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;;;; Copyright (c) 2024, Peter McGoron
;;;;
;;;; Redistribution and use in source and binary forms, with or without
;;;; modification, are permitted provided that the following conditions
;;;; are met:
;;;;
;;;; 1) Redistributions of source code must retain the above copyright
;;;; notice, this list of conditions and the following disclaimer.
;;;; 2) Redistributions in binary form must reproduce the above copyright
;;;; notice, this list of conditions and the following disclaimer
;;;; in the documentation and/or other materials provided with the
;;;; distribution.
;;;;
;;;; THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
;;;; "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
;;;; LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
;;;; FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
;;;; COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
;;;; INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
;;;; BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
;;;; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
;;;; CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
;;;; LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
;;;; ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
;;;; POSSIBILITY OF SUCH DAMAGE.
;;;;
;;;; This implements functional balanced binary search trees (AA Trees).
;;;;
;;;; (NEW <=>) makes a new AA Tree root.
;;;; (<=> KEY1 KEY2) must return one of the symbols in '(< = >), which
;;;; denote that KEY1 is less than, equivalent to, or greater than KEY2.
;;;;
;;;; (SEARCH TREE KEY) searches the tree for the first node equivalent
;;;; to KEY. If successful, it returns (VALUES 'FOUND KEY VAL). Otherwise
;;;; it returns (VALUES 'NOT-FOUND '() '()).
;;;;
;;;; (INSERT TREE KEY VAL) inserts KEY into TREE with value VAL. If
;;;; a key equivalent to KEY is already in TREE, the old key is replaced with
;;;; KEY and the old value is replaced with VAL, and the function returns
;;;; (VALUES 'FOUND OLDKEY OLDVAL). If the value is not found, then
;;;; the function returns (VALUES 'NOT-FOUND OLDKEY OLDVAL).
;;;;
;;;; (DELETE TREE KEY) deletes the node equivalent to KEY, if it exists.
;;;; It returns
(define-namespace aatree
(begin
(define-record-type :aatree-node
(new-node key value left right level)
node?
(key get-key key-set!)
(value get-value value-set!)
(left get-left left-set!)
(right get-right right-set!)
(level get-level-raw level-set!))
(define-record-type :aatree
(%aatree <=> root)
aatree?
(<=> get-<=>)
(root get-root set-root!))
(define leaf '())
(define (new <=>) (%aatree <=> leaf))
(define leaf? null?)
(define (get-level r)
(if (leaf? r)
0
(get-level-raw r)))
; Option-like accessors
(define (maybe-right t)
(if (null? t)
leaf
(get-right t)))
(define (maybe-left t)
(if (null? t)
leaf
(get-left t)))
; right rotation
; a b
; / \ / \
; b c -> d a
; / \ / \
; d e e c
(define (skew A)
(let* ((B (maybe-left A))
(E (maybe-right B)))
(if (and (not (leaf? A))
(eq? (get-level B)
(get-level A)))
(begin
(right-set! B A)
(left-set! A E)
B)
A)))
; left rotation
; a c
; / \ / \
; b c -> a e
; / \ / \
; d e b d
;
(define (split A)
(let* ((C (maybe-right A))
(E (maybe-right C))
(D (maybe-left C)))
(if (and (not (leaf? A))
(not (leaf? C))
(eq? (get-level E)
(get-level A)))
(begin
(left-set! C A)
(right-set! A D)
(level-set! C (+ (get-level C) 1))
C)
A)))
(define (search* <=> tree key)
(if (leaf? tree)
(values 'not-found '() '())
(let ((nodekey (get-key tree)))
(case (<=> key nodekey)
((<) (search* <=> (get-left tree) key))
((>) (search* <=> (get-right tree) key))
((=) (values 'found nodekey (get-value tree)))))))
(define (search tree key) (search* (get-<=> tree) (get-root tree) key))
(define (insert* <=> node key val)
(if (leaf? node)
(values (new-node key val '() '() 1) 'not-found '() '())
(case (<=> key (get-key node))
((=) (let ((oldval (get-value node)))
(value-set! node val)
(values node 'found (get-key node) oldval)))
((<) (let-values (((newnode . rest)
(insert* <=> (get-left node) key val)))
(left-set! node newnode)
(apply values (cons (split (skew node)) rest))))
((>) (let-values (((newnode . rest)
(insert* <=> (get-right node) key val)))
(right-set! node newnode)
(apply values (cons (split (skew node)) rest))))
(else (error "comparision must return <, =, or >")))))
(define (insert tree key val)
(let-values (((new-root . rest) (insert* (get-<=> tree) (get-root tree) key val)))
(set-root! tree new-root)
(apply values rest)))
(define (delete* <=> tree key)
(if (leaf? tree)
(values tree 'not-found '() '())
(let ((process (lambda (t)
(if (leaf? t)
t
(let* ((level (get-level t))
(level-l (get-level (get-left t)))
(level-r (get-level (get-right t)))
(new-level (- level 1)))
(if (or (< level-l new-level)
(< level-r new-level))
(begin
(if (> level-r new-level)
(level-set! (get-right t)
(min level-r new-level)))
(level-set! t new-level)
(set! t (skew t))
(right-set! t (skew (get-right t)))
(right-set! (get-right t)
(skew (get-right (get-right t))))
(set! t (split t))
(right-set! t (split (get-right t)))))
t)))))
(case (<=> key (get-key tree))
((<) (let-values (((newnode . rest)
(delete* <=> (get-left tree) key)))
(left-set! tree newnode)
(apply values (cons (process tree) rest))))
((>) (let-values (((newnode . rest)
(delete* <=> (get-right tree) key)))
(right-set! tree newnode)
(apply values (cons (process tree) rest))))
((=) (letrec
((del-min (lambda (t)
(if (leaf? (get-left t))
(values (get-right t) 'found (get-key t)
(get-value t))
(let-values (((newnode . rest)
(del-min (get-left t))))
(if (leaf? (get-right t))
(error "imbalanced tree"))
(left-set! t newnode)
(apply values
(cons (process t) rest)))))))
(if (leaf? (get-right tree))
(if (not (leaf? (get-left tree)))
(error "imbalanced tree")
(values '() 'found (get-key tree) (get-value tree)))
(let-values (((newnode status key value)
(del-min (get-right tree)))
((oldvalue) (get-value tree))
((oldkey) (get-key tree)))
(key-set! tree key)
(value-set! tree value)
(right-set! tree newnode)
(values (process tree) status oldkey oldvalue)))))))))
(define (delete tree key)
(let-values (((new-root . rest) (delete* (get-<=> tree)
(get-root tree)
key)))
(set-root! tree new-root)
(apply values rest)))
(define (aatree->alist* node alist)
(if (leaf? node)
alist
(let ((alist-right (aatree->alist* (get-right node) alist)))
(aatree->alist* (get-left node)
(cons (cons (get-key node)
(get-value node))
alist-right)))))
(define (aatree->alist tree) (aatree->alist* (get-root tree) '()))
(define (alist->aatree* node <=> alist)
(if (null? alist)
node
(let ((pair (car alist)))
(let-values (((node . _)
(insert node <=>
(car pair)
(cdr pair))))
(alist->aatree* node (cdr alist) <=>)))))
(define (alist->aatree tree)
(alist->aatree* (get-root tree) (get-<=> tree) '())))
(export
aatree?
new
aatree->alist alist->aatree
search insert delete))
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