ML Sorting Examples

Here is some example scheme code for list sorting. These functions assume input is a simple list of elements (no list nesting).

Selection sorting.
Remember that selection sort works by finding the smallest element and moving it into the first position, then finding the second smallest and moving it into the second position. In general, on the ith iteration, it finds the ith smallest element and move it to position i. This works as follows:
(selection '(6 4 5 3 9 3))
(cons 3 (selection '(6 4 5 9 3))
(cons 3 (cons 3 (selection '(6 4 5 9))
(cons 3 (cons 3 (cons 4 (selection '(6 5 9))
(cons 3 (cons 3 (cons 4 (cons 5 (selection '(6 9))
(cons 3 (cons 3 (cons 4 (cons 5 (cons 6 (selection '(9))
(cons 3 (cons 3 (cons 4 (cons 5 (cons 6 (cons 9 (selection '())
Which becomes (3 3 4 5 6 9).
The scheme code below does exactly this:

(* put the smallest element at the front of the  current list 
    call selection on the list minus the smallest  element *)
fun selectionsort([]) = []
	| selectionsort(first::rest) =
		let
		    val s = smallest(rest, first);
		in
		    s::selectionsort(remove(first::rest,s))
		end;

(* remove the first occurance of atom A from L *)
fun remove([], _) = []
	| remove(first::rest, item) =
		if first = item then
			rest
		else
			first::remove(rest, item);

(* looks for the smallest element in the list
    atom A is the current smallest *)
    
fun smallest([], a) = a
	| smallest(first::rest, a) = 
		if first < a then 
			smallest(rest, first)
		else smallest(rest, a);
(*
- use "selection.sml";
val it = [1,2,3,5] : int list
- remove([1,2,3,4,5], 5);
val it = [1,2,3,4] : int list
- remove([1,2,3,4,5], 1);
val it = [2,3,4,5] : int list
- smallest([3,14,5,1,18,2], 3);
val it = 1 : int
*)

Merge Sort
Mergesort divides the list in half everytime, calling itself on each half. As the recursion unwinds, it merges the sorted list into a new sorted list. For example, assume we have the following:

            (mergesort '(6 4 5 7 8 9 3 4))
                     /                    \
    (mergesort '(6 4 5 7))                 (mergesort '(8 9 3 4))
            /             \                   /             \
(mergesort '(6 4)) (mergesort '(5 7)) (mergesort '(8 9)) (mergesort '(3 4))

At the lowest tree leves, the two element get sorted and the recursion unwinds.

            (mergesort '(6 4 5 7 8 9 3 4))
                     /                    \
    (mergesort '(6 4 5 7))                 (mergesort '(8 9 3 4))
            /             \                   /             \
         (4 6)          (5 7)              (8 9))          (3 4))

The next level of recursion:

            (mergesort '(6 4 5 7 8 9 3 4))
                     /                    \
              (4 5 6 7))                 (3 4 8 9)
              /     \                     /    \
           (4 6)    (5 7)              (8 9))  (3 4))

And the final level with merges into the overall list

                         (3 4 4 5 6 7 8 9)
                          /             \
                      (4 5 6 7)      (3 4 8 9)

Of course, this is not the actual order that things happen but it give the overall algorithm flavor. In scheme, we can write functions to split lists, and merge sorted lists. Once we have these, the mergesort itself is fairly short.

fun count([]) = 0
|	count(first::rest) = 1 + count(rest);

fun sub(lis: int list, start, stop, ctr) =
	if lis = nil then lis
	else if ctr < start then
		 sub(tl(lis), start, stop, ctr+1)
	else if ctr > stop then
		 []
	else 
		 hd(lis)::sub(tl(lis),start, stop, ctr+1);

fun split(lis: int list) = 
	let
		val len = count(lis)
	in
		if len = 0 then lis::[lis]
		else if len = 1 then lis::[[]]
		else sub(lis, 1, len div 2, 1)::[sub(lis, (len div 2)+1, len, 1)]
	end;
	
fun mergelists(lis: int list, m:int list) = 
	if lis = nil then
		 m
	else if m = nil then
		 lis
	else if hd(lis) < hd(m) then
		 hd(lis)::mergelists(tl(lis), m)
	else
		 hd(m)::mergelists(lis, tl(m));

fun mergesort(lis : int list) =
	if lis = nil then
		 []
	else if tl(lis) = nil then
		 lis
	else if tl(tl(lis)) = nil then
		 mergelists ([hd(lis)], tl(lis))
	else  
		 mergelists(mergesort(hd(split(lis))),
				 mergesort(hd(tl(split(lis)))));

(*
- use "mergesort.sml";
- mergesort([34,1,7,18,90, 2,0]);
val it = [0,1,2,7,18,34,90] : int list
-
*)

Binary Tree sort
The binary tree sort works by building a binary search tree from the input and then traversing it. A binary search tree has the following property at each tree node with value V:

any nodes in the left subtree have values < V
any nodes in the right subtree have values >= V

If we can build such a tree from the input, than a inorder visit finds the elements in the correct order.


; We are going to represent each node in a binary tree as:
;  (value (left child)(right child))  If the node has no left 
;  and/or right child, this is represented with ().  So,
;  (2 (1()())(3()()))  is the tree:   2
;                                   /   \
;                                  1     3


(define (binarysort L)
  (if (null? L) '()
     (traverse (btree L))
))

(define (btree L)
  (if (= (length L) 1) (leaf (car L))
        (binsert (btree (cdr L)) (car L))
  )
)


(define (binsert T A)     ; insert A into the tree
   (cond  ( (null? T) (leaf A) )        ; insert here
          ( (> (car T) A) (list (car T) 
                                (binsert (car (cdr T)) A)
                                (car (cdr (cdr T))))
           )  ; left subtree 
          ( else (list (car T)
                       (car (cdr T)) 
                       (binsert (car (cdr (cdr T))) A))     ; right subtree
          )
    )
)

(define (leaf A)          ; add a leaf to the tree (A ()())
   (list A '() '())
)

(define (traverse L)   ; output sorted list by traversing the tree 
  (cond ( (null? L) L)
        ( else
           (append (traverse (car (cdr L)))
                   (cons (car L)(traverse (car (cdr (cdr L))))))
        )
  )
)

(define (length L)
  (if (null? L) 0
     (+ 1 (length (cdr L)))
  )
)