Insertion sort: the “natural” sort

Our first and simplest sort is insertion sort. Insertion sort takes a list and returns a new, sorted list that’s a permutation of its input.

Insertion sort depends on two simple recursive functions: insertion_sort does the sorting, and insert_sorted takes an element v and a sorted list l and puts v in the right place. Here’s the algorithm:

-- insertion sort on a list l
insertion_sort(l):
    if l is empty, return the empty list
    split l into the first element x and the rest l'
    return insert_sorted(x, insertion_sort(l'))

So simple! The intuition is that res is always sorted, and each time we insert x into res, we get a new, longer sorted list back. If we do that for every element in l, we’re done. The critical question is: how does insert_sorted work?

-- insert_sorted takes an element v and a SORTED list l
-- returns a new list, with v in the right place
insert_sorted(v, l):
    if l is empty:
        return the list containing just v
    x = first element of l
    if v < x:
        return v followed by l
    otherwise:
        return x followed by insert_sorted(v, l'), where l' is the rest of l after y

Let’s see it in Python code: