Count number of subsets having a particular XOR value

4.1

Given an array arr[] of n numbers and a number K, find the number of subsets of arr[] having XOR of elements as K

Examples:

Input:   arr[]  = {6, 9, 4, 2}, k = 6
Output:  2
The subsets are {4, 2} and {6}

Input:   arr[]  = {1, 2, 3, 4, 5}, k = 4
Output:  4
The subsets are {1, 5}, {4}, {1, 2, 3, 4}
                and {2, 3, 5}

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Brute Force approach O(2n): One naive approach is to generate all the 2n subsets and count all the subsets having XOR value K, but this approach will not be efficient for large values of n.

Dynamic Programming Approach O(n*m):
We define a number m such that m = pow(2,(log2(max(arr))+1))­ – 1. This number is actually the maximum value any XOR subset will acquire. We get this number by counting bits in largest number. We create a 2D array dp[n+1][m+1], such that dp[i][j] equals to the number of subsets having XOR value j from subsets of arr[0…i-1].

We fill the dp array as following:

  1. We initialize all values of dp[i][j] as 0.
  2. Set value of dp[0][0] = 1 since XOR of an empty set is 0.
  3. Iterate over all the values of arr[i] from left to right and for each arr[i], iterate over all the possible values of XOR i.e from 0 to m (both inclusive) and fill the dp array asfollowing:
           for i = 1 to n:
                 for j = 0 to m:
                       dp[i][j] = dp[i­-1][j] + dp[i­-1][j^arr[i-1]]
    This can be explained as, if there is a subset arr[0…i­-2] with XOR value j, then there also exists a subset arr[0…i-1] with XOR value j. also if there exists a subset arr[0….i-2] with XOR value j^arr[i] then clearly there exist a subset arr[0…i-1] with XOR value j, as j ^ arr[i-1] ^ arr[i-1] = j.
  4. Counting the number of subsets with XOR value k: Since dp[i][j] is the number of subsets having j as XOR value from the subsets of arr[0..i-1], then the number of subsets from set arr[0..n] having XOR value as K will be dp[n][K]
// arr dynamic programming solution to finding the number
// of subsets having xor of their elements as k
#include<bits/stdc++.h>
using namespace std;

// Returns count of subsets of arr[] with XOR value equals
// to k.
int subsetXOR(int arr[], int n, int k)
{
    // Find maximum element in arr[]
    int max_ele = arr[0];
    for (int i=1; i<n; i++)
       if (arr[i] > max_ele)
           max_ele = arr[i];

    // Maximum possible XOR value
    int m = (1 << (int)(log2(max_ele) + 1) ) - 1;

    // The value of dp[i][j] is the number of subsets having
    // XOR of their elements as j from the set arr[0...i-1]
    int dp[n+1][m+1];

    // Initializing all the values of dp[i][j] as 0
    for (int i=0; i<=n; i++)
        for (int j=0; j<=m; j++)
            dp[i][j] = 0;

    // The xor of empty subset is 0
    dp[0][0] = 1;

    // Fill the dp table
    for (int i=1; i<=n; i++)
        for (int j=0; j<=m; j++)
            dp[i][j] = dp[i-1][j] + dp[i-1][j^arr[i-1]];

    //  The answer is the number of subset from set
    //  arr[0..n-1] having XOR of elements as k
    return dp[n][k];
}

// Driver program to test above function
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int k = 4;
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << "Count of subsets is " << subsetXOR(arr, n, k);
    return 0;
}

Output :

Count of subsets is 4

This article is contributed by Pranay Pandey. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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