Triples with Bitwise AND Equal To Zero

Problem Description

Given an integer array nums, return the number of AND triples. An AND triple is a triple of indices (i, j, k) such that nums[i] & nums[j] & nums[k] == 0, where & represents the bitwise-AND operator.

Key Insights

• The problem requires finding all possible combinations of indices (i, j, k) in the array such that the bitwise AND of the elements at these indices equals zero.
• The constraints on the input size (up to 1000 elements) suggest that an O(n^3) brute force solution may be feasible but could be optimized.
• Understanding bitwise operations is crucial, as they will determine when the AND operation results in zero.

Space and Time Complexity

Time Complexity: O(n^3) in the brute force solution, but can potentially be optimized with a different approach. Space Complexity: O(1) for the brute force approach, as no additional data structures are required beyond a few variables.

Solution

A brute force approach involves iterating through all possible combinations of indices (i, j, k) and counting the number of times the condition nums[i] & nums[j] & nums[k] == 0 is satisfied. This can be implemented with three nested loops.

1. Use three nested loops to generate all combinations of indices (i, j, k).
2. For each combination, calculate the bitwise AND of nums[i], nums[j], and nums[k].
3. Increment a counter whenever the result equals zero.

This straightforward approach ensures that all combinations are checked, but it may not be the most efficient for larger datasets.

Code Solutions