Binaryset.ML

(*
 * @TAG(OTHER_PRINCETON_OSS)
 *)
(* Binaryset -- modified for Moscow ML
 * from SML/NJ library v. 0.2
 *
 * COPYRIGHT (c) 1993 by AT&T Bell Laboratories.
 * See file mosml/copyrght/copyrght.att for details.
 *
 * This code was adapted from Stephen Adams' binary tree implementation
 * of applicative integer sets.
 *
 *    Copyright 1992 Stephen Adams.
 *
 *    This software may be used freely provided that:
 *      1. This copyright notice is attached to any copy, derived work,
 *         or work including all or part of this software.
 *      2. Any derived work must contain a prominent notice stating that
 *         it has been altered from the original.
 *
 *   Name(s): Stephen Adams.
 *   Department, Institution: Electronics & Computer Science,
 *      University of Southampton
 *   Address:  Electronics & Computer Science
 *             University of Southampton
 *         Southampton  SO9 5NH
 *         Great Britian
 *   E-mail:   sra@ecs.soton.ac.uk
 *
 *   Comments:
 *
 *     1.  The implementation is based on Binary search trees of Bounded
 *         Balance, similar to Nievergelt & Reingold, SIAM J. Computing
 *         2(1), March 1973.  The main advantage of these trees is that
 *         they keep the size of the tree in the node, giving a constant
 *         time size operation.
 *
 *     2.  The bounded balance criterion is simpler than N&R's alpha.
 *         Simply, one subtree must not have more than `weight' times as
 *         many elements as the opposite subtree.  Rebalancing is
 *         guaranteed to reinstate the criterion for weight>2.23, but
 *         the occasional incorrect behaviour for weight=2 is not
 *         detrimental to performance.
 *
 *     3.  There are two implementations of union.  The default,
 *         hedge_union, is much more complex and usually 20% faster.  I
 *         am not sure that the performance increase warrants the
 *         complexity (and time it took to write), but I am leaving it
 *         in for the competition.  It is derived from the original
 *         union by replacing the split_lt(gt) operations with a lazy
 *         version. The `obvious' version is called old_union.
 *
 *     4.  Most time is spent in T', the rebalancing constructor.  If my
 *         understanding of the output of *<file> in the sml batch
 *         compiler is correct then the code produced by NJSML 0.75
 *         (sparc) for the final case is very disappointing.  Most
 *         invocations fall through to this case and most of these cases
 *         fall to the else part, i.e. the plain contructor,
 *         T(v,ln+rn+1,l,r).  The poor code allocates a 16 word vector
 *         and saves lots of registers into it.  In the common case it
 *         then retrieves a few of the registers and allocates the 5
 *         word T node.  The values that it retrieves were live in
 *         registers before the massive save.
 *
 *   Modified to functor to support general ordered values
 *)

signature BINARYSET =
sig
type 'item set

exception NotFound

val empty        : ('item * 'item -> order) -> 'item set
val singleton    : ('item * 'item -> order) -> 'item -> 'item set
val add          : 'item set * 'item -> 'item set
val addList      : 'item set * 'item list -> 'item set
val retrieve     : 'item set * 'item -> 'item
val peek         : 'item set * 'item -> 'item option
val isEmpty      : 'item set -> bool
val equal        : 'item set * 'item set -> bool
val isSubset     : 'item set * 'item set -> bool
val member       : 'item set * 'item -> bool
val delete       : 'item set * 'item -> 'item set
val numItems     : 'item set ->  int
val union        : 'item set * 'item set -> 'item set
val intersection : 'item set * 'item set -> 'item set
val difference   : 'item set * 'item set -> 'item set
val listItems    : 'item set -> 'item list
val app          : ('item -> unit) -> 'item set -> unit
val revapp       : ('item -> unit) -> 'item set -> unit
val foldr        : ('item * 'b -> 'b) -> 'b -> 'item set -> 'b
val foldl        : ('item * 'b -> 'b) -> 'b -> 'item set -> 'b
val find         : ('item -> bool) -> 'item set -> 'item option
end

(*
   ['item set] is the type of sets of ordered elements of type 'item.
   The ordering relation on the elements is used in the representation
   of the set.  The result of combining two sets with different
   underlying ordering relations is undefined.  The implementation
   uses ordered balanced binary trees.

   [empty ordr] creates a new empty set with the given ordering
   relation.

   [singleton ordr i] creates the singleton set containing i, with the
   given ordering relation.

   [add(s, i)] adds item i to set s.

   [addList(s, xs)] adds all items from the list xs to the set s.

   [retrieve(s, i)] returns i if it is in s; raises NotFound otherwise.

   [peek(s, i)] returns SOME i if i is in s; returns NONE otherwise.

   [isEmpty s] returns true if and only if the set is empty.

   [equal(s1, s2)] returns true if and only if the two sets have the
   same elements.

   [isSubset(s1, s2)] returns true if and only if s1 is a subset of s2.

   [member(s, i)] returns true if and only if i is in s.

   [delete(s, i)] removes item i from s.  Raises NotFound if i is not in s.

   [numItems s] returns the number of items in set s.

   [union(s1, s2)] returns the union of s1 and s2.

   [intersection(s1, s2)] returns the intersectionof s1 and s2.

   [difference(s1, s2)] returns the difference between s1 and s2 (that
   is, the set of elements in s1 but not in s2).

   [listItems s] returns a list of the items in set s, in increasing
   order.

   [app f s] applies function f to the elements of s, in increasing
   order.

   [revapp f s] applies function f to the elements of s, in decreasing
   order.

   [foldl f e s] applies the folding function f to the entries of the
   set in increasing order.

   [foldr f e s] applies the folding function f to the entries of the
   set in decreasing order.

   [find p s] returns SOME i, where i is an item in s which satisfies
   p, if one exists; otherwise returns NONE.
*)


structure Binaryset :> BINARYSET =
struct

datatype 'item set = SET of ('item * 'item -> order) * 'item tree
and 'item tree =
    E
  | T of {elt   : 'item,
	  cnt   : int,
	  left  : 'item tree,
	  right : 'item tree}

fun treeSize E = 0
  | treeSize (T{cnt,...}) = cnt

fun numItems (SET(_, t)) = treeSize t

fun isEmpty (SET(_, E)) = true
  | isEmpty _           = false

fun mkT(v,n,l,r) = T{elt=v,cnt=n,left=l,right=r}

(* N(v,l,r) = T(v,1+treeSize(l)+treeSize(r),l,r) *)
fun N(v,E,E) = mkT(v,1,E,E)
  | N(v,E,r as T{cnt=n,...}) = mkT(v,n+1,E,r)
  | N(v,l as T{cnt=n,...}, E) = mkT(v,n+1,l,E)
  | N(v,l as T{cnt=n,...}, r as T{cnt=m,...}) = mkT(v,n+m+1,l,r)

fun single_L (a,x,T{elt=b,left=y,right=z,...}) = N(b,N(a,x,y),z)
  | single_L _ = raise Match
fun single_R (b,T{elt=a,left=x,right=y,...},z) = N(a,x,N(b,y,z))
  | single_R _ = raise Match
fun double_L (a,w,T{elt=c,left=T{elt=b,left=x,right=y,...},right=z,...}) =
      N(b,N(a,w,x),N(c,y,z))
  | double_L _ = raise Match
fun double_R (c,T{elt=a,left=w,right=T{elt=b,left=x,right=y,...},...},z) =
      N(b,N(a,w,x),N(c,y,z))
  | double_R _ = raise Match

(*
**  val weight = 3
**  fun wt i = weight * i
*)
fun wt (i : int) = i + i + i

fun T' (v,E,E) = mkT(v,1,E,E)
  | T' (v,E,r as T{left=E,right=E,...}) = mkT(v,2,E,r)
  | T' (v,l as T{left=E,right=E,...},E) = mkT(v,2,l,E)

  | T' (p as (_,E,T{left=T _,right=E,...})) = double_L p
  | T' (p as (_,T{left=E,right=T _,...},E)) = double_R p

    (* these cases almost never happen with small weight*)
  | T' (p as (_,E,T{left=T{cnt=ln,...},right=T{cnt=rn,...},...})) =
        if ln<rn then single_L p else double_L p
  | T' (p as (_,T{left=T{cnt=ln,...},right=T{cnt=rn,...},...},E)) =
        if ln>rn then single_R p else double_R p

  | T' (p as (_,E,T{left=E,...})) = single_L p
  | T' (p as (_,T{right=E,...},E)) = single_R p

  | T' (p as (v,l as T{elt=lv,cnt=ln,left=ll,right=lr},
          r as T{elt=rv,cnt=rn,left=rl,right=rr})) =
      if rn >= wt ln (*right is too big*)
        then
          let val rln = treeSize rl
              val rrn = treeSize rr
          in
            if rln < rrn then single_L p else double_L p
          end
      else if ln >= wt rn (*left is too big*)
        then
          let val lln = treeSize ll
              val lrn = treeSize lr
          in
            if lrn < lln then single_R p else double_R p
          end
      else mkT(v,ln+rn+1,l,r)

fun addt cmpKey t x =
    let fun h E = mkT(x,1,E,E)
	  | h (T{elt=v,left=l,right=r,cnt}) =
	    case cmpKey(x,v) of
		LESS    => T'(v, h l, r)
	      | GREATER => T'(v, l, h r)
	      | EQUAL   => mkT(x,cnt,l,r)
    in h t end

fun concat3 cmpKey E v r = addt cmpKey r v
  | concat3 cmpKey l v E = addt cmpKey l v
  | concat3 cmpKey (l as T{elt=v1,cnt=n1,left=l1,right=r1})
                   v
		   (r as T{elt=v2,cnt=n2,left=l2,right=r2}) =
    if wt n1 < n2 then T'(v2, concat3 cmpKey l v l2, r2)
    else if wt n2 < n1 then T'(v1, l1, concat3 cmpKey r1 v r)
    else N(v,l,r)

fun split_lt cmpKey E x = E
  | split_lt cmpKey (T{elt=v,left=l,right=r,...}) x =
      case cmpKey(v,x) of
        GREATER => split_lt cmpKey l x
      | LESS    => concat3 cmpKey l v (split_lt cmpKey r x)
      | _ => l

fun split_gt cmpKey E x = E
  | split_gt cmpKey (T{elt=v,left=l,right=r,...}) x =
      case cmpKey(v,x) of
        LESS    => split_gt cmpKey r x
      | GREATER => concat3 cmpKey (split_gt cmpKey l x) v r
      | _       => r

fun min (T{elt=v,left=E,...}) = v
  | min (T{left=l,...}) = min l
  | min _ = raise Match

fun delmin (T{left=E,right=r,...}) = r
  | delmin (T{elt=v,left=l,right=r,...}) = T'(v,delmin l,r)
  | delmin _ = raise Match

fun delete' (E,r) = r
  | delete' (l,E) = l
  | delete' (l,r) = T'(min r,l,delmin r)

fun concat E s = s
  | concat s E = s
  | concat (t1 as T{elt=v1,cnt=n1,left=l1,right=r1})
           (t2 as T{elt=v2,cnt=n2,left=l2,right=r2}) =
	   if wt n1 < n2 then T'(v2, concat t1 l2, r2)
	   else if wt n2 < n1 then T'(v1, l1, concat r1 t2)
  	   else T'(min t2,t1, delmin t2)

fun hedge_union cmpKey s E = s
  | hedge_union cmpKey E s = s
  | hedge_union cmpKey (T{elt=v,left=l1,right=r1,...})
                       (s2 as T{elt=v2,left=l2,right=r2,...}) =
    let fun trim lo hi E = E
	  | trim lo hi (s as T{elt=v,left=l,right=r,...}) =
	    if cmpKey(v,lo) = GREATER
		then if cmpKey(v,hi) = LESS then s else trim lo hi l
	    else trim lo hi r

	fun uni_bd s E _ _ = s
	  | uni_bd E (T{elt=v,left=l,right=r,...}) lo hi =
	    concat3 cmpKey (split_gt cmpKey l lo) v (split_lt cmpKey r hi)
	  | uni_bd (T{elt=v,left=l1,right=r1,...})
		   (s2 as T{elt=v2,left=l2,right=r2,...}) lo hi =
	    concat3 cmpKey (uni_bd l1 (trim lo v s2) lo v)
	                   v (uni_bd r1 (trim v hi s2) v hi)
          (* inv:  lo < v < hi *)

    (* all the other versions of uni and trim are
     * specializations of the above two functions with
     *     lo=-infinity and/or hi=+infinity
     *)

	fun trim_lo _ E = E
	  | trim_lo lo (s as T{elt=v,right=r,...}) =
	    case cmpKey(v,lo) of
		GREATER => s
	      | _       => trim_lo lo r

	fun trim_hi _ E = E
	  | trim_hi hi (s as T{elt=v,left=l,...}) =
	    case cmpKey(v,hi) of
		LESS => s
	      | _    => trim_hi hi l

	fun uni_hi s E _ = s
	  | uni_hi E (T{elt=v,left=l,right=r,...}) hi =
	    concat3 cmpKey l v (split_lt cmpKey r hi)
	  | uni_hi (T{elt=v,left=l1,right=r1,...})
		   (s2 as T{elt=v2,left=l2,right=r2,...}) hi =
	    concat3 cmpKey (uni_hi l1 (trim_hi v s2) v)
	                   v (uni_bd r1 (trim v hi s2) v hi)

	fun uni_lo s E _ = s
	  | uni_lo E (T{elt=v,left=l,right=r,...}) lo =
	    concat3 cmpKey (split_gt cmpKey l lo) v r
	  | uni_lo (T{elt=v,left=l1,right=r1,...})
		   (s2 as T{elt=v2,left=l2,right=r2,...}) lo =
	    concat3 cmpKey (uni_bd l1 (trim lo v s2) lo v)
	                   v (uni_lo r1 (trim_lo v s2) v)
    in
	concat3 cmpKey (uni_hi l1 (trim_hi v s2) v)
                     v (uni_lo r1 (trim_lo v s2) v)
    end

  (* The old_union version is about 20% slower than
   *  hedge_union in most cases
   *)
fun old_union _ E s2 = s2
  | old_union _ s1 E = s1
  | old_union cmpKey (T{elt=v,left=l,right=r,...}) s2 =
      let val l2 = split_lt cmpKey s2 v
          val r2 = split_gt cmpKey s2 v
      in
	  concat3 cmpKey (old_union cmpKey l l2) v (old_union cmpKey r r2)
      end

exception NotFound

fun empty cmpKey = SET(cmpKey, E)

fun singleton cmpKey x = SET(cmpKey, T{elt=x,cnt=1,left=E,right=E})

fun addList (SET(cmpKey, t), l) =
    SET(cmpKey, List.foldl (fn (i,s) => addt cmpKey s i) t l)

fun add (SET(cmpKey, t), x) = SET(cmpKey, addt cmpKey t x)

fun peekt cmpKey t x =
    let fun pk E = NONE
	  | pk (T{elt=v,left=l,right=r,...}) =
	    case cmpKey(x,v) of
		LESS    => pk l
	      | GREATER => pk r
	      | _       => SOME v
    in pk t end;

fun membert cmpKey t x =
    case peekt cmpKey t x of NONE => false | _ => true

fun peek (SET(cmpKey, t), x) = peekt cmpKey t x;
fun member arg = case peek arg of NONE => false | _ => true

local
    (* true if every item in t is in t' *)
  fun treeIn cmpKey (t,t') =
      let fun isIn E = true
	    | isIn (T{elt,left=E,right=E,...}) =
	      membert cmpKey t' elt
	    | isIn (T{elt,left,right=E,...}) =
              membert cmpKey t' elt andalso isIn left
	    | isIn (T{elt,left=E,right,...}) =
              membert cmpKey t' elt  andalso isIn right
	    | isIn (T{elt,left,right,...}) =
              membert cmpKey t' elt andalso isIn left andalso isIn right
      in isIn t end
in
fun isSubset (SET(_, E),_) = true
  | isSubset (_,SET(_, E)) = false
  | isSubset (SET(cmpKey, t as T{cnt=n,...}),
	      SET(_,      t' as T{cnt=n',...})) =
    (n<=n') andalso treeIn cmpKey (t,t')

fun equal (SET(_,E), SET(_, E)) = true
  | equal (SET(cmpKey, t as T{cnt=n,...}),
	   SET(_,      t' as T{cnt=n',...})) =
    (n=n') andalso treeIn cmpKey (t,t')
  | equal _ = false
end

fun retrieve arg =
    case peek arg of NONE => raise NotFound | SOME v => v

fun delete (SET(cmpKey, t), x) =
    let fun delt E = raise NotFound
	  | delt (t as T{elt=v,left=l,right=r,...}) =
	    case cmpKey(x,v) of
		LESS    => T'(v, delt l, r)
	      | GREATER => T'(v, l, delt r)
	      | _       => delete'(l,r)
    in SET(cmpKey, delt t) end;

fun union (SET(cmpKey, t1), SET(_, t2)) =
    SET(cmpKey, hedge_union cmpKey t1 t2)

fun intersection (SET(cmpKey, t1), SET(_, t2)) =
    let fun intert E _ = E
	  | intert _ E = E
	  | intert t (T{elt=v,left=l,right=r,...}) =
	    let val l2 = split_lt cmpKey t v
		val r2 = split_gt cmpKey t v
	    in
		case peekt cmpKey t v of
		    NONE => concat (intert l2 l) (intert r2 r)
		  | _    => concat3 cmpKey (intert l2 l) v (intert r2 r)
	    end
    in SET(cmpKey, intert t1 t2) end

fun difference (SET(cmpKey, t1), SET(_, t2)) =
    let fun difft E s = E
	  | difft s E  = s
	  | difft s (T{elt=v,left=l,right=r,...}) =
	    let val l2 = split_lt cmpKey s v
		val r2 = split_gt cmpKey s v
	    in
		concat (difft l2 l) (difft r2 r)
	    end
    in SET(cmpKey, difft t1 t2) end

fun foldr f b (SET(_, t)) =
    let fun foldf E b = b
	  | foldf (T{elt,left,right,...}) b =
	    foldf left (f(elt, foldf right b))
    in foldf t b end

fun foldl f b (SET(_, t)) =
    let fun foldf E b = b
	  | foldf (T{elt,left,right,...}) b =
	    foldf right (f(elt, foldf left b))
    in foldf t b end

fun listItems set = foldr (op::) [] set

fun revapp f (SET(_, t)) =
    let fun apply E = ()
	  | apply (T{elt,left,right,...}) =
	    (apply right; ignore (f elt); apply left)
    in apply t end

fun app f (SET(_, t)) =
    let fun apply E = ()
	  | apply (T{elt,left,right,...}) =
	    (apply left; ignore (f elt); apply right)
    in apply t end

fun find p (SET(_, t)) =
    let fun findt E = NONE
	  | findt (T{elt,left,right,...}) =
	    if p elt then SOME elt
	    else case findt left of
		NONE => findt right
	      | a    => a
    in findt t end

end;