Find X-Sum of All K-Long Subarrays I

Find X-Sum of All K-Long Subarrays I

1. What is this problem about?

The Find X-Sum of All K-Long Subarrays I interview question introduces a custom metric called the "X-Sum." For any subarray, the X-Sum is calculated by:

1. Counting the frequency of each element.
2. Picking the top $x$ most frequent elements (using the value as a tie-breaker if frequencies are equal).
3. Calculating the sum of these elements multiplied by their frequencies. If there are fewer than $x$ unique elements, you sum all of them. Your goal is to find the X-Sum for every contiguous subarray of length k.

2. Why is this asked in interviews?

Microsoft and Google use the Find X-Sum coding problem to test a candidate's ability to handle Sliding Window tasks combined with Priority Queues. It requires you to maintain a dynamic frequency map and extract top-K statistics as the window slides across the array. It evaluations your proficiency in Heap (Priority Queue) interview patterns.

3. Algorithmic pattern used

This problem follows the Sliding Window and Heap-based Top-K pattern.

1. Windowing: Slide a window of size k over the array.
2. Frequency Map: Use a hash map to maintain counts of elements currently inside the window.
3. X-Sum Calculation:
   • Collect all (frequency, value) pairs from the map.
   • Use a Max-Heap or Sort to find the top $x$ pairs.
   • Multiply and sum.
4. Optimization: Since $k$ and $x$ are small in version I, re-calculating from the map for each window is acceptable ($O(N \cdot K)$).

4. Example explanation

Array: [1, 1, 2, 2, 3, 3], $k=4, x=2$.

• Window 1: [1, 1, 2, 2]. Freqs: {1:2, 2:2}. Top 2: (2, 2) and (2, 1). X-Sum: $2(2) + 2(1) = 6$.
• Window 2: [1, 2, 2, 3]. Freqs: {1:1, 2:2, 3:1}. Top 2 are (2, 2) and (1, 3) (since 3 > 1). X-Sum: $2(2) + 1(3) = 7$. Result: [6, 7, ...]

5. Common mistakes candidates make

• Tie-breaking: Forgetting to use the element's value as a secondary sorting criterion for the "top $x$" selection.
• Inefficient Map handling: Not properly updating the map (decrementing or removing) when an element leaves the sliding window.
• Complexity: Over-complicating the frequency tracking when simple sorting of the map's keys is sufficient for small constraints.

6. Interview preparation tip

Practice "Top-K" variations. For version I, sorting the frequency map is fine. For harder versions, you will need two heaps (one for top $x$, one for the rest) to update the X-Sum in $O(\log ( ext{unique elements}))$ per slide.