;;; "wttree.scm" Weight balanced trees -*-Scheme-*-
;;; Copyright (c) 1993-1994 Stephen Adams
;;;
;;; References:
;;;
;;; Stephen Adams, Implemeting Sets Efficiently in a Functional
;;; Language, CSTR 92-10, Department of Electronics and Computer
;;; Science, University of Southampton, 1992
;;;
;;;
;;; Copyright (c) 1993-94 Massachusetts Institute of Technology
;;;
;;; This material was developed by the Scheme project at the
;;; Massachusetts Institute of Technology, Department of Electrical
;;; Engineering and Computer Science. Permission to copy and modify
;;; this software, to redistribute either the original software or a
;;; modified version, and to use this software for any purpose is
;;; granted, subject to the following restrictions and understandings.
;;;
;;; 1. Any copy made of this software must include this copyright
;;; notice in full.
;;;
;;; 2. Users of this software agree to make their best efforts (a) to
;;; return to the MIT Scheme project any improvements or extensions
;;; that they make, so that these may be included in future releases;
;;; and (b) to inform MIT of noteworthy uses of this software.
;;;
;;; 3. All materials developed as a consequence of the use of this
;;; software shall duly acknowledge such use, in accordance with the
;;; usual standards of acknowledging credit in academic research.
;;;
;;; 4. MIT has made no warranty or representation that the operation
;;; of this software will be error-free, and MIT is under no
;;; obligation to provide any services, by way of maintenance, update,
;;; or otherwise.
;;;
;;; 5. In conjunction with products arising from the use of this
;;; material, there shall be no use of the name of the Massachusetts
;;; Institute of Technology nor of any adaptation thereof in any
;;; advertising, promotional, or sales literature without prior
;;; written consent from MIT in each case.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Weight Balanced Binary Trees
;;
;;
;;
;; This file has been modified from the MIT-Scheme library version to
;; make it more standard. The main changes are
;;
;; . The whole thing has been put in a LET as R4RS Scheme has no module
;; system.
;; . The MIT-Scheme define structure operations have been written out by
;; hand.
;;
;; It has been tested on MIT-Scheme, scheme48 and scm4e1
;;
;; If your system has a compiler and you want this code to run fast, you
;; should do whatever is necessary to inline all of the structure accessors.
;;
;; This is MIT-Scheme's way of saying that +, car etc should all be inlined.
;;
;;(declare (usual-integrations))
;;;
;;; Interface to this package.
;;;
;;; ONLY these procedures (and TEST at the end of the file) will be
;;; (re)defined in your system.
;;;
;@
(define make-wt-tree-type #f)
(define number-wt-type #f)
(define string-wt-type #f)
;@
(define make-wt-tree #f)
(define singleton-wt-tree #f)
(define alist->wt-tree #f)
(define wt-tree/empty? #f)
(define wt-tree/size #f)
(define wt-tree/add #f)
(define wt-tree/delete #f)
(define wt-tree/add! #f)
(define wt-tree/delete! #f)
(define wt-tree/member? #f)
(define wt-tree/lookup #f)
(define wt-tree/split< #f)
(define wt-tree/split> #f)
(define wt-tree/union #f)
(define wt-tree/intersection #f)
(define wt-tree/difference #f)
(define wt-tree/subset? #f)
(define wt-tree/set-equal? #f)
(define wt-tree/fold #f)
(define wt-tree/for-each #f)
(define wt-tree/index #f)
(define wt-tree/index-datum #f)
(define wt-tree/index-pair #f)
(define wt-tree/rank #f)
(define wt-tree/min #f)
(define wt-tree/min-datum #f)
(define wt-tree/min-pair #f)
(define wt-tree/delete-min #f)
(define wt-tree/delete-min! #f)
;; This LET sets all of the above variables.
(let ()
;; We use the folowing MIT-Scheme operation on fixnums (small
;; integers). R4RS compatible (but less efficient) definitions.
;; You should replace these with something that is efficient in your
;; system.
(define fix:fixnum? (lambda (x) (and (exact? x) (integer? x))))
(define fix:+ +)
(define fix:- -)
(define fix:< <)
(define fix:<= <=)
(define fix:> >)
(define fix:* *)
;; A TREE-TYPE is a collection of those procedures that depend on the
;; ordering relation.
;; MIT-Scheme structure definition
;;(define-structure
;; (tree-type
;; (conc-name tree-type/)
;; (constructor %make-tree-type))
;; (key<? #F read-only true)
;; (alist->tree #F read-only true)
;; (add #F read-only true)
;; (insert! #F read-only true)
;; (delete #F read-only true)
;; (delete! #F read-only true)
;; (member? #F read-only true)
;; (lookup #F read-only true)
;; (split-lt #F read-only true)
;; (split-gt #F read-only true)
;; (union #F read-only true)
;; (intersection #F read-only true)
;; (difference #F read-only true)
;; (subset? #F read-only true)
;; (rank #F read-only true)
;;)
;; Written out by hand, using vectors:
;;
;; If possible, you should teach your system to print out something
;; like #[tree-type <] instread of the whole vector.
(define tag:tree-type (string->symbol "#[(runtime wttree)tree-type]"))
(define (%make-tree-type key<? alist->tree
add insert!
delete delete!
member? lookup
split-lt split-gt
union intersection
difference subset?
rank )
(vector tag:tree-type
key<? alist->tree add insert!
delete delete! member? lookup
split-lt split-gt union intersection
difference subset? rank ))
(define (tree-type? tt)
(and (vector? tt)
(eq? (vector-ref tt 0) tag:tree-type)))
(define (tree-type/key<? tt) (vector-ref tt 1))
(define (tree-type/alist->tree tt) (vector-ref tt 2))
(define (tree-type/add tt) (vector-ref tt 3))
(define (tree-type/insert! tt) (vector-ref tt 4))
(define (tree-type/delete tt) (vector-ref tt 5))
(define (tree-type/delete! tt) (vector-ref tt 6))
(define (tree-type/member? tt) (vector-ref tt 7))
(define (tree-type/lookup tt) (vector-ref tt 8))
(define (tree-type/split-lt tt) (vector-ref tt 9))
(define (tree-type/split-gt tt) (vector-ref tt 10))
(define (tree-type/union tt) (vector-ref tt 11))
(define (tree-type/intersection tt) (vector-ref tt 12))
(define (tree-type/difference tt) (vector-ref tt 13))
(define (tree-type/subset? tt) (vector-ref tt 14))
(define (tree-type/rank tt) (vector-ref tt 15))
;; User level tree representation.
;;
;; WT-TREE is a wrapper for trees of nodes.
;;
;;MIT-Scheme:
;;(define-structure
;; (wt-tree
;; (conc-name tree/)
;; (constructor %make-wt-tree))
;; (type #F read-only true)
;; (root #F read-only false))
;; If possible, you should teach your system to print out something
;; like #[wt-tree] instread of the whole vector.
(define tag:wt-tree (string->symbol "#[(runtime wttree)wt-tree]"))
(define (%make-wt-tree type root)
(vector tag:wt-tree type root))
(define (wt-tree? t)
(and (vector? t)
(eq? (vector-ref t 0) tag:wt-tree)))
(define (tree/type t) (vector-ref t 1))
(define (tree/root t) (vector-ref t 2))
(define (set-tree/root! t v) (vector-set! t 2 v))
;; Nodes are the thing from which the real trees are built. There are
;; lots of these and the uninquisitibe user will never see them, so
;; they are represented as untagged to save the slot that would be
;; used for tagging structures.
;; In MIT-Scheme these were all DEFINE-INTEGRABLE
(define (make-node k v l r w) (vector w l k r v))
(define (node/k node) (vector-ref node 2))
(define (node/v node) (vector-ref node 4))
(define (node/l node) (vector-ref node 1))
(define (node/r node) (vector-ref node 3))
(define (node/w node) (vector-ref node 0))
(define empty 'empty)
(define (empty? x) (eq? x 'empty))
(define (node/size node)
(if (empty? node) 0 (node/w node)))
(define (node/singleton k v) (make-node k v empty empty 1))
(define (with-n-node node receiver)
(receiver (node/k node) (node/v node) (node/l node) (node/r node)))
;;
;; Constructors for building node trees of various complexity
;;
(define (n-join k v l r)
(make-node k v l r (fix:+ 1 (fix:+ (node/size l) (node/size r)))))
(define (single-l a.k a.v x r)
(with-n-node r
(lambda (b.k b.v y z) (n-join b.k b.v (n-join a.k a.v x y) z))))
(define (double-l a.k a.v x r)
(with-n-node r
(lambda (c.k c.v r.l z)
(with-n-node r.l
(lambda (b.k b.v y1 y2)
(n-join b.k b.v
(n-join a.k a.v x y1)
(n-join c.k c.v y2 z)))))))
(define (single-r b.k b.v l z)
(with-n-node l
(lambda (a.k a.v x y) (n-join a.k a.v x (n-join b.k b.v y z)))))
(define (double-r c.k c.v l z)
(with-n-node l
(lambda (a.k a.v x l.r)
(with-n-node l.r
(lambda (b.k b.v y1 y2)
(n-join b.k b.v
(n-join a.k a.v x y1)
(n-join c.k c.v y2 z)))))))
;; (define-integrable wt-tree-ratio 5)
(define wt-tree-ratio 5)
(define (t-join k v l r)
(define (simple-join) (n-join k v l r))
(let ((l.n (node/size l))
(r.n (node/size r)))
(cond ((fix:< (fix:+ l.n r.n) 2) (simple-join))
((fix:> r.n (fix:* wt-tree-ratio l.n))
;; right is too big
(let ((r.l.n (node/size (node/l r)))
(r.r.n (node/size (node/r r))))
(if (fix:< r.l.n r.r.n)
(single-l k v l r)
(double-l k v l r))))
((fix:> l.n (fix:* wt-tree-ratio r.n))
;; left is too big
(let ((l.l.n (node/size (node/l l)))
(l.r.n (node/size (node/r l))))
(if (fix:< l.r.n l.l.n)
(single-r k v l r)
(double-r k v l r))))
(else
(simple-join)))))
;;
;; Node tree procedures that are independent of key<?
;;
(define (node/min node)
(cond ((empty? node) (error:empty 'min))
((empty? (node/l node)) node)
(else (node/min (node/l node)))))
(define (node/delmin node)
(cond ((empty? node) (error:empty 'delmin))
((empty? (node/l node)) (node/r node))
(else (t-join (node/k node) (node/v node)
(node/delmin (node/l node)) (node/r node)))))
(define (node/concat2 node1 node2)
(cond ((empty? node1) node2)
((empty? node2) node1)
(else
(let ((min-node (node/min node2)))
(t-join (node/k min-node) (node/v min-node)
node1 (node/delmin node2))))))
(define (node/inorder-fold procedure base node)
(define (fold base node)
(if (empty? node)
base
(with-n-node node
(lambda (k v l r)
(fold (procedure k v (fold base r)) l)))))
(fold base node))
(define (node/for-each procedure node)
(if (not (empty? node))
(with-n-node node
(lambda (k v l r)
(node/for-each procedure l)
(procedure k v)
(node/for-each procedure r)))))
(define (node/height node)
(if (empty? node)
0
(+ 1 (max (node/height (node/l node))
(node/height (node/r node))))))
(define (node/index node index)
(define (loop node index)
(let ((size.l (node/size (node/l node))))
(cond ((fix:< index size.l) (loop (node/l node) index))
((fix:> index size.l) (loop (node/r node)
(fix:- index (fix:+ 1 size.l))))
(else node))))
(let ((bound (node/size node)))
(if (or (< index 0)
(>= index bound)
(not (fix:fixnum? index)))
(slib:error 'bad-range-argument index 'node/index)
(loop node index))))
(define (error:empty owner)
(slib:error "Operation requires non-empty tree:" owner))
(define (local:make-wt-tree-type key<?)
;; MIT-Scheme definitions:
;;(declare (integrate key<?))
;;(define-integrable (key>? x y) (key<? y x))
(define (key>? x y) (key<? y x))
(define (node/find k node)
;; Returns either the node or #f.
;; Loop takes D comparisons where D is the depth of the tree
;; rather than the traditional compare-low, compare-high which
;; takes on average 1.5(D-1) comparisons
(define (loop this best)
(cond ((empty? this) best)
((key<? k (node/k this)) (loop (node/l this) best))
(else (loop (node/r this) this))))
(let ((best (loop node #f)))
(cond ((not best) #f)
((key<? (node/k best) k) #f)
(else best))))
(define (node/rank k node rank)
(cond ((empty? node) #f)
((key<? k (node/k node)) (node/rank k (node/l node) rank))
((key>? k (node/k node))
(node/rank k (node/r node)
(fix:+ 1 (fix:+ rank (node/size (node/l node))))))
(else (fix:+ rank (node/size (node/l node))))))
(define (node/add node k v)
(if (empty? node)
(node/singleton k v)
(with-n-node node
(lambda (key val l r)
(cond ((key<? k key) (t-join key val (node/add l k v) r))
((key<? key k) (t-join key val l (node/add r k v)))
(else (n-join key v l r)))))))
(define (node/delete x node)
(if (empty? node)
empty
(with-n-node node
(lambda (key val l r)
(cond ((key<? x key) (t-join key val (node/delete x l) r))
((key<? key x) (t-join key val l (node/delete x r)))
(else (node/concat2 l r)))))))
(define (node/concat tree1 tree2)
(cond ((empty? tree1) tree2)
((empty? tree2) tree1)
(else
(let ((min-node (node/min tree2)))
(node/concat3 (node/k min-node) (node/v min-node) tree1
(node/delmin tree2))))))
(define (node/concat3 k v l r)
(cond ((empty? l) (node/add r k v))
((empty? r) (node/add l k v))
(else
(let ((n1 (node/size l))
(n2 (node/size r)))
(cond ((fix:< (fix:* wt-tree-ratio n1) n2)
(with-n-node r
(lambda (k2 v2 l2 r2)
(t-join k2 v2 (node/concat3 k v l l2) r2))))
((fix:< (fix:* wt-tree-ratio n2) n1)
(with-n-node l
(lambda (k1 v1 l1 r1)
(t-join k1 v1 l1 (node/concat3 k v r1 r)))))
(else
(n-join k v l r)))))))
(define (node/split-lt node x)
(cond ((empty? node) empty)
((key<? x (node/k node))
(node/split-lt (node/l node) x))
((key<? (node/k node) x)
(node/concat3 (node/k node) (node/v node) (node/l node)
(node/split-lt (node/r node) x)))
(else (node/l node))))
(define (node/split-gt node x)
(cond ((empty? node) empty)
((key<? (node/k node) x)
(node/split-gt (node/r node) x))
((key<? x (node/k node))
(node/concat3 (node/k node) (node/v node)
(node/split-gt (node/l node) x) (node/r node)))
(else (node/r node))))
(define (node/union tree1 tree2)
(cond ((empty? tree1) tree2)
((empty? tree2) tree1)
(else
(with-n-node tree2
(lambda (ak av l r)
(let ((l1 (node/split-lt tree1 ak))
(r1 (node/split-gt tree1 ak)))
(node/concat3 ak av (node/union l1 l) (node/union r1 r))))))))
(define (node/difference tree1 tree2)
(cond ((empty? tree1) empty)
((empty? tree2) tree1)
(else
(with-n-node tree2
(lambda (ak av l r)
(let ((l1 (node/split-lt tree1 ak))
(r1 (node/split-gt tree1 ak)))
av
(node/concat (node/difference l1 l)
(node/difference r1 r))))))))
(define (node/intersection tree1 tree2)
(cond ((empty? tree1) empty)
((empty? tree2) empty)
(else
(with-n-node tree2
(lambda (ak av l r)
(let ((l1 (node/split-lt tree1 ak))
(r1 (node/split-gt tree1 ak)))
(if (node/find ak tree1)
(node/concat3 ak av (node/intersection l1 l)
(node/intersection r1 r))
(node/concat (node/intersection l1 l)
(node/intersection r1 r)))))))))
(define (node/subset? tree1 tree2)
(or (empty? tree1)
(and (fix:<= (node/size tree1) (node/size tree2))
(with-n-node tree1
(lambda (k v l r)
v
(cond ((key<? k (node/k tree2))
(and (node/subset? l (node/l tree2))
(node/find k tree2)
(node/subset? r tree2)))
((key>? k (node/k tree2))
(and (node/subset? r (node/r tree2))
(node/find k tree2)
(node/subset? l tree2)))
(else
(and (node/subset? l (node/l tree2))
(node/subset? r (node/r tree2))))))))))
;;; Tree interface: stripping off or injecting the tree types
(define (tree/map-add tree k v)
(%make-wt-tree (tree/type tree)
(node/add (tree/root tree) k v)))
(define (tree/insert! tree k v)
(set-tree/root! tree (node/add (tree/root tree) k v)))
(define (tree/delete tree k)
(%make-wt-tree (tree/type tree)
(node/delete k (tree/root tree))))
(define (tree/delete! tree k)
(set-tree/root! tree (node/delete k (tree/root tree))))
(define (tree/split-lt tree key)
(%make-wt-tree (tree/type tree)
(node/split-lt (tree/root tree) key)))
(define (tree/split-gt tree key)
(%make-wt-tree (tree/type tree)
(node/split-gt (tree/root tree) key)))
(define (tree/union tree1 tree2)
(%make-wt-tree (tree/type tree1)
(node/union (tree/root tree1) (tree/root tree2))))
(define (tree/intersection tree1 tree2)
(%make-wt-tree (tree/type tree1)
(node/intersection (tree/root tree1) (tree/root tree2))))
(define (tree/difference tree1 tree2)
(%make-wt-tree (tree/type tree1)
(node/difference (tree/root tree1) (tree/root tree2))))
(define (tree/subset? tree1 tree2)
(node/subset? (tree/root tree1) (tree/root tree2)))
(define (alist->tree alist)
(define (loop alist node)
(cond ((null? alist) node)
((pair? alist) (loop (cdr alist)
(node/add node (caar alist) (cdar alist))))
(else
(slib:error 'wrong-type-argument alist "alist" 'alist->tree))))
(%make-wt-tree my-type (loop alist empty)))
(define (tree/get tree key default)
(let ((node (node/find key (tree/root tree))))
(if node
(node/v node)
default)))
(define (tree/rank tree key) (node/rank key (tree/root tree) 0))
(define (tree/member? key tree)
(and (node/find key (tree/root tree))
#t))
(define my-type #F)
(set! my-type
(%make-tree-type
key<? ; key<?
alist->tree ; alist->tree
tree/map-add ; add
tree/insert! ; insert!
tree/delete ; delete
tree/delete! ; delete!
tree/member? ; member?
tree/get ; lookup
tree/split-lt ; split-lt
tree/split-gt ; split-gt
tree/union ; union
tree/intersection ; intersection
tree/difference ; difference
tree/subset? ; subset?
tree/rank ; rank
))
my-type)
(define (guarantee-tree tree procedure)
(if (not (wt-tree? tree))
(slib:error 'wrong-type-argument
tree "weight-balanced tree" procedure)))
(define (guarantee-tree-type type procedure)
(if (not (tree-type? type))
(slib:error 'wrong-type-argument
type "weight-balanced tree type" procedure)))
(define (guarantee-compatible-trees tree1 tree2 procedure)
(guarantee-tree tree1 procedure)
(guarantee-tree tree2 procedure)
(if (not (eq? (tree/type tree1) (tree/type tree2)))
(slib:error "The trees" tree1 'and tree2 'have 'incompatible 'types
(tree/type tree1) 'and (tree/type tree2))))
;;;______________________________________________________________________
;;;
;;; Export interface
;;;
(set! make-wt-tree-type local:make-wt-tree-type)
(set! make-wt-tree
(lambda (tree-type)
(%make-wt-tree tree-type empty)))
(set! singleton-wt-tree
(lambda (type key value)
(guarantee-tree-type type 'singleton-wt-tree)
(%make-wt-tree type (node/singleton key value))))
(set! alist->wt-tree
(lambda (type alist)
(guarantee-tree-type type 'alist->wt-tree)
((tree-type/alist->tree type) alist)))
(set! wt-tree/empty?
(lambda (tree)
(guarantee-tree tree 'wt-tree/empty?)
(empty? (tree/root tree))))
(set! wt-tree/size
(lambda (tree)
(guarantee-tree tree 'wt-tree/size)
(node/size (tree/root tree))))
(set! wt-tree/add
(lambda (tree key datum)
(guarantee-tree tree 'wt-tree/add)
((tree-type/add (tree/type tree)) tree key datum)))
(set! wt-tree/delete
(lambda (tree key)
(guarantee-tree tree 'wt-tree/delete)
((tree-type/delete (tree/type tree)) tree key)))
(set! wt-tree/add!
(lambda (tree key datum)
(guarantee-tree tree 'wt-tree/add!)
((tree-type/insert! (tree/type tree)) tree key datum)))
(set! wt-tree/delete!
(lambda (tree key)
(guarantee-tree tree 'wt-tree/delete!)
((tree-type/delete! (tree/type tree)) tree key)))
(set! wt-tree/member?
(lambda (key tree)
(guarantee-tree tree 'wt-tree/member?)
((tree-type/member? (tree/type tree)) key tree)))
(set! wt-tree/lookup
(lambda (tree key default)
(guarantee-tree tree 'wt-tree/lookup)
((tree-type/lookup (tree/type tree)) tree key default)))
(set! wt-tree/split<
(lambda (tree key)
(guarantee-tree tree 'wt-tree/split<)
((tree-type/split-lt (tree/type tree)) tree key)))
(set! wt-tree/split>
(lambda (tree key)
(guarantee-tree tree 'wt-tree/split>)
((tree-type/split-gt (tree/type tree)) tree key)))
(set! wt-tree/union
(lambda (tree1 tree2)
(guarantee-compatible-trees tree1 tree2 'wt-tree/union)
((tree-type/union (tree/type tree1)) tree1 tree2)))
(set! wt-tree/intersection
(lambda (tree1 tree2)
(guarantee-compatible-trees tree1 tree2 'wt-tree/intersection)
((tree-type/intersection (tree/type tree1)) tree1 tree2)))
(set! wt-tree/difference
(lambda (tree1 tree2)
(guarantee-compatible-trees tree1 tree2 'wt-tree/difference)
((tree-type/difference (tree/type tree1)) tree1 tree2)))
(set! wt-tree/subset?
(lambda (tree1 tree2)
(guarantee-compatible-trees tree1 tree2 'wt-tree/subset?)
((tree-type/subset? (tree/type tree1)) tree1 tree2)))
(set! wt-tree/set-equal?
(lambda (tree1 tree2)
(and (wt-tree/subset? tree1 tree2)
(wt-tree/subset? tree2 tree1))))
(set! wt-tree/fold
(lambda (combiner-key-datum-result init tree)
(guarantee-tree tree 'wt-tree/fold)
(node/inorder-fold combiner-key-datum-result
init
(tree/root tree))))
(set! wt-tree/for-each
(lambda (action-key-datum tree)
(guarantee-tree tree 'wt-tree/for-each)
(node/for-each action-key-datum (tree/root tree))))
(set! wt-tree/index
(lambda (tree index)
(guarantee-tree tree 'wt-tree/index)
(let ((node (node/index (tree/root tree) index)))
(and node (node/k node)))))
(set! wt-tree/index-datum
(lambda (tree index)
(guarantee-tree tree 'wt-tree/index-datum)
(let ((node (node/index (tree/root tree) index)))
(and node (node/v node)))))
(set! wt-tree/index-pair
(lambda (tree index)
(guarantee-tree tree 'wt-tree/index-pair)
(let ((node (node/index (tree/root tree) index)))
(and node (cons (node/k node) (node/v node))))))
(set! wt-tree/rank
(lambda (tree key)
(guarantee-tree tree 'wt-tree/rank)
((tree-type/rank (tree/type tree)) tree key)))
(set! wt-tree/min
(lambda (tree)
(guarantee-tree tree 'wt-tree/min)
(node/k (node/min (tree/root tree)))))
(set! wt-tree/min-datum
(lambda (tree)
(guarantee-tree tree 'wt-tree/min-datum)
(node/v (node/min (tree/root tree)))))
(set! wt-tree/min-pair
(lambda (tree)
(guarantee-tree tree 'wt-tree/min-pair)
(let ((node (node/min (tree/root tree))))
(cons (node/k node) (node/v node)))))
(set! wt-tree/delete-min
(lambda (tree)
(guarantee-tree tree 'wt-tree/delete-min)
(%make-wt-tree (tree/type tree)
(node/delmin (tree/root tree)))))
(set! wt-tree/delete-min!
(lambda (tree)
(guarantee-tree tree 'wt-tree/delete-min!)
(set-tree/root! tree (node/delmin (tree/root tree)))))
;; < is a lexpr. Many compilers can open-code < so the lambda is faster
;; than passing <.
(set! number-wt-type (local:make-wt-tree-type (lambda (u v) (< u v))))
(set! string-wt-type (local:make-wt-tree-type string<?))
'done)
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