Question

Given an array with integers, with each integer being at most n positions away from its final position, what would be the best sorting algorithm?

I’ve been thinking for a while about this and I can’t seem to get a good strategy to start dealing with this problem. Can someone please guide me?

Answer 1

I’d split the list (of size N) into 2n sublists (using zero-based indexing):

list 0: elements 0, 2n, 4n, …
list 1: elements 1, 2n+1, 4n+1, …
…
list 2n-1: elements 2n-1, 4n-1, …

Each of these lists is obviously sorted.

Now merge these lists (repeatedly merging 2 lists at a time, or using a min heap with one element of each of these lists).

That’s all. Time complexity is O(N log(n)).

This is easy in Python:

>>> a = [1, 0, 5, 4, 3, 2, 6, 8, 9, 7, 12, 13, 10, 11]
>>> n = max(abs(i - x) for i, x in enumerate(a))
>>> n
3
>>> print(*heapq.merge(*(a[i::2 * n] for i in range(2 * n))))
0 1 2 3 4 5 6 7 8 9 10 11 12 13