Introduction

Finding the top `k` sums of two sorted arrays is a classic problem in computer science and competitive programming. The problem involves two sorted arrays from which we construct a new sorted dataset of sums of elements, one from each array, and then efficiently identify the top `k` sums. This article delves into the technical aspects of this task, outlining different approaches to solving it, while discussing their complexity and providing a concrete example.

Problem Definition

Given two sorted arrays `A` and `B` of sizes `m` and `n` respectively, the task is to find the top `k` sums formed by adding an element from `A` to an element from `B`.

Example:

Consider the arrays:

• `A = [1, 2, 3]`
• `B = [4, 5, 6]`

If `k = 3`, the top k sums would be `9, 8, and 8` (i.e., the pairings `(3+6, 3+5, 2+6)`).

Approaches

1. Naïve Approach

Description:

• Generate all possible sums from both arrays.
• Sort the resulting sums.
• Extract the largest `k` sums.

Complexity:

• Time Complexity: $O(m \times n \log(m \times n))$, as generating all combinations takes $O(m \times n)$ and sorting them requires an additional logarithmic factor.
• Space Complexity: $O(m \times n)$ for storing all sums.

2. Min-Heap Approach

Description:

• Utilize a min-heap to efficiently track the largest sums.
• Initially, push the largest element possible from each array combination into the heap.
• Extract the maximum, push the next possible candidate from the arrays, and repeat until `k` sums are extracted.

Technical Explanation:

• Initialize a min-heap (priority queue) to keep track of the combinations.
• Start by pushing pairs `(A[i], B[n-1], i, n-1)` for each `i` in `0` to `k-1`, only if `k` does not exceed `m`.
• Pop the heap's top, which holds the current maximum sum, and insert new pairs by decrementing the second array index.
• If `(i,j)` is the index used at the top of heap, push `(A[i]+B[j-1], i, j-1)` if `j > 0`.

Complexity:

• Time Complexity: $O(k \log k)$, where push and pop operations occur `k` times on the heap.
• Space Complexity: $O(k)$ to maintain the heap.

Example Walkthrough

Consider `A = [1, 3, 5]` and `B = [2, 4, 6]` with `k = 4`.

Heap Initialization:

• Start with the combinations: `(5, 6)`, `(3, 6)`, `(1, 6)`.
• Insert sums: `11, 9, 7` with indices `(2, 2)`, `(1, 2)`, `(0, 2)` into the heap.

Steps:

1. Extract: `(5, 6)`, Total = 11 (indices `2,2`).
2. Push Next Candidate: `(5, 4)`, Total = 9 (indices `2, 1`).
3. Extract: `(3, 6)`, Total = 9 (indices `1,2`).
4. Push Next Candidate: `(3, 4)`, Total = 7 (indices `1, 1`).
5. Extract: `(5, 4)`, Total = 9 (indices `2,1`).
6. Push Next Candidate: `(5, 2)`, Total = 7 (indices `2, 0`).
7. …and so on until `k` sums are extracted.

Final Top `k` sums:

• `[11, 9, 9, 8]`

Analysis Table

Additional Details

Edge Cases

1. Small `k`: When `k` is smaller than the sizes of both arrays, using a heap offers clear time savings.
2. Large Arrays: The heap approach scales well with large arrays, where a complete sort of all sums is impractical or infeasible due to time and space constraints.
3. Equal Elements: Handling situations where elements in either array are equals and their sums appear several times within the top results.

Conclusion

The problem of finding the top `k` sums from two sorted arrays offers significant insight into efficiency and optimization strategies. While the naive approach is intuitive, leveraging a priority queue can dramatically improve performance, especially in scenarios involving large datasets. Understanding and selecting the right approach based on problem constraints is vital for efficient problem-solving.