Construct the Minimum Bitwise Array II

Problem Description

You are given an array nums consisting of n prime integers. You need to construct an array ans of length n, such that, for each index i, the bitwise OR of ans[i] and ans[i] + 1 is equal to nums[i], i.e. ans[i] OR (ans[i] + 1) == nums[i]. Additionally, you must minimize each value of ans[i] in the resulting array. If it is not possible to find such a value for ans[i] that satisfies the condition, then set ans[i] = -1.

Key Insights

• The expression x OR (x + 1) results in a number where all bits below the highest set bit in x are set to 1.
• For a prime number p, it can be represented as p = 2^k + r where 0 <= r < 2^k. Thus, the optimal ans[i] can often be derived from the binary representation of p.
• If p is odd, ans[i] must be a power of two minus one to ensure that the OR operation yields the prime.
• If p is even, it cannot be satisfied because primes are odd except for 2.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(n)

Solution

To solve the problem, we iterate through each number in the input array nums. For each prime number:

1. Check if it is equal to 2. If so, assign -1 to ans[i] since there is no valid ans for 2.
2. For odd primes, find the largest power of two less than nums[i] to formulate ans[i].
3. If no valid ans[i] can be derived, set it to -1.

This approach takes advantage of bit manipulation to derive the correct values efficiently.

Code Solutions