Number of subsets with a given AND value

Given an array arr of length N and an integer X, the task is to find the number of subsets whose AND value is X.

Examples:

Input: arr[] = {2, 3, 2} X = 2
Output: 6
All possible subsets and there AND values are:
{2} = 2
{3} = 3
{2} = 2
{2, 3} = 2 & 3 = 2
{3, 2} = 3 & 2 = 2
{2, 2} = 2 & 2 = 2
{2, 3, 2} = 2 & 3 & 2 = 2



Input: arr[] = {0, 0, 0}, X = 0
Output: 7

Approach: A simple approach is to solve the problem by generating all the possible subsets and then by counting the number of subsets with the given AND value. However, for smaller values of array elements, it can be solved using dynamic programming.
Let’s look at the recurrence relation first.

dp[i][curr_and] = dp[i + 1][curr_and] + dp[i + 1][curr_and & arr[i]]

The above recurrence relation can be defined as the number of subsets of sub-array arr[i…N-1] such that ANDing them with curr_and will yield the required AND value.
The recurrence relation is justified as there are only paths. Either, take the current element and AND it with curr_and or ignore it and move forward.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
#define maxN 20
#define maxM 64
  
// To store states of DP
int dp1[maxN][maxM];
bool v1[maxN][maxM];
  
// Function to return the required count
int findCnt(int* arr, int i, int curr, int n, int m)
{
    // Base case
    if (i == n) {
        return (curr == m);
    }
  
    // If the state has been solved before
    // return the value of the state
    if (v1[i][curr])
        return dp1[i][curr];
  
    // Setting the state as solved
    v1[i][curr] = 1;
  
    // Recurrence relation
    dp1[i][curr]
        = findCnt(arr, i + 1, curr, n, m)
          + findCnt(arr, i + 1, (curr & arr[i]), n, m);
  
    return dp1[i][curr];
}
  
// Driver code
int main()
{
    int arr[] = { 0, 0, 0 };
    int n = sizeof(arr) / sizeof(int);
    int m = 0;
  
    cout << findCnt(arr, 0, ((1 << 6) - 1), n, m);
  
    return 0;
}

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Java

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// Java implementation of the approach
class GFG 
{
static int maxN = 20;
static int maxM = 64;
  
// To store states of DP
static int [][]dp1 = new int[maxN][maxM];
static boolean [][]v1 = new boolean[maxN][maxM];
  
// Function to return the required count
static int findCnt(int []arr, int i, 
                   int curr, int n, int m)
{
    // Base case
    if (i == n) 
    {
        return (curr == m ? 1 : 0);
    }
  
    // If the state has been solved before
    // return the value of the state
    if (v1[i][curr])
        return dp1[i][curr];
  
    // Setting the state as solved
    v1[i][curr] = true;
  
    // Recurrence relation
    dp1[i][curr] = findCnt(arr, i + 1, curr, n, m) +
                   findCnt(arr, i + 1, (curr & arr[i]), n, m);
  
    return dp1[i][curr];
}
  
// Driver code
public static void main(String []args) 
{
    int arr[] = { 0, 0, 0 };
    int n = arr.length;
    int m = 0;
  
    System.out.println(findCnt(arr, 0, ((1 << 6) - 1), n, m));
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 implementation of the approach 
import numpy as np
  
maxN = 20
maxM = 64
  
# To store states of DP 
dp1 = np.zeros((maxN, maxM)); 
v1 = np.zeros((maxN, maxM)); 
  
# Function to return the required count 
def findCnt(arr, i, curr, n, m) :
  
    # Base case 
    if (i == n) :
        return (curr == m); 
  
    # If the state has been solved before 
    # return the value of the state 
    if (v1[i][curr]) :
        return dp1[i][curr]; 
  
    # Setting the state as solved 
    v1[i][curr] = 1
  
    # Recurrence relation 
    dp1[i][curr] = findCnt(arr, i + 1, curr, n, m) + \
                   findCnt(arr, i + 1, (curr & arr[i]), n, m); 
  
    return dp1[i][curr]; 
  
# Driver code 
if __name__ == "__main__"
      
    arr = [ 0, 0, 0 ]; 
    n = len(arr); 
    m = 0
  
    print(findCnt(arr, 0, ((1 << 6) - 1), n, m)); 
  
# This code is contributed by AnkitRai01

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C#

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// C# implementation of the approach
using System;
  
class GFG 
{
static int maxN = 20;
static int maxM = 64;
  
// To store states of DP
static int [,]dp1 = new int[maxN, maxM];
static bool [,]v1 = new bool[maxN, maxM];
  
// Function to return the required count
static int findCnt(int []arr, int i, 
                   int curr, int n, int m)
{
    // Base case
    if (i == n) 
    {
        return (curr == m ? 1 : 0);
    }
  
    // If the state has been solved before
    // return the value of the state
    if (v1[i, curr])
        return dp1[i, curr];
  
    // Setting the state as solved
    v1[i, curr] = true;
  
    // Recurrence relation
    dp1[i, curr] = findCnt(arr, i + 1, curr, n, m) +
                   findCnt(arr, i + 1, (curr & arr[i]), n, m);
  
    return dp1[i, curr];
}
  
// Driver code
public static void Main(String []args) 
{
    int []arr = { 0, 0, 0 };
    int n = arr.Length;
    int m = 0;
  
    Console.WriteLine(findCnt(arr, 0, ((1 << 6) - 1), n, m));
}
}
  
// This code is contributed by Rajput-Ji

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Output:

7


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