277 lines
11 KiB
Prolog
277 lines
11 KiB
Prolog
/*************************************************************************
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* *
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* YAP Prolog *
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* *
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* Yap Prolog was developed at NCCUP - Universidade do Porto *
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* *
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* Copyright L.Damas, V.S.Costa and Universidade do Porto 1985-1997 *
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* *
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**************************************************************************
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* *
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* File: splay.yap *
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* Last rev: 5/12/99 *
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* mods: *
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* comments: Vijay Saraswat's implementation of Splay trees *
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* *
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*************************************************************************/
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/**
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* @file splay.yap
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* @author Vijay Saraswat
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* @date Wed Nov 18 01:12:49 2015
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*
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* @brief "Self-adjusting Binary Search Trees
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*
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*
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*/
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:- module(splay,[
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splay_access/5,
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splay_insert/4,
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splay_del/3,
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splay_init/1,
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splay_join/3,
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splay_split/5]).
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/** @defgroup Splay_Trees Splay Trees
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@ingroup library
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@{
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Splay trees are explained in the paper "Self-adjusting Binary Search
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Trees", by D.D. Sleator and R.E. Tarjan, JACM, vol. 32, No.3, July 1985,
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p. 668. They are designed to support fast insertions, deletions and
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removals in binary search trees without the complexity of traditional
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balanced trees. The key idea is to allow the tree to become
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unbalanced. To make up for this, whenever we \ find a node, we move it up
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to the top. We use code by Vijay Saraswat originally posted to the Prolog
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mailing-list.
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Date: Sun 22 Mar 87 03:40:22-EST
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>From: vijay <Vijay.Saraswat@C.CS.CMU.EDU>
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Subject: Splay trees in LP languages.
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There have hardly been any interesting programs in this Digest for a
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long while now. Here is something which may stir the slothful among
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you! I present Prolog programs for implementing self-adjusting binary
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search trees, using splaying. These programs should be among the most
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efficient Prolog programs for maintaining binary search trees, with
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dynamic insertion and deletion.
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The algorithm is taken from: "Self-adjusting Binary Search Trees",
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D.D. Sleator and R.E. Tarjan, JACM, vol. 32, No.3, July 1985, p. 668.
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(See Tarjan's Turing Award lecture in this month's CACM for a more
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informal introduction).
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-----------------------------------------
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The operations provided by the program are:
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1. access(i,t): (implemented by the call access(V, I, T, New))
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"If item i is in tree t, return a pointer to its location;
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otherwise return a pointer to the null node."
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In our implementation, in the call access(V, I, T, New),
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V is unifies with `null' if the item is not there, else
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with `true' if it is there, in which case I is also
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unified with that item.
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2. insert(i,t): (implemented by the call insert(I, T, New))
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"Insert item i in tree t, assuming that it is not there already."
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(In our implementation, i is not inserted if it is already
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there: rather it is unified with the item already in the tree.)
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3. delete(i,t): (implemented by the call del(I, T, New))
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"Delete item i from tree t, assuming that it is present."
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(In our implementation, the call fails if the item is not in
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the tree.)
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4. join(t1,t2): (Implemented by the call join(T1, T2, New))
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"Combine trees t1 and t2 into a single tree containing
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all items from both trees, and return the resulting
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tree. This operation assumes that all items in t1 are
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less than all those in t2 and destroys both t1 and t2."
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5. split(i,t): (implemented by the call split(I, T, Left, Right))
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"Construct and return two trees t1 and t2, where t1
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contains all items in t less than i, and t2 contains all
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items in t greater than i. This operations destroys t."
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The basic workhorse is the routine bst(Op, Item, Tree, NewTree), which
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returns in NewTree a binary search tree obtained by searching for Item
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in< Tree and splaying. OP controls what must happen if Item is not
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found in the Tree. If Op = access(V), then V is unified with null if
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the item is not found in the tree, and with true if it is; in the
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latter case Item is also unified with the item found in the tree. In
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% the first case, splaying is done at the node at which the discovery
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% was made that Item was not in the tree, and in the second case
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% splaying is done at the node at which Item is found. If Op=insert,
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% then Item is inserted in the tree if it is not found, and splaying is
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% done at the new node; if the item is found, then splaying is done at
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% the node at which it is found.
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% A node is simply an n/3 structure: n(NodeValue, LeftSon, RightSon).
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% NodeValue could be as simple as an integer, or it could be a (Key,
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% Value) pair.
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% A node is simply an n/3 structure: n(NodeValue, LeftSon, RightSon).
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% NodeValue could be as simple as an integer, or it could be a (Key,
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% Value) pair.
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% Here are the top-level axioms. The algorithm for del/3 is the first
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% algorithm mentioned in the JACM paper: namely, first access the
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% element to be deleted, thus bringing it to the root, and then join its
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% sons. (join/4 is discussed later.)
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*/
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/*
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@pred splay_access(- _Return_,+ _Key_,? _Val_,+ _Tree_,- _NewTree_)
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v
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If item _Key_ is in tree _Tree_, return its _Val_ and
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unify _Return_ with `true`. Otherwise unify _Return_ with
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`null`. The variable _NewTree_ unifies with the new tree.
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*/
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/** @pred splay_del(+ _Item_,+ _Tree_,- _NewTree_)
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Delete item _Key_ from tree _Tree_, assuming that it is present
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already. The variable _Val_ unifies with a value for key _Key_,
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and the variable _NewTree_ unifies with the new tree. The predicate
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will fail if _Key_ is not present.
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*/
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/** @pred splay_init(- _NewTree_)
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Initialize a new splay tree.
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*/
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/** @pred splay_insert(+ _Key_,? _Val_,+ _Tree_,- _NewTree_)
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Insert item _Key_ in tree _Tree_, assuming that it is not
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there already. The variable _Val_ unifies with a value for key
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_Key_, and the variable _NewTree_ unifies with the new
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tree. In our implementation, _Key_ is not inserted if it is
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already there: rather it is unified with the item already in the tree.
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*/
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/** @pred splay_join(+ _LeftTree_,+ _RighTree_,- _NewTree_)
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Combine trees _LeftTree_ and _RighTree_ into a single
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tree _NewTree_ containing all items from both trees. This operation
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assumes that all items in _LeftTree_ are less than all those in
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_RighTree_ and destroys both _LeftTree_ and _RighTree_.
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*/
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/** @pred splay_split(+ _Key_,? _Val_,+ _Tree_,- _LeftTree_,- _RightTree_)
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Construct and return two trees _LeftTree_ and _RightTree_,
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where _LeftTree_ contains all items in _Tree_ less than
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_Key_, and _RightTree_ contains all items in _Tree_
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greater than _Key_. This operations destroys _Tree_.
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*/
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splay_access(V, Item, Val, Tree, NewTree):-
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bst(access(V), Item, Val, Tree, NewTree).
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splay_insert(Item, Val,Tree, NewTree):-
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bst(insert, Item, Val, Tree, NewTree).
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splay_del(Item, Tree, NewTree):-
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bst(access(true), Item, Val, Tree, n(Item, Val, Left, Right)),
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splay_join(Left, Right, NewTree).
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splay_join(Left, Right, New):-
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join(L-L, Left, Right, New).
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splay_split(Item, Val, Tree, Left, Right):-
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bst(access(true), Item, Val, Tree, n(Item, Val, Left, Right)).
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% We now consider the definition of bst. We use the notion of
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% `difference-bsts'. There are two types of difference-bsts, a left one
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% and a right one. The left one is of the form T-L where T is a bst and
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% L is the *right* son of the node with the largest value in T. The
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% right one is of the form T-R where T is a binary search tree and R is
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% the *left* son of the node with the smallest value in T. An empty bst
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% is denoted by a variable. Hence L-L denotes the empty left (as well as
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% right) difference bst.
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% As discussed in the JACM paper, we start with a notion of a left
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% fragment and a right fragment of the new bst to be constructed.
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% Intially, the two fragments are empty.
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bst(Op, Item, Val, Tree, NewTree):-
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bst(Op, Item, Val, L-L, Tree, R-R, NewTree).
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% We now consider the base cases. The empty tree is a variable: hence it
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% will unify with the atom null. (A non-empty tree is a n/3 structure,
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% which will not unify with null). If Item was being *access*ed, then it
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% was not found in the tree, and hence Null=null. On the other hand, if
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% the Item is found at the root, then the call terminates, with the New
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% Tree being set up appropriately.
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% The base clauses are:
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bst(access(Null), _Item, _, _L, null, _R, _Tree):- !, Null = null.
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bst(access(true), Item, Val, Left-A, n(Item0, Val0, A, B), Right-B, n(Item, Val, Left, Right)) :- Item == Item0, !, Val = Val0.
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bst(insert, Item, Val, Left-A, T, Right-B, n(Item0, Val, Left, Right)) :-
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(var(T) ; T = n(Item0, _Val0, A, B), Item == Item0), !, Item = Item0.
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% We now consider the zig case, namely that we have reached a node such
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% that the required Item is either to the left of the current node and
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% the current node is a leaf, or the required item is the left son of
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% the current node. Depending upon the Op, the appropriate action is
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% taken:
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bst(access(Null), Item, _, Left-L, n(X, VX, null, B), Right-B, n(X, VX, Left, Right)) :-
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Item @< X, !, Null = null.
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bst(Op, Item, Val, Left, n(X, VX, n(Item, Val, A1, A2), B), R-n(X, VX, NR,B), New):-
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Item @< X, !,
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bst(Op, Item, Val, Left, n(Item, Val, A1, A2), R-NR, New).
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% The recursive cases are straightforward:
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% Zig-Zig:
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bst(Op, Item, Val, Left, n(X, VX, n(Y, VY, Z, B), C), R-n(Y, VY, NR, n(X, VX, B, C)), New):-
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Item @< X, Item @< Y, !,
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bst(Op, Item, Val, Left, Z, R-NR, New).
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% Zig-Zag:
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bst(Op, Item, Val, L-n(Y, VY, A, NL), n(X, _VX, n(Y, VY, A, Z), C), R-n(X, _NX, NR, C), New):-
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Item @< X, Y @< Item,!,
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bst(Op, Item, Val, L-NL, Z, R-NR, New).
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% The symmetric cases for the right sons of the current node
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% are straightforward too:
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% Zag
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bst(access(Null), Item, _, Left-B, n(X, VX, B, null), Right-_R, n(X, VX, Left, Right)):-
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X @< Item, !, Null = null. % end of the road.
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bst(Op, Item, Val, L-n(X, VX, B, NL), n(X, VX, B, n(Item, Val, A1, A2)), Right, New):-
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X @< Item, !,
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bst(Op, Item, Val, L-NL, n(Item, Val, A1, A2), Right, New).
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% Zag-Zag
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bst(Op, Item, Val, L-n(Y, VY, n(X, VX, C, B), NL), n(X, VX, C, n(Y, VY, B, Z)), Right, New):-
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X @< Item, Y @<Item,!,
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bst(Op, Item, Val, L-NL, Z, Right, New).
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% Zag-Zig
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bst(Op, Item, Val, L-n(X, VX, A, NL), n(X, VX, A, n(Y, VY, Z, C)), R-n(Y, VY, NR, C), New):-
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X @< Item, Item @< Y,!,
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bst(Op, Item, Val, L-NL, Z, R-NR, New).
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% We now consider the definition of join. To join Left to Right, it is
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% sufficient to splay at the rightmost vertex in Left, and make Right
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% its Right son. To build NewTree, we initially start with an empty left
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join(Left-A, n(X, VX, A, var), Right, n(X, VX, Left, Right)):-!.
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join(Left-n(X, VX, A, B), n(X, VX, A, n(Y, VY, B, var)), Right, n(Y, VY, Left, Right)):- !.
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join(Left-n(Y, VY, n(X, VX, C, B), NL), n(X, VX, C, n(Y, VY, B, n(Z, VZ, A1, A2))), Right, New):-
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join(Left-NL, n(Z, VZ,A1, A2), Right, New).
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splay_init(_).
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