Selection sort#

Selection sort works by finding the minimum element of the list each time and swapping elements to get the list sorted. Where insertion sort creates a new list, selection sort is an in place algorithm.

Here’s some pseudocode:

-- selection sort - sort a list l of length n in place
selection_sort(l):
for each index i in 0 ... n - 2 inclusive:
    j = minimal element in range i ... n - 1 inclusive
    swap i and j

Why does this work? Let’s imagine we’re sorting a list [4,3,1,2]. What happens?

  • First, i is 0. We find the minimal element in the range 0 to n-1, i.e., of the whole list. The minimum is 1, and it’s at index 2. So we swap indices 0 and 2, and our list is now [1,3,4,2].

  • Now i is 1. We find the minimal element in the range 1 to n-1, i.e., leaving out index 0, which we know is sorted. The minimum is 2, and it’s at index 3. So we swap indices 1 and 3 and our list is now [1,2,4,3].

  • Now i is 2. We find the leftmost minimal element in the range 2 to n-1, i.e., leaving out indices 0 and 1, which are know are sorted. The minimum is 3, which is at index 3 (after the last swap). So we swap indices 2 and 3 and our list is now [1,2,3,4].

  • We’re done! We don’t need a loop for the last index, because everything to the left is sorted and was less than this element.

The intuition for correctness here is that at the end of iteration i, everything up to i is sorted, and everything to the right is no less than everything up to i.

For performance, selection sort does a lot of comparisons: each element is compared to the rest of the list. If the list is of length n, we’ll do n ** 2 comparisons. That turns out to be on the high side for sorting. But thw good news is that we’ll do at most n-1 swaps—one per iteration. (If i and j are the same—i.e., the minimal element is already in the right spot—we don’t need to actually swap anything.) So if it was cheap to compare elements but expensive to swap them, selection sort could be a great fit.

Let’s see it in Python: