Overview

In programming interviews, a variety of questions test a candidate’s problem-solving abilities, understanding of data structures, and algorithmic skills. One interesting problem is searching in a sorted array to find an index `i` such that `X[i] = i`. This task appears simple but requires a creative approach to solve efficiently, especially when considering large datasets. The focus is on understanding the properties of the array and optimizing search strategies.

Problem Statement

You are given a sorted array `X` of distinct integers. The goal is to find an index `i` such that `X[i] = i`. Since the array is sorted, use this property to design an efficient solution.

Brute Force Approach

A straightforward solution is the brute force method, iterating through each index of the array to check whether `X[i] = i`. Though simple and easy to implement, this approach is inefficient with a time complexity of $O(n)$, where `n` is the number of elements in the array. It's not suitable for large datasets where performance is a concern.

• Calculate the middle index `mid` as `(low + high) // 2`.
• If `X[mid]` equals `mid`, return `mid` as the fixed point.
• If `X[mid]` is less than `mid`, explore the right half; set `low` to `mid + 1`.
• If `X[mid]` is greater than `mid`, explore the left half; set `high` to `mid - 1`.
• Brute Force: Iterate through each element:
  • `0 != -10`, `1 != -5`, `2 != 0`, `3 == 3` (Found, return 3).
• Binary Search:
  • Initial `low = 0`, `high = 6`; `mid = 3`.
  • Check `X[3] == 3` (Found, return 3).
• No Solution: If no such index exists, both approaches must adequately return `None`.
• Single Element: Handle cases where `n = 1`. If `X[0] == 0`, return 0; otherwise, return `None`.
• Negative Integers: Ensure the solution handles arrays containing negative numbers explicitly, as they won't have a fixed point unless `0` is included.
• This approach assumes distinct integers; if the array isn't distinct, additional logic is needed.
• For multidimensional arrays or unsorted datasets, the approach significantly complicates and might require entirely different strategies.