The problem of generating a sorted array of the sums of all pairs from a given sorted array is intriguing both from a theoretical and practical standpoint. In computer science, understanding the complexity and feasibility of such operations can have significant implications on performance and optimization.

Understanding the Problem

Given a sorted array `A` of size `n`, the task is to compute a new sorted array `S` that consists of the sums of all possible pairs `(A[i], A[j])` where `0 ≤ i < j < n`. The challenge is to achieve this in $O(n^2)$ time complexity.

Approach to the Solution

Naive Method

A straightforward approach to solve this problem would be to:

1. Initialize an empty list `S`.
2. Iterate over all pairs `(i, j)` with `i < j` and add the sum `A[i] + A[j]` to `S`.
3. Sort the list `S`.

While easy to implement, this method does not perform in $O(n^2)$ time since sorting the sums after computing them could increase the time complexity to $O(n^2 \log n)$, especially when using typical comparison-based sorting algorithms.

Optimized Approach

To ensure we stick to $O(n^2)$, leveraging the properties of the sorted array `A` is crucial. Note the following observations:

1. Pair Sums and Sortedness: Since the array `A` is sorted, the smallest sum is `A[0] + A[1]` and the largest is `A[n-2] + A[n-1]`. Any mid-range sum will be naturally bounded by these extremes.
2. Two-Pointer Technique: Given that `A` is sorted, using the two-pointer technique can help in efficiently generating sums within possible bounds:
   • Start with one pointer `i` at the beginning of the array and another pointer `j` at the next index (`i + 1`).
   • Calculate sums `A[i] + A[j]`, adjust pointers based on desired conditions, and move through possible combinations.

This technique ensures that each sum is generated in $O(1)$ time, and iterating over all pairs happens within the $O(n^2)$ bounds naturally.

Here is a pseudo-code snippet of the above approach: