Title: Counting Element Pairs to Make Their AND Having B[i] Set Bits

Introduction

In programming, we often encounter problems where we need to manipulate bitwise operations on integers. One such scenario is when we need to count the number of pairs of elements such that their bitwise AND operation has a specified number of set bits. In this article, we will discuss an efficient algorithm to solve this problem.

Problem Statement

Given an array of integers A with length N and an array of integers B with length M, where B[i] represents the number of set bits the bitwise AND of any two elements of A should have. The task is to find the number of pairs of elements in A that satisfies the condition.

Approach

To count the number of pairs, we can first generate a list of all possible AND values for any two elements. We can use nested loops to compare each element with every other element and calculate its bitwise AND. This will result in an array of values with the number of set bits in each value.

Next, we can use a hash table to keep track of the count of these values. Finally, we can iterate over the B array and count the number of pairs that satisfy the condition.

Pseudo Code

pairs_count = 0
hash_table = {}

for i in range(N):
    for j in range(i + 1, N):
        and_value = A[i] & A[j]
        bit_count = bin(and_value).count('1')
        if bit_count not in hash_table:
            hash_table[bit_count] = 1
        else:
            hash_table[bit_count] += 1

for i in range(M):
    if B[i] in hash_table:
        pairs_count += hash_table[B[i]]

return pairs_count

Time Complexity

The time complexity of the above algorithm is O(N^2 + Mk), where k is the maximum number of set bits in any AND value. The nested loop to generate all possible AND values takes O(N^2) time. The loop to count the number of matching pairs takes O(Mk) time.

Conclusion

In this article, we discussed an approach to count the number of element pairs that have a specified number of set bits in their bitwise AND operation. This problem involves manipulating bitwise operations on integers and can be efficiently solved using hash tables. The time complexity of the algorithm can be improved by using more advanced data structures like trie or bitsets for hashing.