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(* binary-set-fn.sml * * COPYRIGHT (c) 1993 by AT&T Bell Laboratories. See COPYRIGHT file 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 *) functor BinarySetFn (K : ORD_KEY) : ORD_SET = struct structure Key = K type item = K.ord_key datatype set = E | T of { elt : item, cnt : int, left : set, right : set } fun numItems E = 0 | numItems (T{cnt,...}) = cnt fun isEmpty 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+numItems(l)+numItems(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 = numItems rl val rrn = numItems 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 = numItems ll val lrn = numItems lr in if lrn < lln then single_R p else double_R p end else mkT(v,ln+rn+1,l,r) fun add (E,x) = mkT(x,1,E,E) | add (set as T{elt=v,left=l,right=r,cnt},x) = case K.compare(x,v) of LESS => T'(v,add(l,x),r) | GREATER => T'(v,l,add(r,x)) | EQUAL => mkT(x,cnt,l,r) fun add' (s, x) = add(x, s) fun concat3 (E,v,r) = add(r,v) | concat3 (l,v,E) = add(l,v) | concat3 (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(l,v,l2),r2) else if wt n2 < n1 then T'(v1,l1,concat3(r1,v,r)) else N(v,l,r) fun split_lt (E,x) = E | split_lt (T{elt=v,left=l,right=r,...},x) = case K.compare(v,x) of GREATER => split_lt(l,x) | LESS => concat3(l,v,split_lt(r,x)) | _ => l fun split_gt (E,x) = E | split_gt (T{elt=v,left=l,right=r,...},x) = case K.compare(v,x) of LESS => split_gt(r,x) | GREATER => concat3(split_gt(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) local fun trim (lo,hi,E) = E | trim (lo,hi,s as T{elt=v,left=l,right=r,...}) = if K.compare(v,lo) = GREATER then if K.compare(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(split_gt(l,lo),v,split_lt(r,hi)) | uni_bd (T{elt=v,left=l1,right=r1,...}, s2 as T{elt=v2,left=l2,right=r2,...},lo,hi) = concat3(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 K.compare(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 K.compare(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(l,v,split_lt(r,hi)) | uni_hi (T{elt=v,left=l1,right=r1,...}, s2 as T{elt=v2,left=l2,right=r2,...},hi) = concat3(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(split_gt(l,lo),v,r) | uni_lo (T{elt=v,left=l1,right=r1,...}, s2 as T{elt=v2,left=l2,right=r2,...},lo) = concat3(uni_bd(l1,trim(lo,v,s2),lo,v),v,uni_lo(r1,trim_lo(v,s2),v)) fun uni (s,E) = s | uni (E,s) = s | uni (T{elt=v,left=l1,right=r1,...}, s2 as T{elt=v2,left=l2,right=r2,...}) = concat3(uni_hi(l1,trim_hi(v,s2),v), v, uni_lo(r1,trim_lo(v,s2),v)) in val hedge_union = uni 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 (T{elt=v,left=l,right=r,...},s2) = let val l2 = split_lt(s2,v) val r2 = split_gt(s2,v) in concat3(old_union(l,l2),v,old_union(r,r2)) end val empty = E fun singleton x = T{elt=x,cnt=1,left=E,right=E} fun addList (s,l) = List.foldl (fn (i,s) => add(s,i)) s l val add = add fun member (set, x) = let fun pk E = false | pk (T{elt=v, left=l, right=r, ...}) = ( case K.compare(x,v) of LESS => pk l | EQUAL => true | GREATER => pk r (* end case *)) in pk set end local (* true if every item in t is in t' *) fun treeIn (t,t') = let fun isIn E = true | isIn (T{elt,left=E,right=E,...}) = member(t',elt) | isIn (T{elt,left,right=E,...}) = member(t',elt) andalso isIn left | isIn (T{elt,left=E,right,...}) = member(t',elt) andalso isIn right | isIn (T{elt,left,right,...}) = member(t',elt) andalso isIn left andalso isIn right in isIn t end in fun isSubset (E,_) = true | isSubset (_,E) = false | isSubset (t as T{cnt=n,...},t' as T{cnt=n',...}) = (n<=n') andalso treeIn (t,t') fun equal (E,E) = true | equal (t as T{cnt=n,...},t' as T{cnt=n',...}) = (n=n') andalso treeIn (t,t') | equal _ = false end local fun next ((t as T{right, ...})::rest) = (t, left(right, rest)) | next _ = (E, []) and left (E, rest) = rest | left (t as T{left=l, ...}, rest) = left(l, t::rest) in fun compare (s1, s2) = let fun cmp (t1, t2) = (case (next t1, next t2) of ((E, _), (E, _)) => EQUAL | ((E, _), _) => LESS | (_, (E, _)) => GREATER | ((T{elt=e1, ...}, r1), (T{elt=e2, ...}, r2)) => ( case Key.compare(e1, e2) of EQUAL => cmp (r1, r2) | order => order (* end case *)) (* end case *)) in cmp (left(s1, []), left(s2, [])) end end fun delete (E,x) = raise LibBase.NotFound | delete (set as T{elt=v,left=l,right=r,...},x) = case K.compare(x,v) of LESS => T'(v,delete(l,x),r) | GREATER => T'(v,l,delete(r,x)) | _ => delete'(l,r) val union = hedge_union fun intersection (E, _) = E | intersection (_, E) = E | intersection (s, T{elt=v,left=l,right=r,...}) = let val l2 = split_lt(s,v) val r2 = split_gt(s,v) in if member(s,v) then concat3(intersection(l2,l),v,intersection(r2,r)) else concat(intersection(l2,l),intersection(r2,r)) end fun difference (E,s) = E | difference (s,E) = s | difference (s, T{elt=v,left=l,right=r,...}) = let val l2 = split_lt(s,v) val r2 = split_gt(s,v) in concat(difference(l2,l),difference(r2,r)) end fun map f set = let fun map'(acc, E) = acc | map'(acc, T{elt,left,right,...}) = map' (add (map' (acc, left), f elt), right) in map' (E, set) end fun app apf = let fun apply E = () | apply (T{elt,left,right,...}) = (apply left;apf elt; apply right) in apply end fun foldl f b set = let fun foldf (E, b) = b | foldf (T{elt,left,right,...}, b) = foldf (right, f(elt, foldf (left, b))) in foldf (set, b) end fun foldr f b set = let fun foldf (E, b) = b | foldf (T{elt,left,right,...}, b) = foldf (left, f(elt, foldf (right, b))) in foldf (set, b) end fun listItems set = foldr (op::) [] set fun filter pred set = foldl (fn (item, s) => if (pred item) then add(s, item) else s) empty set fun partition pred set = foldl (fn (item, (s1, s2)) => if (pred item) then (add(s1, item), s2) else (s1, add(s2, item)) ) (empty, empty) set fun find p E = NONE | find p (T{elt,left,right,...}) = (case find p left of NONE => if (p elt) then SOME elt else find p right | a => a (* end case *)) fun exists p E = false | exists p (T{elt, left, right,...}) = (exists p left) orelse (p elt) orelse (exists p right) end (* BinarySetFn *)

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