Find X Value of Array II - Solution & Explanation

Problem Overview: You are given an array and must compute the array's X value, which depends on pairwise relationships between elements. The challenge becomes harder because the array changes over time, so you must update values and recompute the X metric efficiently without recalculating every pair from scratch.

Approach 1: Brute Force Pair Enumeration (O(n²) time, O(1) space)

The most direct solution recomputes the X value by iterating over every pair of indices (i, j) with i < j. For each pair, compute the contribution defined by the problem’s formula and add it to the running total. This guarantees correctness but becomes impractical for large arrays because each update forces a full recomputation. With n elements, the pair count grows to n(n-1)/2, leading to quadratic time. This approach mainly helps verify correctness during initial implementation.

Approach 2: Sorted Structure + Prefix Math (O(n log n) time, O(n) space)

Instead of recomputing every pair, maintain elements in sorted order and track prefix aggregates such as counts and sums. When processing an element, you can compute its total contribution against previously processed elements using mathematical relationships rather than explicit pair iteration. Binary search or balanced structures keep the array ordered, while prefix sums quickly calculate aggregated differences. This reduces repeated work and converts many pair operations into constant-time arithmetic. The main cost comes from maintaining order during updates.

Approach 3: Segment Tree with Frequency and Sum Tracking (O(n log n) time, O(n) space)

The scalable solution uses a segment tree indexed by value range. Each node stores two pieces of information: the number of elements in that range and the sum of those elements. When inserting or updating a value, query the tree to compute how many elements are smaller or larger and the total of those groups. Using these aggregates, you derive the element’s total contribution to the X value in logarithmic time. Updates simply adjust the counts and sums along the tree path. This approach combines array processing with mathematical aggregation, keeping every operation at O(log n).

Recommended for interviews: Start by explaining the brute force pair enumeration to show you understand how the X value is derived. Then transition to the segment tree solution. Interviewers expect you to recognize that repeated pair recomputation is too slow and replace it with aggregated counts and sums stored in a tree structure, reducing each update or query to logarithmic time.