Given an integer array, nums, sort it in ascending order and return the sorted array.

You must implement the sorting algorithm yourself; do not use any built-in sorting functions. The solution must run in $O(n \log n)$ time and use the minimum possible extra space.

Constraints:

• $1 \leq$ nums.length $\leq 10^3$

• $-10^3 \leq$ nums[i] $\leq 10^3$

The problem requires sorting an array in ascending order with $O(n \log n)$ time complexity and the minimum possible space complexity, without using built-in sort functions. Considering the constraints and requirements, we can use any of the following sorting algorithms:

1. Merge sort: It guarantees $O(n \log n)$ time complexity in all cases. Its space complexity is $O(n)$ due to the auxiliary array needed for merging. Even though the space complexity is not $O(1)$, it is a common and robust approach for sorting.

2. Quick sort: On average, it achieves $O(n \log n)$ time complexity and $O(\log n)$ space complexity (for the recursion stack), making it very efficient. It’s often an in-place sort. However, its worst-case time complexity is $O(n^2)$, which is not promising if not implemented carefully with good pivot selection, potentially failing the $O(n \log n)$ requirement under specific input distributions.

3. Heap sort: It offers $O(n \log n)$ time complexity in all cases and $O(1)$ space complexity, making it an excellent choice for strict space requirements. It’s an in-place sorting algorithm.

Given the explicit requirement for minimum space complexity possible and a guaranteed $O(n \log n)$ time complexity, heap sort stands out as the best choice. However, it is essential to note that heap sort is not a stable sorting algorithm. This means that when two elements have equal values, their relative order in the array might not be preserved after sorting. In situations where stability is required (e.g., sorting by multiple criteria), heap sort might not be the ideal choice. It is highly efficient for applications where space is a critical concern and stability is not a requirement.

In the solution with heap sort, the input array, nums, is first converted into a max heap, so the largest element is always at the root (the first position of the array). Once the max heap is built, the algorithm repeatedly swaps the root with the last element of the max heap, effectively moving the largest value to its correct position at the end. After each swap, the heap size is reduced by one, and the remaining elements are restructured to maintain the heap property. This process continues until all elements are placed in order, producing a sorted array in ascending order.