Find X-Sum of All K-Long Subarrays II

Find X-Sum of All K-Long Subarrays II

1. What is this problem about?

The Find X-Sum of All K-Long Subarrays II interview question is the advanced version of the X-Sum problem. The core rules are the same: for every sliding window of size k, identify the top x most frequent elements and calculate their weighted sum. However, the constraints are significantly larger ($N = 10^5$), meaning you cannot afford to sort the frequency map for every window. You must find a way to update the "Top-X" set in $O(\log N)$ time as the window slides.

2. Why is this asked in interviews?

Bloomberg and other finance-focused companies ask the Find X-Sum coding problem to evaluate a candidate's ability to manage dynamic ranked data. It tests your knowledge of Ordered Sets or Dual-Heap systems. It is a rigorous test of data structure implementation and efficiency, as you must maintain two balanced sets of elements while handling frequent additions and removals.

3. Algorithmic pattern used

This problem is solved using the Sliding Window with Two Sorted Sets pattern.

1. Dual Structure: Maintain two sorted sets (or balanced BSTs):
   • TopX: Contains the (frequency, value) pairs of the top $x$ elements.
   • Rest: Contains all other (frequency, value) pairs currently in the window.
2. Running Sum: Maintain a currentXSum variable that only tracks the sum of elements currently in the TopX set.
3. Update Logic:
   • When an element $E$ enters/leaves:
   • Remove the old (freq, E) from whichever set it was in.
   • Update the frequency.
   • Add the new (freq, E) back into the system.
   • Rebalance: If TopX has fewer than $x$ elements, move the best from Rest to TopX. If Rest has an element better than the worst in TopX, swap them.
4. Efficiency: Each update takes $O(\log ( ext{unique elements}))$.

4. Example explanation

$k=4, x=1$.

• Window: [1, 1, 2, 2]. Freqs: 1:2, 2:2.
• TopX: {(2, 2)} (2 wins tie over 1). Rest: {(2, 1)}. XSum = 2*2 = 4.
• Slide: 1 leaves, 3 enters.
• New Freqs: 1:1, 2:2, 3:1.
• TopX still has (2, 2). Rest now has (1, 1), (1, 3). XSum = 4. Result: [4, 4, ...]

5. Common mistakes candidates make

• Wrong Rebalancing: Failing to update the currentXSum when an element is moved between TopX and Rest.
• Tie-breaking: Inconsistent comparison logic between the two sets (always use frequency first, then value).
• $O(N \cdot X)$ approach: Using a list and linear insertion, which will time out for large $x$.

6. Interview preparation tip

Master the use of two TreeMaps or two std::sets to maintain "Top K" statistics. This pattern is similar to the "Median from Data Stream" problem and is a powerful tool for dynamic optimization questions.