Problem Statement

The   merge sort   is a recursive sort of order   nlog(n). It is notable for having a worst case and average complexity of   O(nlog(n)),   and a best case complexity of   O(n)   (for pre-sorted input). The basic idea is to split the collection into smaller groups by halving it until the groups only have one element or no elements   (which are both entirely sorted groups). Then merge the groups back together so that their elements are in order. This is how the algorithm gets its   divide and conquer   description.

Write a function to sort a collection of integers using the merge sort.

The merge sort algorithm comes in two parts: The functions in pseudocode look like this:

Note:   better performance can be expected if, rather than recursing until   length(m) ≤ 1,   an insertion sort is used for   length(m)   smaller than some threshold larger than   1.   However, this complicates the example code, so it is not shown here.

Let's start with the solution:

fun merge cmp ([], ys) = ys
  | merge cmp (xs, []) = xs
  | merge cmp (xs as x::xs', ys as y::ys') =
      case cmp (x, y) of GREATER => y :: merge cmp (xs, ys')
                       | _       => x :: merge cmp (xs', ys)
;
fun merge_sort cmp [] = []
  | merge_sort cmp [x] = [x]
  | merge_sort cmp xs = let
      val ys = List.take (xs, length xs div 2)
      val zs = List.drop (xs, length xs div 2)
    in
      merge cmp (merge_sort cmp ys, merge_sort cmp zs)
    end
;
merge_sort Int.compare [8,6,4,2,1,3,5,7,9]