Find a Value of a Mysterious Function Closest to Target

Find a Value of a Mysterious Function Closest to Target

What is this problem about?

The Find a Value of a Mysterious Function Closest to Target coding problem involves an array of integers and a bitwise AND function defined over a range. The "mysterious function" $f(i, j)$ calculates the bitwise AND of all elements from index $i$ to $j$. You are given a target value and need to find a range $[i, j]$ such that the absolute difference between $f(i, j)$ and target is minimized.

Why is this asked in interviews?

This "Hard" difficulty problem is used by companies like American Express to test advanced Bit Manipulation interview pattern skills and optimization techniques. Since the number of sub-ranges is $O(N^2)$, a brute-force check is impossible for large arrays. It evaluations your ability to realize that bitwise AND is a monotonic non-increasing operation—as you include more numbers, the result can only stay the same or decrease.

Algorithmic pattern used

The problem is solved using a Sliding Window or a Set-based DP approach. The key observation is that for any fixed starting position, there are only at most 30 unique values of bitwise AND possible (since each value change must turn at least one bit from 1 to 0).

1. Maintain a set of all possible AND values ending at the current index.
2. For each new element, AND it with all values in the previous set to get the new set.
3. Keep track of the value closest to the target throughout this process.

Example explanation

Array: [9, 12, 3], Target: 3.

1. Index 0: Set = {9}. Closest diff: $|9-3| = 6$.
2. Index 1: New value 12. AND with previous: 9 & 12 = 8. Set = {12, 8}. Closest diff: $|8-3| = 5$.
3. Index 2: New value 3. AND with previous: 12 & 3 = 0, 8 & 3 = 0. Set = {3, 0}. Closest diff: $|3-3| = 0$. Result: 0.

Common mistakes candidates make

• Brute force: Calculating AND for all $N^2$ subarrays, which leads to $O(N^2)$ or $O(N^3)$ complexity.
• Ignoring bitwise properties: Not realizing that the number of unique AND prefixes is extremely small (logarithmic).
• Complexity confusion: Trying to use a Segment Tree for range AND queries, which is $O(N \log^2 N)$ and still requires a smart way to choose the ranges.

Interview preparation tip

Bitwise AND and OR have a unique property: they are "lossy." Once a bit is 0 in an AND sequence, it can never become 1 again. This means the results of range operations change very few times, which is a powerful shortcut for many Array interview pattern problems.