Triples with Bitwise AND Equal To Zero

Triples with Bitwise AND Equal To Zero

What is this problem about?

The "Triples with Bitwise AND Equal To Zero coding problem" is an intensive bitwise manipulation task. You are given an array of integers and need to find the number of triples $(i, j, k)$ such that nums[i] & nums[j] & nums[k] == 0. The indices $i, j, k$ do not have to be distinct. With an array size up to 1000, a brute-force $O(n^3)$ approach would perform a billion operations, which is likely to time out.

Why is this asked in interviews?

This "Triples with Bitwise AND Equal To Zero interview question" is a favorite at companies like Microsoft and Philips for testing a candidate's ability to use "Hash Table frequency" to optimize a multi-variable search. It evaluates if you can break a 3-variable problem into a 2-variable step and a 1-variable step, which is a classic optimization strategy in both math and computer science.

Algorithmic pattern used

The "Array, Hash Table, Bit Manipulation interview pattern" is used to optimize the solution to $O(n^2 + n \cdot 2^{16})$. First, we use a hash map or an array to store the frequency of all possible results of nums[i] & nums[j]. Then, we iterate through the original array and, for each num, we iterate through all possible AND results in our hash map. If (num & and_result) == 0, we add the frequency to our total count.

Example explanation

Array: [2, 1, 3]

1. Calculate pairs (i & j):
   • 2&2=2, 2&1=0, 2&3=2
   • 1&2=0, 1&1=1, 1&3=1
   • 3&2=2, 3&1=1, 3&3=3
2. Frequencies: {0:2, 1:3, 2:3, 3:1}
3. Check against each num:
   • For 2: 2&0=0 (count+2), 2&1=0 (count+3), 2&2=2, 2&3=2.
   • For 1: 1&0=0 (count+2), 1&1=1, 1&2=0 (count+3), 1&3=1.
   • For 3: 3&0=0 (count+2), 3&1=1, 3&2=2, 3&3=3. Total Triples = 2+3+2+3+2 = 12.

Common mistakes candidates make

A common error in the "Triples with Bitwise AND Equal To Zero coding problem" is forgetting that the indices can be the same. Another mistake is using a HashMap for the frequency counts, which can be slow; using a fixed-size array of $2^{16}$ (since numbers are up to $2^{16}-1$) is much faster and more idiomatic for bitwise problems.

Interview preparation tip

When you encounter an $O(n^3)$ or $O(n^4)$ problem, always look for a way to precompute part of the result in a hash table. This "Trade Space for Time" strategy is one of the most powerful tools in algorithm design.