Construct the Minimum Bitwise Array II

Construct the Minimum Bitwise Array II

What is this problem about?

The Construct the Minimum Bitwise Array II interview question is the optimized version of the previous problem. Given an array of primes, find the smallest x such that x OR (x + 1) = nums[i]. With larger constraints, you can no longer use a linear search. You must derive the bit-level structure of x directly from nums[i].

Why is this asked in interviews?

This Construct the Minimum Bitwise Array II coding problem is asked to see if a candidate can move from simulation to observation. It tests your ability to manipulate specific bits and understand the "trailing ones" property of binary numbers. It is a classic "optimization through observation" problem favored by companies like Aon.

Algorithmic pattern used

This follows the Array, Bit Manipulation interview pattern.

• Let's look at x OR (x + 1). If x ends in a sequence of ones (e.g., ...0111), then x+1 will be ...1000.
• The OR result will be ...1111.
• This means nums[i] must be formed by taking some number x and turning its lowest 0 into a 1.
• To make x as small as possible, we should find the rightmost block of consecutive 1s in nums[i] and turn the last 1 in that block into a 0.

Example explanation

Suppose nums[i] = 31 (binary 11111).

1. The rightmost (and only) block of 1s is at the end.
2. Turn the last 1 into 0: 11110 (binary 30).
3. Check: 30 OR 31 = 31. This is the smallest x. Suppose nums[i] = 11 (binary 1011).
4. Rightmost block of 1s is the last two bits (11).
5. Turn the last 1 of that block (bit at index 1) into 0: 1001 (binary 9).
6. Check: 9 OR 10 = 1001 OR 1010 = 1011 (11).

Common mistakes candidates make

• Linear Search: Using the O(N) approach from the Easy version, which will result in TLE.
• Wrong Bit Flip: Flipping the wrong bit in the sequence of 1s. Only flipping the least significant bit of the trailing ones guarantees the minimum x.
• Not handling 2: The prime number 2 is the only prime that cannot be represented this way (it ends in 10, not ...01).

Interview preparation tip

Practice finding the "lowest set bit" (x & -x) and the "lowest unset bit" (~x & (x + 1)). These are fundamental building blocks for efficient bitwise algorithms.