Very Quick Sort: Analysis of Time and Space Complexity

This code implements a variation of quick sort called 'very_quick_sort'.

First, given an array 'a' and two ranges [l, r] and [p, q], it sorts elements in 'a' from 'l' to 'r' where the element range is from 'p' to 'q'.

Next, it finds the middle element 'mid' in array 'a' and places elements in 'a' greater than 'mid' into the first half of array 'b', while elements less than or equal to 'mid' go into the second half of 'b'.

Then, elements from array 'b' are copied back into array 'a'.

Finally, it recursively calls 'very_quick_sort' on the first and second halves of array 'a'.

Analysis of Complexity:

In the worst case, where elements in array 'a' are all equal, or completely ascending or descending, each partition only splits the array into one element and n-1 elements, effectively reducing only one element per partition.

Therefore, the number of recursive calls to 'very_quick_sort' is n-1.

Each partition has a time complexity of O(n) as it involves traversing array 'a' for partitioning.

Consequently, the overall time complexity is O(n^2).

The space complexity is O(n) because an auxiliary array 'b' of size 'n' is used.

In conclusion, this algorithm has a time complexity of O(n^2) and a space complexity of O(n).