Find X Value of Array II

Find X Value of Array II

1. What is this problem about?

The Find X Value of Array II interview question is the high-performance version of the $X$-value search. The constraints are intensified, or the queries are made dynamic (e.g., the array changes, or you need to answer many different targets). This version typically requires an $O(\log N)$ or $O(1)$ query time after some sophisticated preprocessing.

2. Why is this asked in interviews?

Rubrik uses the Find X Value of Array II coding problem to evaluate advanced knowledge of Segment Trees or Fenwick Trees. It tests if you can maintain aggregate statistics (like sums and counts) over a range of values, allowing you to perform the "check" function of a binary search in sub-linear time. It's a test of data structure engineering.

3. Algorithmic pattern used

This problem follows the Binary Search on a Segment Tree or Prefix Sums with Sorting pattern.

1. Sorting: Sort the array to make the "check" function easier.
2. Preprocessing: Use prefix sums to calculate $\sum nums[i]$ for any range in $O(1)$.
3. Search: For a given $X$, you can use binary search to find how many elements are $> X$. The sum $\sum (nums[i] - X)$ can then be calculated as RangeSum - (Count * X).
4. Tree Structure: If the array is updated, a Segment Tree can store the sum and count of elements within specific value ranges.

4. Example explanation

Target sum of differences = 100.

• Use a Segment Tree where nodes store (sum, count) of values in their range.
• Traverse the tree: if the right child's "effective sum" is greater than 100, the $X$ must be in the right child. If not, subtract the right child's contribution and search the left. This "Walk on Segment Tree" finds the exact $X$ in $O(\log ( ext{MaxValue}))$.

5. Common mistakes candidates make

• Standard Binary Search TLE: Not realizing that $Q$ queries each taking $O(N \log N)$ will be too slow. You must optimize the check to $O(1)$ or $O(\log N)$.
• Coordinate Compression: Failing to handle very large or sparse values if using a tree structure.
• Complex Math: Over-complicating the algebra needed to extract $X$ from the range sum formula.

6. Interview preparation tip

Master the "Walk on Tree" technique. Instead of doing a binary search using a segment tree (which is $O(\log^2 N)$), you can binary search inside the segment tree traversal to achieve $O(\log N)$. This is a top-tier Segment Tree interview pattern.