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