高度平衡树 -- AVL 树

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Scheme 的表达, 优雅.

#lang scheme

( define nil '() )
( define ( root tree )( car tree ) )
( define ( left-tree tree )( cadr tree ) )
( define ( right-tree tree )( caddr tree ) )
( define ( height tree )
( cond [ ( null? tree ) 0 ]
[ else ( cadddr tree ) ] ) )

( define ( make-leaf elem )( list elem nil nil 1 ) )

( define ( make-avl-tree root left right )
( list root left right ( + 1 ( max ( height left )
( height right ) ) ) ) )

( define ( contains-elem? elem tree )
( cond [ ( null? tree ) false ]
[ ( = elem ( root tree ) ) true ]
[ ( < elem ( root tree ) )
( contains-elem? elem ( left-tree tree ) ) ]
[ ( > elem ( root tree ) )
( contains-elem? elem ( right-tree tree ) ) ] ) )

( define ( rotate-left-left tree )
( cond [ ( null? tree ) tree ]
[ else ( make-avl-tree ( root ( left-tree tree ) )
( left-tree ( left-tree tree ) )
( make-avl-tree ( root tree )
( right-tree ( left-tree tree ) )
( right-tree tree ) ) ) ] ) )

( define ( rotate-right-right tree )
( cond [ ( null? tree ) tree ]
[ else ( make-avl-tree ( root ( right-tree tree ) )
( make-avl-tree ( root tree )
( left-tree tree )
( left-tree ( right-tree tree ) ) )
( right-tree ( right-tree tree ) ) ) ] ) )

( define ( rotate-right-left tree )
( cond [ ( null? tree ) tree ]
[ else ( make-avl-tree ( left-tree ( right-tree tree ) )
( make-avl-tree ( root tree )
( left-tree tree )
( left-tree ( left-tree ( right-tree tree ) ) ) )
( make-avl-tree ( root ( right-tree tree ) )
( right-tree ( left-tree ( right-tree tree ) ) )
( right-tree ( right-tree tree ) ) ) ) ] ) )

( define ( rotate-left-right tree )
( cond [ ( null? tree ) tree ]
[ else ( make-avl-tree ( root ( right-tree ( left-tree tree ) ) )
( make-avl-tree ( root ( left-tree tree ) )
( left-tree ( left-tree tree ) )
( left-tree ( right-tree ( left-tree tree ) ) ) )
( make-avl-tree ( root tree )
( right-tree ( right-tree ( left-tree tree ) ) )
( right-tree tree ) ) ) ] ) )

( define ( balance-avl-tree tree )
( define ( factor tree )
( - ( height ( right-tree tree ) )
( height ( left-tree tree ) ) ) )
( let ( [ f ( factor tree ) ] )
( cond [ ( = f 2 )
( cond [ ( < ( factor ( right-tree tree ) ) 0 )
( rotate-right-left tree ) ]
[ else ( rotate-right-right tree ) ] ) ]
[ ( = f -2 )
( cond [ ( > ( factor ( left-tree tree ) ) 0 )
( rotate-left-right tree ) ]
[ else ( rotate-left-left tree ) ] ) ]
[ else tree ] ) ) )

( define ( insert-elem elem tree )
( define ( insert-in-son elem tree )
( cond [ ( null? tree )
( make-leaf elem ) ]
[ ( < elem ( root tree ) )
( let* ( [ newLeftTree ( insert-in-son elem ( left-tree tree ) ) ]
[ newAVLTree ( make-avl-tree ( root tree )
newLeftTree
( right-tree tree ) ) ] )
( balance-avl-tree newAVLTree ) ) ]
[ ( > elem ( root tree ) )
( let* ( [ newRightTree ( insert-in-son elem ( right-tree tree ) ) ]
[ newAVLTree ( make-avl-tree ( root tree )
( left-tree tree )
newRightTree ) ] )
( balance-avl-tree newAVLTree ) ) ]
[ else tree ] ) )
( cond [ ( contains-elem? elem tree ) tree ]
[ else ( insert-in-son elem tree ) ] ) )

( define ( delete-elem elem tree )
( define ( delete-left-most tree )
( cond [ ( left-empty? tree ) tree ]
[ else ( let* ( [ leftMost ( delete-left-most ( left-tree tree ) ) ]
[ newRightTree ( make-avl-tree ( root tree )
( right-tree leftMost )
( right-tree tree ) ) ] )
( make-avl-tree ( root leftMost )
nil
( balance-avl-tree newRightTree ) ) ) ] ) )
( define ( delete-in-son elem tree )
( cond [ ( < elem ( root tree ) )
( let* ( [ newLeftTree ( delete-in-son elem ( left-tree tree ) ) ]
[ newAVLTree ( make-avl-tree ( root tree )
newLeftTree
( right-tree tree ) ) ] )
( balance-avl-tree newAVLTree ) ) ]
[ ( > elem ( root tree ) )
( let* ( [ newRightTree ( delete-in-son elem ( right-tree tree ) ) ]
[ newAVLTree ( make-avl-tree ( root tree )
( left-tree tree )
newRightTree ) ] )
( balance-avl-tree newAVLTree ) ) ]
[ ( = elem ( root tree ) )
( cond [ ( and ( right-empty? tree )
( left-empty? tree ) )
nil ]
[ ( right-empty? tree )
( left-tree tree ) ]
[ ( left-empty? tree )
( right-tree tree ) ]
[ else ( let ( [ leftMost ( delete-left-most ( right-tree tree ) ) ] )
( make-avl-tree ( root leftMost )
( left-tree tree )
( right-tree leftMost ) ) ) ] ) ] ) )
( define ( left-empty? tree )( null? ( left-tree tree ) ) )
( define ( right-empty? tree )( null? ( right-tree tree ) ) )
( cond [ ( contains-elem? elem tree )
( delete-in-son elem tree ) ]
[ else tree ] ) )

( define ( list->avl elems )
( define ( iter elems tree )
( cond [ ( null? elems ) tree ]
[ else ( iter ( cdr elems )
( insert-elem ( car elems ) tree ) ) ] ) )
( cond [ ( null? elems ) '() ]
[ else ( let( [ avl ( make-leaf ( car elems ) ) ] )

( iter ( cdr elems ) avl ) ) ] ) )