Count the Number of Beautiful Subarrays

Problem Description

You are given a 0-indexed integer array nums. In one operation, you can choose two different indices i and j such that 0 <= i, j < nums.length, choose a non-negative integer k such that the k-th bit (0-indexed) in the binary representation of nums[i] and nums[j] is 1, and subtract 2^k from nums[i] and nums[j]. A subarray is beautiful if it is possible to make all of its elements equal to 0 after applying the above operation any number of times. Return the number of beautiful subarrays in the array nums.

Key Insights

• A subarray can be transformed into zeros if all its elements can be reduced to zero using the allowed operations.
• The operations depend on the bits that are set to 1 in the binary representation of the numbers in the subarray.
• A bit position k can only be used to zero out numbers if at least two numbers in the subarray have that bit set.
• We need to find contiguous subarrays where all elements share common bits that can be manipulated together.

Space and Time Complexity

Time Complexity: O(n^2) in the worst case, where n is the length of the array, as we may need to check every subarray. Space Complexity: O(1), as we are using a constant amount of extra space.

Solution

To solve the problem, we can use a two-pointer technique or a nested loop to explore all possible subarrays. For each subarray, we can count the bits set to 1 for each number and check whether there are overlapping bits that allow us to reduce the entire subarray to zero. If there are enough overlapping bits in the current subarray, we can mark it as beautiful.

1. Use two pointers to explore all subarrays.
2. For each subarray, maintain a frequency count of bits that are set to 1.
3. Check if for every bit position, we have at least two elements contributing to it.
4. Count valid subarrays.

Code Solutions