSolutionSet
                .insert(vertices[i - 1]);
            findAllIndependentSets(
                i + 1,
                setSize,
                tempSolutionSet);
            tempSolutionSet
                .erase(vertices[i - 1]);
        }
    }
    independentSets
        .insert(tempSolutionSet);
}
 
// Driver Program
int main()
{
    int V = 3, E = 0;
 
    for (int i = 1; i <= V; i++)
        vertices.push_back(i);
 
    vector<pair<int, int> > inputEdges;
 
    pair<int, int> edge;
    int x, y;
    for (int i = 0; i < E; i++) {
          cout<<i<<endl;
        edge.first = inputEdges[i].first;
        edge.second = inputEdges[i].second;
        edges[edge] = 1;
        int t = edge.first;
        edge.first = edge.second;
        edge.second = t;
        edges[edge] = 1;
    }
 
    set<int> tempSolutionSet;
 
    findAllIndependentSets(1,
                           V,
                           tempSolutionSet);
 
    printAllIndependentSets();
 
    printMaximalIndependentSets();
 
    return 0;
}

Python3




# Python3 Program to print the
# independent sets and
# maximal independent sets
# of the given graph
 
# To store all the independent
# sets of the graph
independentSets=set()
 
# To store all maximal independent
# sets in the graph
maximalIndependentSets=set()
 
edges=dict()
vertices=[]
 
# Function to print all independent sets
def printAllIndependentSets():
    for itr in independentSets:
        print("{",end=" ")
        for itr2 in itr:
            print(itr2,end= " ")
         
        print("}",end='')
     
    print()
 
 
# Function to extract all
# maximal independent sets
def printMaximalIndependentSets():
    maxCount = 0;localCount = 0
    for itr in independentSets:
        localCount = 0
        for itr2 in itr:
            localCount+=1
         
        if (localCount > maxCount):
            maxCount = localCount
     
    for itr in independentSets:
 
        localCount = 0
        tempMaximalSet=set()
 
        for itr2 in itr:
            localCount+=1
            tempMaximalSet.add(itr2)
         
        if (localCount == maxCount):
            maximalIndependentSets.add(frozenset(tempMaximalSet))
     
    for itr in maximalIndependentSets :
        print("{",end=" ")
        for itr2 in itr:
            print(itr2,end=" ")
         
        print("}",end="")
     
    print()
 
 
# Function to check if a
# node is safe node.
def isSafeForIndependentSet(vertex, tempSolutionSet):
    for itr in tempSolutionSet:
        if (itr, vertex) in edges:
            return False
 
    return True
 
 
# Recursive function to find
# all independent sets
def findAllIndependentSets(currV, setSize, tempSolutionSet):
    for i in range(currV,setSize+1):
        if (isSafeForIndependentSet(vertices[i - 1], tempSolutionSet)) :
            tempSolutionSet.add(vertices[i - 1])
            findAllIndependentSets(i + 1, setSize, tempSolutionSet)
            tempSolutionSet.remove(vertices[i - 1])
         
    independentSets.add(frozenset(tempSolutionSet))
 
 
# Driver Program
if __name__ == '__main__':
    V = 3; E = 0
 
    for i in range(1,V+1):
        vertices.append(i)
 
    inputEdges=[]
 
    for i in range(E):
        edges[inputEdges[i]]=True
        edges[(inputEdges[i][1],inputEdges[i][0])]=True
     
 
    tempSolutionSet=set()
 
    findAllIndependentSets(1, V, tempSolutionSet)
 
    printAllIndependentSets()
 
    printMaximalIndependentSets()
Output: 
{ }{ 1 }{ 1 2 }{ 1 2 3 }{ 1 3 }{ 2 }{ 2 3 }{ 3 }
{ 1 2 3 }

 

Time Complexity: O(2 ^ N)

Auxiliary Space: O(2 ^ N)




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