142 lines
4.7 KiB
Racket
142 lines
4.7 KiB
Racket
#lang racket
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;; (C) Copyright Collin J. Doering 2014
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;;
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;; This program is free software: you can redistribute it and/or modify
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;; it under the terms of the GNU General Public License as published by
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;; the Free Software Foundation, either version 3 of the License, or
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;; (at your option) any later version.
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;;
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;; This program is distributed in the hope that it will be useful,
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;; but WITHOUT ANY WARRANTY; without even the implied warranty of
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;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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;; GNU General Public License for more details.
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;;
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;; You should have received a copy of the GNU General Public License
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;; along with this program. If not, see <http://www.gnu.org/licenses/>.
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;; File: avl-tree.rkt
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;; Author: Collin J. Doering <collin.doering@rekahsoft.ca>
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;; Date: Sep 2, 2014
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(require "comparable.rkt")
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;; Structure representing a Binary Search Tree
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(struct bst ())
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(struct bst-empty bst () #:transparent)
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(struct bst-node bst (val left right) #:transparent)
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;; Make a leaf in a binary tree
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(define (bst-make-leaf i)
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(bst-node i (bst-empty) (bst-empty)))
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;; ----------------------------------------------------------------------
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;; First a naive approach to binary search trees
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;; Specifically these functions do not maintain any balance
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;; of the bst and thus are inefficient in many cases
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;; Naive bst insert (not balanced)
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(define/match (bst-insert-naive t i)
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[((bst-empty) _) (bst-node i (bst-empty) (bst-empty))]
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[((bst-node v (bst-empty) (bst-empty)) _)
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(if (gte i v)
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(bst-node v (bst-empty) (bst-make-leaf i))
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(bst-node v (bst-make-leaf i) (bst-empty)))]
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[((bst-node v l r) _) #:when (gte i v)
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(bst-node v l (bst-insert-naive r i))]
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[((bst-node v l r) _) #:when (lt i v)
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(bst-node v (bst-insert-naive l i) l)])
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;; Naive bst delete (not balanced)
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;; (define (bst-delete-naive t i)
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;; (match t
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;; [(bst-empty) t]
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;; [(bst-node v l r) #:when (> i v)
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;; (bst-node v l (bst-delete-naive r i))]
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;; [(bst-node v l r) #:when (lt i v)
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;; (bst-node v (bst-delete-naive l i) r)]
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;; [(bst-node v l r) #:when (= i v)
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;; (match* (l r)
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;; [((bst-empty) (bst-empty)) (bst-empty)]
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;; [((bst-node v1 l1 r1) (bst-empty)) ???]
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;; [((bst-empty) (bst-node v1 l1 r1)) ???]
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;; [((bst-node v1 l1 r1) (bst-node v2 l2 r2)) ???])]))
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;; ----------------------------------------------------------------------
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;; Below is a more effiecient implementation of BST's; specifically using
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;; the AVL binary seach tree algorithm.
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;; See: https://en.wikipedia.org/wiki/AVL_tree
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;; Calculate AVL score for a particular node
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(define/match (avl-score t)
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[((bst-empty)) 0]
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[((bst-node v l r)) (- (bst-height l) (bst-height r))])
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;; Rotate a bst
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(define/match (bst-rotate dir t)
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[((quote left-right) _) (bst-rotate 'left (bst-rotate 'right t))]
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[((quote right-left) _) (bst-rotate 'right (bst-rotate 'left t))]
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[((quote left) (bst-node v1 l1 (bst-node v2 l2 r2)))
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(bst-node v2 (bst-node v1 l1 l2) r2)]
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[((quote right) (bst-node v1 (bst-node v2 l2 r2) r1))
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(bst-node v2 l2 (bst-node v1 r2 r1))])
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;; Balance AVL binary tree
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(define (avl-balance t)
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(let ([score (avl-score t)])
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(match t
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[(bst-node v l r) #:when (= score 2)
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(cond [(< (avl-score l) 0) (bst-rotate 'right-left t)]
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[else (bst-rotate 'right t)])]
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[(bst-node v l r) #:when (= score -2)
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(cond [(> (avl-score r) 0) (bst-rotate 'left-right t)]
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[else (bst-rotate 'left t)])]
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[_ t])))
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;; Insert into AVL binary tree
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(define (bst-insert t i)
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(match t
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[(bst-empty) (bst-make-leaf i)]
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[(bst-node v l r) #:when (eql i v) t]
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[(bst-node v l r) #:when (gte i v)
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(let* ([r-not (bst-insert r i)]
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[rt (bst-node v l r-not)])
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(avl-balance rt))]
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[(bst-node v l r) #:when (lt i v)
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(let* ([l-not (bst-insert l i)]
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[rt (bst-node v l-not r)])
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(avl-balance rt))]))
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;; Delete item from AVL binary tree
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(define (bst-delete t i)
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'undefined)
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(define (bst-search t i)
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(match t
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[(bst-empty) #f]
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[(bst-node v _ _) #:when (eql i v) #t]
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[(bst-node v _ r) #:when (gt i v) (bst-search r i)]
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[(bst-node v l _) #:when (lt i v) (bst-search l i)]))
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(define (bst-flatten t)
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(match t
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[(bst-empty) '()]
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[(bst-node v l r) `(,@(bst-flatten l) ,v ,@(bst-flatten r))]))
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(define (bst-height t)
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(match t
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[(bst-empty) 0]
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[(bst-node _ l (bst-empty)) (+ 1 (bst-height l))]
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[(bst-node _ (bst-empty) r) (+ 1 (bst-height r))]
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[(bst-node _ l r) (+ 1 (max (bst-height l) (bst-height r)))]))
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(define (list->bst xs)
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(foldr (lambda (x acc)
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(bst-insert acc x))
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(bst-empty) xs))
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(define (bst-sort xs)
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(bst-flatten (list->bst xs)))
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