Special Array With X Elements Greater Than or Equal X - Solution & Explanation

Problem Overview: You receive an integer array nums. The goal is to find a number x such that exactly x elements in the array are greater than or equal to x. If such a value exists, return it. Otherwise return -1. The value of x must be consistent with the count of elements meeting the condition.

Approach 1: Sorting and Linear Scan (Time: O(n log n), Space: O(1) or O(n) depending on sort)

Sort the array in non-decreasing order using a standard sorting algorithm. After sorting, iterate through the array and treat each position as a potential boundary where x elements remain on the right. If the array has n elements and you are at index i, then there are n - i elements that are >= nums[i]. Check whether n - i could be the special value by verifying that nums[i] >= n - i and the previous element (if it exists) is < n - i. The first position that satisfies this condition gives the correct x. Sorting groups smaller values on the left and larger ones on the right, which makes counting elements greater than or equal to a candidate value straightforward.

Approach 2: Binary Search on Result (Time: O(n log n), Space: O(1))

The possible value of x ranges from 0 to n. Instead of testing all values linearly, apply binary search on this range. For each candidate x, iterate through the array and count how many elements are >= x. If the count equals x, you found the special value. If the count is larger than x, move the search range higher; if smaller, move it lower. The binary search reduces the number of candidate checks while the counting step validates the condition. This technique is known as binary search on the answer, a common pattern for problems where the result lies within a numeric range.

Both approaches operate directly on the array and rely on counting elements that satisfy a threshold condition. The sorting approach leverages order to compute counts quickly from indices, while the binary search approach repeatedly evaluates candidate answers.

Recommended for interviews: The sorting and linear scan solution is usually the expected answer. It is simple, deterministic, and easy to reason about after ordering the array. Binary search on the result demonstrates stronger algorithmic thinking and familiarity with the "search on answer" pattern, but the sorting approach is typically faster to implement during interviews while still achieving optimal practical performance.