Sort Integers by Binary Reflection - Solution & Explanation

Problem Overview: You are given a list of integers and need to sort them based on their binary reflection. For each number, convert it to binary, reverse the bit sequence, interpret the reversed bits as a new integer, and sort the array according to this reflected value.

Approach 1: Brute Force Reflection During Comparison (O(n log n * b) time, O(1) extra space)

The most direct solution uses a custom comparator during sorting. Every time two numbers are compared, convert each integer to its binary form, reverse the bit string, convert it back to a decimal value, and compare those reflected values. The sorting algorithm repeatedly performs these conversions, which adds overhead proportional to the number of bits b. This approach works but is inefficient because the same reflection may be recomputed many times during the sort.

Approach 2: Custom Sorting with Precomputed Reflection Keys (O(n log n) time, O(n) space)

A more efficient method computes the binary reflection for each integer once and uses it as a sorting key. Iterate through the array, convert each number to binary, reverse the bits, and store the resulting integer. Then apply a standard sort where the key is this precomputed reflection value. Most languages support this pattern using a key function or custom comparator. Because each reflection is calculated once, the overall complexity becomes O(n log n) for sorting plus O(n * b) preprocessing.

The core operation is the bit reflection step. You can perform it by converting the number to a binary string and reversing it, or by repeatedly extracting the least significant bit and building the reversed value using bit operations. The bitwise method avoids string allocations and can be slightly faster, though both approaches are acceptable for an easy-level problem.

This problem mainly tests your understanding of array manipulation and custom comparators in sorting. The key insight is that the original value does not determine order directly; the order is based on a transformed representation of each element.

Recommended for interviews: Use the precomputed reflection key approach with custom sorting. Start by explaining the brute force comparator to demonstrate the core idea, then optimize by computing reflections once and sorting with a key function. Interviewers expect you to recognize unnecessary repeated work inside comparators and move that logic outside the sort.