Mergesort is a simple sorting algorithm that is guaranteed to be log-linear. It is stable in the sense that the order of two equal elements always stay the same after sorting. I give as follows a typical functional style of implementation of mergesort on lists.

First, let us introduce abbreviations for the list constructors list0_nil and list0_cons:

#define :: list0_cons // writing [::] for list0_cons #define cons0 list0_cons // writing [cons0] for list0_cons #define nil0 list0_nil // writing [nil0] for list0_nil

We next implement a function template merge to merge two given ordered lists into a single ordered one:

// typedef lte (a:t@ype) = (a, a) -> bool // fun{ a:t@ype } merge ( xs: list0 a, ys: list0 a, lte: lte a ) : list0 a = ( case+ xs of | cons0 (x, xs1) => ( case+ ys of | cons0 (y, ys1) => if x lte y then cons0{a}(x, merge<a> (xs1, ys, lte)) else cons0{a}(y, merge<a> (xs, ys1, lte)) // end of [if] | nil0 () => xs ) // end of [cons0] | nil0 () => ys ) (* end of [merge] *) //

The following function template mergesort implements the standard mergesort algorithm:

fun{ a:t@ype } mergesort ( xs: list0(a), lte: lte(a) ) : list0 a = let // val n = list0_length<a>(xs) // fun msort ( xs: list0(a), n: int, lte: lte(a) ) : list0 a = if n >= 2 then split(xs, n, lte, n/2, nil0) else xs // and split ( xs: list0(a), n: int, lte: lte(a), i: int, xsf: list0(a) ) : list0 a = if i > 0 then let val-cons0 (x, xs) = xs in split (xs, n, lte, i-1, cons0{a}(x, xsf)) end else let val xsf = list0_reverse<a>(xsf) // make sorting stable! val xsf = msort (xsf, n/2, lte) and xs = msort (xs, n-n/2, lte) in merge<a> (xsf, xs, lte) end // end of [if] // in msort (xs, n, lte) end // end of [mergesort]

Note that the function template merge is not tail-recursive as the call to merge in its body is not a tail-call. This can be a serious problem in practice: It is almost certain that a stack overflow is to occur if the above implementation of mergesort is employed to sort a list that is very long (e.g., containing 1,000,000 elements or more). I will later give a tail-recursive implementation of the merge function in ATS that makes use of linear types.

Please find the entire code in this section plus some additional code for testing on-line.