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yap-6.3/library/rbtrees.yap
2012-08-28 20:19:10 -05:00

1231 lines
33 KiB
Prolog

/*
This code implements Red-Black trees as described in:
"Introduction to Algorithms", Second Edition
Cormen, Leiserson, Rivest, and Stein,
MIT Press
Author: Vitor Santos Costa
*/
:- module(rbtrees,
[rb_new/1,
rb_empty/1, % ?T
rb_lookup/3, % +Key, -Value, +T
rb_update/4, % +T, +Key, +NewVal, -TN
rb_update/5, % +T, +Key, ?OldVal, +NewVal, -TN
rb_rewrite/3, % +T, +Key, +NewVal
rb_rewrite/4, % +T, +Key, ?OldVal, +NewVal
rb_apply/4, % +T, +Key, :G, -TN
rb_lookupall/3, % +Key, -Value, +T
rb_insert/4, % +T0, +Key, ?Value, -TN
rb_insert_new/4, % +T0, +Key, ?Value, -TN
rb_delete/3, % +T, +Key, -TN
rb_delete/4, % +T, +Key, -Val, -TN
rb_visit/2, % +T, -Pairs
rb_visit/3,
rb_keys/2, % +T, +Keys
rb_keys/3,
rb_map/2,
rb_map/3,
rb_partial_map/4,
rb_accumulate/4,
rb_clone/3,
rb_clone/4,
rb_min/3,
rb_max/3,
rb_del_min/4,
rb_del_max/4,
rb_next/4,
rb_previous/4,
rb_fold/4,
rb_key_fold/4,
list_to_rbtree/2,
ord_list_to_rbtree/2,
is_rbtree/1,
rb_size/2,
rb_in/3
]).
/** <module> Red black trees
Red-Black trees are balanced search binary trees. They are named because
nodes can be classified as either red or black. The code we include is
based on "Introduction to Algorithms", second edition, by Cormen,
Leiserson, Rivest and Stein. The library includes routines to insert,
lookup and delete elements in the tree.
A Red black tree is represented as a term t(Nil, Tree), where Nil is the
Nil-node, a node shared for each nil-node in the tree. Any node has the
form colour(Left, Key, Value, Right), where _colour_ is one of =red= or
=black=.
@author Vitor Santos Costa, Jan Wielemaker
*/
:- meta_predicate rb_map(+,2,-),
rb_partial_map(+,+,2,-),
rb_apply(+,+,2,-).
/*
:- use_module(library(type_check)).
:- type rbtree(K,V) ---> t(tree(K,V),tree(K,V)).
:- type tree(K,V) ---> black(tree(K,V),K,V,tree(K,V))
; red(tree(K,V),K,V,tree(K,V))
; ''.
:- type cmp ---> (=) ; (<) ; (>).
:- pred rb_new(rbtree(_K,_V)).
:- pred rb_empty(rbtree(_K,_V)).
:- pred rb_lookup(K,V,rbtree(K,V)).
:- pred lookup(K,V, tree(K,V)).
:- pred lookup(cmp, K, V, tree(K,V)).
:- pred rb_min(rbtree(K,V),K,V).
:- pred min(tree(K,V),K,V).
:- pred rb_max(rbtree(K,V),K,V).
:- pred max(tree(K,V),K,V).
:- pred rb_next(rbtree(K,V),K,pair(K,V),V).
:- pred next(tree(K,V),K,pair(K,V),V,tree(K,V)).
*/
% create an empty tree.
%% rb_new(-T) is det.
%
% Create a new Red-Black tree.
%
% @deprecated Use rb_empty/1.
rb_new(t(Nil,Nil)) :- Nil = black('',_,_,'').
rb_new(K,V,t(Nil,black(Nil,K,V,Nil))) :- Nil = black('',_,_,'').
%% rb_empty(?T) is semidet.
%
% Succeeds if T is an empty Red-Black tree.
rb_empty(t(Nil,Nil)) :- Nil = black('',_,_,'').
%% rb_lookup(+Key, -Value, +T) is semidet.
%
% Backtrack through all elements with key Key in the Red-Black
% tree T, returning for each the value Value.
rb_lookup(Key, Val, t(_,Tree)) :-
lookup(Key, Val, Tree).
lookup(_, _, black('',_,_,'')) :- !, fail.
lookup(Key, Val, Tree) :-
arg(2,Tree,KA),
compare(Cmp,KA,Key),
lookup(Cmp,Key,Val,Tree).
lookup(>, K, V, Tree) :-
arg(1,Tree,NTree),
lookup(K, V, NTree).
lookup(<, K, V, Tree) :-
arg(4,Tree,NTree),
lookup(K, V, NTree).
lookup(=, _, V, Tree) :-
arg(3,Tree,V).
%% rb_min(+T, -Key, -Value) is semidet.
%
% Key is the minimum key in T, and is associated with Val.
rb_min(t(_,Tree), Key, Val) :-
min(Tree, Key, Val).
min(red(black('',_,_,_),Key,Val,_), Key, Val) :- !.
min(black(black('',_,_,_),Key,Val,_), Key, Val) :- !.
min(red(Right,_,_,_), Key, Val) :-
min(Right,Key,Val).
min(black(Right,_,_,_), Key, Val) :-
min(Right,Key,Val).
%% rb_max(+T, -Key, -Value) is semidet.
%
% Key is the maximal key in T, and is associated with Val.
rb_max(t(_,Tree), Key, Val) :-
max(Tree, Key, Val).
max(red(_,Key,Val,black('',_,_,_)), Key, Val) :- !.
max(black(_,Key,Val,black('',_,_,_)), Key, Val) :- !.
max(red(_,_,_,Left), Key, Val) :-
max(Left,Key,Val).
max(black(_,_,_,Left), Key, Val) :-
max(Left,Key,Val).
%% rb_next(+T, +Key, -Next,-Value) is semidet.
%
% Next is the next element after Key in T, and is associated with
% Val.
rb_next(t(_,Tree), Key, Next, Val) :-
next(Tree, Key, Next, Val, []).
next(black('',_,_,''), _, _, _, _) :- !, fail.
next(Tree, Key, Next, Val, Candidate) :-
arg(2,Tree,KA),
arg(3,Tree,VA),
compare(Cmp,KA,Key),
next(Cmp, Key, KA, VA, Next, Val, Tree, Candidate).
next(>, K, KA, VA, NK, V, Tree, _) :-
arg(1,Tree,NTree),
next(NTree,K,NK,V,KA-VA).
next(<, K, _, _, NK, V, Tree, Candidate) :-
arg(4,Tree,NTree),
next(NTree,K,NK,V,Candidate).
next(=, _, _, _, NK, Val, Tree, Candidate) :-
arg(4,Tree,NTree),
(
min(NTree, NK, Val)
-> true
;
Candidate = (NK-Val)
).
%% rb_previous(+T, +Key, -Previous, -Value) is semidet.
%
% Previous is the previous element after Key in T, and is
% associated with Val.
rb_previous(t(_,Tree), Key, Previous, Val) :-
previous(Tree, Key, Previous, Val, []).
previous(black('',_,_,''), _, _, _, _) :- !, fail.
previous(Tree, Key, Previous, Val, Candidate) :-
arg(2,Tree,KA),
arg(3,Tree,VA),
compare(Cmp,KA,Key),
previous(Cmp, Key, KA, VA, Previous, Val, Tree, Candidate).
previous(>, K, _, _, NK, V, Tree, Candidate) :-
arg(1,Tree,NTree),
previous(NTree,K,NK,V,Candidate).
previous(<, K, KA, VA, NK, V, Tree, _) :-
arg(4,Tree,NTree),
previous(NTree,K,NK,V,KA-VA).
previous(=, _, _, _, K, Val, Tree, Candidate) :-
arg(1,Tree,NTree),
(
max(NTree, K, Val)
-> true
;
Candidate = (K-Val)
).
%% rb_update(+T, +Key, +NewVal, -TN) is semidet.
%% rb_update(+T, +Key, ?OldVal, +NewVal, -TN) is semidet.
%
% Tree TN is tree T, but with value for Key associated with
% NewVal. Fails if it cannot find Key in T.
rb_update(t(Nil,OldTree), Key, OldVal, Val, t(Nil,NewTree)) :-
update(OldTree, Key, OldVal, Val, NewTree).
rb_update(t(Nil,OldTree), Key, Val, t(Nil,NewTree)) :-
update(OldTree, Key, _, Val, NewTree).
update(black(Left,Key0,Val0,Right), Key, OldVal, Val, NewTree) :-
Left \= [],
compare(Cmp,Key0,Key),
(Cmp == (=)
-> OldVal = Val0,
NewTree = black(Left,Key0,Val,Right)
;
Cmp == (>) ->
NewTree = black(NewLeft,Key0,Val0,Right),
update(Left, Key, OldVal, Val, NewLeft)
;
NewTree = black(Left,Key0,Val0,NewRight),
update(Right, Key, OldVal, Val, NewRight)
).
update(red(Left,Key0,Val0,Right), Key, OldVal, Val, NewTree) :-
compare(Cmp,Key0,Key),
(Cmp == (=)
-> OldVal = Val0,
NewTree = red(Left,Key0,Val,Right)
;
Cmp == (>)
-> NewTree = red(NewLeft,Key0,Val0,Right),
update(Left, Key, OldVal, Val, NewLeft)
;
NewTree = red(Left,Key0,Val0,NewRight),
update(Right, Key, OldVal, Val, NewRight)
).
%% rb_rewrite(+T, +Key, +NewVal) is semidet.
%% rb_rewrite(+T, +Key, ?OldVal, +NewVal) is semidet.
%
% Tree T has value for Key associated with
% NewVal. Fails if it cannot find Key in T.
rb_rewrite(t(_Nil,OldTree), Key, OldVal, Val) :-
rewrite(OldTree, Key, OldVal, Val).
rb_rewrite(t(_Nil,OldTree), Key, Val) :-
rewrite(OldTree, Key, _, Val).
rewrite(Node, Key, OldVal, Val) :-
Node = black(Left,Key0,Val0,Right),
Left \= [],
compare(Cmp,Key0,Key),
(Cmp == (=)
-> OldVal = Val0,
setarg(3, Node, Val)
;
Cmp == (>) ->
rewrite(Left, Key, OldVal, Val)
;
rewrite(Right, Key, OldVal, Val)
).
rewrite(Node, Key, OldVal, Val) :-
Node = red(Left,Key0,Val0,Right),
Left \= [],
compare(Cmp,Key0,Key),
(
Cmp == (=)
->
OldVal = Val0,
setarg(3, Node, Val)
;
Cmp == (>)
->
rewrite(Left, Key, OldVal, Val)
;
rewrite(Right, Key, OldVal, Val)
).
%% rb_apply(+T, +Key, :G, -TN) is semidet.
%
% If the value associated with key Key is Val0 in T, and if
% call(G,Val0,ValF) holds, then TN differs from T only in that Key
% is associated with value ValF in tree TN. Fails if it cannot
% find Key in T, or if call(G,Val0,ValF) is not satisfiable.
rb_apply(t(Nil,OldTree), Key, Goal, t(Nil,NewTree)) :-
apply(OldTree, Key, Goal, NewTree).
%apply(black('',_,_,''), _, _, _) :- !, fail.
apply(black(Left,Key0,Val0,Right), Key, Goal,
black(NewLeft,Key0,Val,NewRight)) :-
Left \= [],
compare(Cmp,Key0,Key),
(Cmp == (=)
-> NewLeft = Left,
NewRight = Right,
call(Goal,Val0,Val)
; Cmp == (>)
-> NewRight = Right,
Val = Val0,
apply(Left, Key, Goal, NewLeft)
;
NewLeft = Left,
Val = Val0,
apply(Right, Key, Goal, NewRight)
).
apply(red(Left,Key0,Val0,Right), Key, Goal,
red(NewLeft,Key0,Val,NewRight)) :-
compare(Cmp,Key0,Key),
( Cmp == (=)
-> NewLeft = Left,
NewRight = Right,
call(Goal,Val0,Val)
; Cmp == (>)
-> NewRight = Right,
Val = Val0,
apply(Left, Key, Goal, NewLeft)
;
NewLeft = Left,
Val = Val0,
apply(Right, Key, Goal, NewRight)
).
%% rb_in(?Key, ?Val, +Tree) is nondet.
%
% True if Key-Val appear in Tree. Uses indexing if Key is bound.
rb_in(Key, Val, t(_,T)) :-
var(Key), !,
enum(Key, Val, T).
rb_in(Key, Val, t(_,T)) :-
lookup(Key, Val, T).
enum(Key, Val, black(L,K,V,R)) :-
L \= '',
enum_cases(Key, Val, L, K, V, R).
enum(Key, Val, red(L,K,V,R)) :-
enum_cases(Key, Val, L, K, V, R).
enum_cases(Key, Val, L, _, _, _) :-
enum(Key, Val, L).
enum_cases(Key, Val, _, Key, Val, _).
enum_cases(Key, Val, _, _, _, R) :-
enum(Key, Val, R).
%% rb_lookupall(+Key, -Value, +T)
%
% Lookup all elements with key Key in the red-black tree T,
% returning the value Value.
rb_lookupall(Key, Val, t(_,Tree)) :-
lookupall(Key, Val, Tree).
lookupall(_, _, black('',_,_,'')) :- !, fail.
lookupall(Key, Val, Tree) :-
arg(2,Tree,KA),
compare(Cmp,KA,Key),
lookupall(Cmp,Key,Val,Tree).
lookupall(>, K, V, Tree) :-
arg(4,Tree,NTree),
rb_lookupall(K, V, NTree).
lookupall(=, _, V, Tree) :-
arg(3,Tree,V).
lookupall(=, K, V, Tree) :-
arg(1,Tree,NTree),
lookupall(K, V, NTree).
lookupall(<, K, V, Tree) :-
arg(1,Tree,NTree),
lookupall(K, V, NTree).
/*******************************
* TREE INSERTION *
*******************************/
% We don't use parent nodes, so we may have to fix the root.
%% rb_insert(+T0, +Key, ?Value, -TN) is det.
%
% Add an element with key Key and Value to the tree T0 creating a
% new red-black tree TN. If Key is a key in T0, the associated
% value is replaced by Value. See also rb_insert_new/4.
rb_insert(t(Nil,Tree0),Key,Val,t(Nil,Tree)) :-
insert(Tree0,Key,Val,Nil,Tree).
insert(Tree0,Key,Val,Nil,Tree) :-
insert2(Tree0,Key,Val,Nil,TreeI,_),
fix_root(TreeI,Tree).
%
% Cormen et al present the algorithm as
% (1) standard tree insertion;
% (2) from the viewpoint of the newly inserted node:
% partially fix the tree;
% move upwards
% until reaching the root.
%
% We do it a little bit different:
%
% (1) standard tree insertion;
% (2) move upwards:
% when reaching a black node;
% if the tree below may be broken, fix it.
% We take advantage of Prolog unification
% to do several operations in a single go.
%
%
% actual insertion
%
insert2(black('',_,_,''), K, V, Nil, T, Status) :- !,
T = red(Nil,K,V,Nil),
Status = not_done.
insert2(red(L,K0,V0,R), K, V, Nil, NT, Flag) :-
( K @< K0
-> NR = R,
NT = red(NL,K0,V0,R),
insert2(L, K, V, Nil, NL, Flag)
; K == K0 ->
NT = red(L,K0,V,R),
Flag = done
;
NT = red(L,K0,V0,NR),
insert2(R, K, V, Nil, NR, Flag)
).
insert2(black(L,K0,V0,R), K, V, Nil, NT, Flag) :-
( K @< K0
-> insert2(L, K, V, Nil, IL, Flag0),
fix_left(Flag0, black(IL,K0,V0,R), NT, Flag)
; K == K0 ->
NT = black(L,K0,V,R),
Flag = done
;
insert2(R, K, V, Nil, IR, Flag0),
fix_right(Flag0, black(L,K0,V0,IR), NT, Flag)
).
% We don't use parent nodes, so we may have to fix the root.
%% rb_insert_new(+T0, +Key, ?Value, -TN) is semidet.
%
% Add a new element with key Key and Value to the tree T0 creating a
% new red-black tree TN. Fails if Key is a key in T0.
rb_insert_new(t(Nil,Tree0),Key,Val,t(Nil,Tree)) :-
insert_new(Tree0,Key,Val,Nil,Tree).
insert_new(Tree0,Key,Val,Nil,Tree) :-
insert_new_2(Tree0,Key,Val,Nil,TreeI,_),
fix_root(TreeI,Tree).
%
% actual insertion, copied from insert2
%
insert_new_2(black('',_,_,''), K, V, Nil, T, Status) :- !,
T = red(Nil,K,V,Nil),
Status = not_done.
insert_new_2(red(L,K0,V0,R), K, V, Nil, NT, Flag) :-
( K @< K0
-> NR = R,
NT = red(NL,K0,V0,R),
insert_new_2(L, K, V, Nil, NL, Flag)
; K == K0 ->
fail
;
NT = red(L,K0,V0,NR),
insert_new_2(R, K, V, Nil, NR, Flag)
).
insert_new_2(black(L,K0,V0,R), K, V, Nil, NT, Flag) :-
( K @< K0
-> insert_new_2(L, K, V, Nil, IL, Flag0),
fix_left(Flag0, black(IL,K0,V0,R), NT, Flag)
; K == K0 ->
fail
;
insert_new_2(R, K, V, Nil, IR, Flag0),
fix_right(Flag0, black(L,K0,V0,IR), NT, Flag)
).
%
% make sure the root is always black.
%
fix_root(black(L,K,V,R),black(L,K,V,R)).
fix_root(red(L,K,V,R),black(L,K,V,R)).
%
% How to fix if we have inserted on the left
%
fix_left(done,T,T,done) :- !.
fix_left(not_done,Tmp,Final,Done) :-
fix_left(Tmp,Final,Done).
%
% case 1 of RB: just need to change colors.
%
fix_left(black(red(Al,AK,AV,red(Be,BK,BV,Ga)),KC,VC,red(De,KD,VD,Ep)),
red(black(Al,AK,AV,red(Be,BK,BV,Ga)),KC,VC,black(De,KD,VD,Ep)),
not_done) :- !.
fix_left(black(red(red(Al,KA,VA,Be),KB,VB,Ga),KC,VC,red(De,KD,VD,Ep)),
red(black(red(Al,KA,VA,Be),KB,VB,Ga),KC,VC,black(De,KD,VD,Ep)),
not_done) :- !.
%
% case 2 of RB: got a knee so need to do rotations
%
fix_left(black(red(Al,KA,VA,red(Be,KB,VB,Ga)),KC,VC,De),
black(red(Al,KA,VA,Be),KB,VB,red(Ga,KC,VC,De)),
done) :- !.
%
% case 3 of RB: got a line
%
fix_left(black(red(red(Al,KA,VA,Be),KB,VB,Ga),KC,VC,De),
black(red(Al,KA,VA,Be),KB,VB,red(Ga,KC,VC,De)),
done) :- !.
%
% case 4 of RB: nothing to do
%
fix_left(T,T,done).
%
% How to fix if we have inserted on the right
%
fix_right(done,T,T,done) :- !.
fix_right(not_done,Tmp,Final,Done) :-
fix_right(Tmp,Final,Done).
%
% case 1 of RB: just need to change colors.
%
fix_right(black(red(Ep,KD,VD,De),KC,VC,red(red(Ga,KB,VB,Be),KA,VA,Al)),
red(black(Ep,KD,VD,De),KC,VC,black(red(Ga,KB,VB,Be),KA,VA,Al)),
not_done) :- !.
fix_right(black(red(Ep,KD,VD,De),KC,VC,red(Ga,Ka,Va,red(Be,KB,VB,Al))),
red(black(Ep,KD,VD,De),KC,VC,black(Ga,Ka,Va,red(Be,KB,VB,Al))),
not_done) :- !.
%
% case 2 of RB: got a knee so need to do rotations
%
fix_right(black(De,KC,VC,red(red(Ga,KB,VB,Be),KA,VA,Al)),
black(red(De,KC,VC,Ga),KB,VB,red(Be,KA,VA,Al)),
done) :- !.
%
% case 3 of RB: got a line
%
fix_right(black(De,KC,VC,red(Ga,KB,VB,red(Be,KA,VA,Al))),
black(red(De,KC,VC,Ga),KB,VB,red(Be,KA,VA,Al)),
done) :- !.
%
% case 4 of RB: nothing to do.
%
fix_right(T,T,done).
%
% simplified processor
%
%
pretty_print(t(_,T)) :-
pretty_print(T,6).
pretty_print(black('',_,_,''),_) :- !.
pretty_print(red(L,K,_,R),D) :-
DN is D+6,
pretty_print(L,DN),
format("~t~a:~d~*|~n",[r,K,D]),
pretty_print(R,DN).
pretty_print(black(L,K,_,R),D) :-
DN is D+6,
pretty_print(L,DN),
format("~t~a:~d~*|~n",[b,K,D]),
pretty_print(R,DN).
rb_delete(t(Nil,T), K, t(Nil,NT)) :-
delete(T, K, _, NT, _).
%% rb_delete(+T, +Key, -TN).
%% rb_delete(+T, +Key, -Val, -TN).
%
% Delete element with key Key from the tree T, returning the value
% Val associated with the key and a new tree TN.
rb_delete(t(Nil,T), K, V, t(Nil,NT)) :-
delete(T, K, V0, NT, _),
V = V0.
%
% I am afraid our representation is not as nice for delete
%
delete(red(L,K0,V0,R), K, V, NT, Flag) :-
K @< K0, !,
delete(L, K, V, NL, Flag0),
fixup_left(Flag0,red(NL,K0,V0,R),NT, Flag).
delete(red(L,K0,V0,R), K, V, NT, Flag) :-
K @> K0, !,
delete(R, K, V, NR, Flag0),
fixup_right(Flag0,red(L,K0,V0,NR),NT, Flag).
delete(red(L,_,V,R), _, V, OUT, Flag) :-
% K == K0,
delete_red_node(L,R,OUT,Flag).
delete(black(L,K0,V0,R), K, V, NT, Flag) :-
K @< K0, !,
delete(L, K, V, NL, Flag0),
fixup_left(Flag0,black(NL,K0,V0,R),NT, Flag).
delete(black(L,K0,V0,R), K, V, NT, Flag) :-
K @> K0, !,
delete(R, K, V, NR, Flag0),
fixup_right(Flag0,black(L,K0,V0,NR),NT, Flag).
delete(black(L,_,V,R), _, V, OUT, Flag) :-
% K == K0,
delete_black_node(L,R,OUT,Flag).
%% rb_del_min(+T, -Key, -Val, -TN)
%
% Delete the least element from the tree T, returning the key Key,
% the value Val associated with the key and a new tree TN.
rb_del_min(t(Nil,T), K, Val, t(Nil,NT)) :-
del_min(T, K, Val, Nil, NT, _).
del_min(red(black('',_,_,_),K,V,R), K, V, Nil, OUT, Flag) :- !,
delete_red_node(Nil,R,OUT,Flag).
del_min(red(L,K0,V0,R), K, V, Nil, NT, Flag) :-
del_min(L, K, V, Nil, NL, Flag0),
fixup_left(Flag0,red(NL,K0,V0,R), NT, Flag).
del_min(black(black('',_,_,_),K,V,R), K, V, Nil, OUT, Flag) :- !,
delete_black_node(Nil,R,OUT,Flag).
del_min(black(L,K0,V0,R), K, V, Nil, NT, Flag) :-
del_min(L, K, V, Nil, NL, Flag0),
fixup_left(Flag0,black(NL,K0,V0,R),NT, Flag).
%% rb_del_max(+T, -Key, -Val, -TN)
%
% Delete the largest element from the tree T, returning the key
% Key, the value Val associated with the key and a new tree TN.
rb_del_max(t(Nil,T), K, Val, t(Nil,NT)) :-
del_max(T, K, Val, Nil, NT, _).
del_max(red(L,K,V,black('',_,_,_)), K, V, Nil, OUT, Flag) :- !,
delete_red_node(L,Nil,OUT,Flag).
del_max(red(L,K0,V0,R), K, V, Nil, NT, Flag) :-
del_max(R, K, V, Nil, NR, Flag0),
fixup_right(Flag0,red(L,K0,V0,NR),NT, Flag).
del_max(black(L,K,V,black('',_,_,_)), K, V, Nil, OUT, Flag) :- !,
delete_black_node(L,Nil,OUT,Flag).
del_max(black(L,K0,V0,R), K, V, Nil, NT, Flag) :-
del_max(R, K, V, Nil, NR, Flag0),
fixup_right(Flag0,black(L,K0,V0,NR), NT, Flag).
delete_red_node(L1,L2,L1,done) :- L1 == L2, !.
delete_red_node(black('',_,_,''),R,R,done) :- !.
delete_red_node(L,black('',_,_,''),L,done) :- !.
delete_red_node(L,R,OUT,Done) :-
delete_next(R,NK,NV,NR,Done0),
fixup_right(Done0,red(L,NK,NV,NR),OUT,Done).
delete_black_node(L1,L2,L1,not_done) :- L1 == L2, !.
delete_black_node(black('',_,_,''),red(L,K,V,R),black(L,K,V,R),done) :- !.
delete_black_node(black('',_,_,''),R,R,not_done) :- !.
delete_black_node(red(L,K,V,R),black('',_,_,''),black(L,K,V,R),done) :- !.
delete_black_node(L,black('',_,_,''),L,not_done) :- !.
delete_black_node(L,R,OUT,Done) :-
delete_next(R,NK,NV,NR,Done0),
fixup_right(Done0,black(L,NK,NV,NR),OUT,Done).
delete_next(red(black('',_,_,''),K,V,R),K,V,R,done) :- !.
delete_next(black(black('',_,_,''),K,V,red(L1,K1,V1,R1)),
K,V,black(L1,K1,V1,R1),done) :- !.
delete_next(black(black('',_,_,''),K,V,R),K,V,R,not_done) :- !.
delete_next(red(L,K,V,R),K0,V0,OUT,Done) :-
delete_next(L,K0,V0,NL,Done0),
fixup_left(Done0,red(NL,K,V,R),OUT,Done).
delete_next(black(L,K,V,R),K0,V0,OUT,Done) :-
delete_next(L,K0,V0,NL,Done0),
fixup_left(Done0,black(NL,K,V,R),OUT,Done).
fixup_left(done,T,T,done).
fixup_left(not_done,T,NT,Done) :-
fixup2(T,NT,Done).
%
% case 1: x moves down, so we have to try to fix it again.
% case 1 -> 2,3,4 -> done
%
fixup2(black(black(Al,KA,VA,Be),KB,VB,red(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),
black(T1,KD,VD,black(Ep,KE,VE,Fi)),done) :- !,
fixup2(red(black(Al,KA,VA,Be),KB,VB,black(Ga,KC,VC,De)),
T1,
_).
%
% case 2: x moves up, change one to red
%
fixup2(red(black(Al,KA,VA,Be),KB,VB,black(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),
black(black(Al,KA,VA,Be),KB,VB,red(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),done) :- !.
fixup2(black(black(Al,KA,VA,Be),KB,VB,black(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),
black(black(Al,KA,VA,Be),KB,VB,red(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),not_done) :- !.
%
% case 3: x stays put, shift left and do a 4
%
fixup2(red(black(Al,KA,VA,Be),KB,VB,black(red(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),
red(black(black(Al,KA,VA,Be),KB,VB,Ga),KC,VC,black(De,KD,VD,black(Ep,KE,VE,Fi))),
done) :- !.
fixup2(black(black(Al,KA,VA,Be),KB,VB,black(red(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),
black(black(black(Al,KA,VA,Be),KB,VB,Ga),KC,VC,black(De,KD,VD,black(Ep,KE,VE,Fi))),
done) :- !.
%
% case 4: rotate left, get rid of red
%
fixup2(red(black(Al,KA,VA,Be),KB,VB,black(C,KD,VD,red(Ep,KE,VE,Fi))),
red(black(black(Al,KA,VA,Be),KB,VB,C),KD,VD,black(Ep,KE,VE,Fi)),
done).
fixup2(black(black(Al,KA,VA,Be),KB,VB,black(C,KD,VD,red(Ep,KE,VE,Fi))),
black(black(black(Al,KA,VA,Be),KB,VB,C),KD,VD,black(Ep,KE,VE,Fi)),
done).
fixup_right(done,T,T,done).
fixup_right(not_done,T,NT,Done) :-
fixup3(T,NT,Done).
%
% case 1: x moves down, so we have to try to fix it again.
% case 1 -> 2,3,4 -> done
%
fixup3(black(red(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)),
black(black(Fi,KE,VE,Ep),KD,VD,T1),done) :- !,
fixup3(red(black(De,KC,VC,Ga),KB,VB,black(Be,KA,VA,Al)),T1,_).
%
% case 2: x moves up, change one to red
%
fixup3(red(black(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)),
black(red(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)),
done) :- !.
fixup3(black(black(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)),
black(red(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)),
not_done):- !.
%
% case 3: x stays put, shift left and do a 4
%
fixup3(red(black(black(Fi,KE,VE,Ep),KD,VD,red(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)),
red(black(black(Fi,KE,VE,Ep),KD,VD,De),KC,VC,black(Ga,KB,VB,black(Be,KA,VA,Al))),
done) :- !.
fixup3(black(black(black(Fi,KE,VE,Ep),KD,VD,red(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)),
black(black(black(Fi,KE,VE,Ep),KD,VD,De),KC,VC,black(Ga,KB,VB,black(Be,KA,VA,Al))),
done) :- !.
%
% case 4: rotate right, get rid of red
%
fixup3(red(black(red(Fi,KE,VE,Ep),KD,VD,C),KB,VB,black(Be,KA,VA,Al)),
red(black(Fi,KE,VE,Ep),KD,VD,black(C,KB,VB,black(Be,KA,VA,Al))),
done).
fixup3(black(black(red(Fi,KE,VE,Ep),KD,VD,C),KB,VB,black(Be,KA,VA,Al)),
black(black(Fi,KE,VE,Ep),KD,VD,black(C,KB,VB,black(Be,KA,VA,Al))),
done).
%
% whole list
%
%% rb_visit(+T, -Pairs)
%
% Pairs is an infix visit of tree T, where each element of Pairs
% is of the form K-Val.
rb_visit(t(_,T),Lf) :-
visit(T,[],Lf).
rb_visit(t(_,T),L0,Lf) :-
visit(T,L0,Lf).
visit(black('',_,_,_),L,L) :- !.
visit(red(L,K,V,R),L0,Lf) :-
visit(L,[K-V|L1],Lf),
visit(R,L0,L1).
visit(black(L,K,V,R),L0,Lf) :-
visit(L,[K-V|L1],Lf),
visit(R,L0,L1).
:- meta_predicate map(?,2,?,?). % this is required.
%% rb_map(+T, :Goal) is semidet.
%
% True if call(Goal, Value) is true for all nodes in T.
rb_map(t(Nil,Tree),Goal,t(Nil,NewTree)) :-
map(Tree,Goal,NewTree,Nil).
map(black('',_,_,''),_,Nil,Nil) :- !.
map(red(L,K,V,R),Goal,red(NL,K,NV,NR),Nil) :-
call(Goal,V,NV), !,
map(L,Goal,NL,Nil),
map(R,Goal,NR,Nil).
map(black(L,K,V,R),Goal,black(NL,K,NV,NR),Nil) :-
call(Goal,V,NV), !,
map(L,Goal,NL,Nil),
map(R,Goal,NR,Nil).
:- meta_predicate rb_map(?,1). % this is not strictly required
:- meta_predicate map(?,1). % this is required.
%% rb_map(+T, :G, -TN) is semidet.
%
% For all nodes Key in the tree T, if the value associated with
% key Key is Val0 in tree T, and if call(G,Val0,ValF) holds, then
% the value associated with Key in TN is ValF. Fails if
% call(G,Val0,ValF) is not satisfiable for all Var0.
rb_map(t(_,Tree),Goal) :-
map(Tree,Goal).
map(black('',_,_,''),_) :- !.
map(red(L,_,V,R),Goal) :-
call(Goal,V), !,
map(L,Goal),
map(R,Goal).
map(black(L,_,V,R),Goal) :-
call(Goal,V), !,
map(L,Goal),
map(R,Goal).
:- meta_predicate rb_fold(3,?,?,?). % this is required.
:- meta_predicate map_acc(?,3,?,?). % this is required.
%% rb_fold(+T, :G, +Acc0, -AccF) is semidet.
%
% For all nodes Key in the tree T, if the value associated with
% key Key is V in tree T, if call(G,V,Acc1,Acc2) holds, then
% if VL is value of the previous node in inorder,
% call(G,VL,_,Acc0) must hold, and
% if VR is the value of the next node in inorder,
% call(G,VR,Acc1,_) must hold.
rb_fold(Goal, t(_,Tree), In, Out) :-
map_acc(Tree, Goal, In, Out).
map_acc(black('',_,_,''), _, Acc, Acc) :- !.
map_acc(red(L,_,V,R), Goal, Left, Right) :-
map_acc(L,Goal, Left, Left1),
once(call(Goal,V, Left1, Right1)),
map_acc(R,Goal, Right1, Right).
map_acc(black(L,_,V,R), Goal, Left, Right) :-
map_acc(L,Goal, Left, Left1),
once(call(Goal,V, Left1, Right1)),
map_acc(R,Goal, Right1, Right).
:- meta_predicate rb_key_fold(4,?,?,?). % this is required.
:- meta_predicate map_key_acc(?,3,?,?). % this is required.
%% rb_key_fold(+T, :G, +Acc0, -AccF) is semidet.
%
% For all nodes Key in the tree T, if the value associated with
% key Key is V in tree T, if call(G,Key,V,Acc1,Acc2) holds, then
% if VL is value of the previous node in inorder,
% call(G,VL,_,Acc0) must hold, and
% if VR is the value of the next node in inorder,
% call(G,VR,Acc1,_) must hold.
rb_key_fold(Goal, t(_,Tree), In, Out) :-
map_key_acc(Tree, Goal, In, Out).
map_key_acc(black('',_,_,''), _, Acc, Acc) :- !.
map_key_acc(red(L,Key,V,R), Goal, Left, Right) :-
map_key_acc(L,Goal, Left, Left1),
once(call(Goal, Key, V, Left1, Right1)),
map_key_acc(R,Goal, Right1, Right).
map_key_acc(black(L,Key,V,R), Goal, Left, Right) :-
map_key_acc(L,Goal, Left, Left1),
once(call(Goal, Key, V, Left1, Right1)),
map_key_acc(R,Goal, Right1, Right).
%% rb_clone(+T, -NT, -Pairs)
%
% "Clone" the red-back tree into a new tree with the same keys as
% the original but with all values set to unbound values. Nodes is
% a list containing all new nodes as pairs K-V.
rb_clone(t(Nil,T),t(Nil,NT),Ns) :-
clone(T,Nil,NT,Ns,[]).
clone(black('',_,_,''),Nil,Nil,Ns,Ns) :- !.
clone(red(L,K,_,R),Nil,red(NL,K,NV,NR),NsF,Ns0) :-
clone(L,Nil,NL,NsF,[K-NV|Ns1]),
clone(R,Nil,NR,Ns1,Ns0).
clone(black(L,K,_,R),Nil,black(NL,K,NV,NR),NsF,Ns0) :-
clone(L,Nil,NL,NsF,[K-NV|Ns1]),
clone(R,Nil,NR,Ns1,Ns0).
rb_clone(t(Nil,T),ONs,t(Nil,NT),Ns) :-
clone(T,Nil,ONs,[],NT,Ns,[]).
clone(black('',_,_,''),Nil,ONs,ONs,Nil,Ns,Ns) :- !.
clone(red(L,K,V,R),Nil,ONsF,ONs0,red(NL,K,NV,NR),NsF,Ns0) :-
clone(L,Nil,ONsF,[K-V|ONs1],NL,NsF,[K-NV|Ns1]),
clone(R,Nil,ONs1,ONs0,NR,Ns1,Ns0).
clone(black(L,K,V,R),Nil,ONsF,ONs0,black(NL,K,NV,NR),NsF,Ns0) :-
clone(L,Nil,ONsF,[K-V|ONs1],NL,NsF,[K-NV|Ns1]),
clone(R,Nil,ONs1,ONs0,NR,Ns1,Ns0).
%% rb_partial_map(+T, +Keys, :G, -TN)
%
% For all nodes Key in Keys, if the value associated with key Key
% is Val0 in tree T, and if call(G,Val0,ValF) holds, then the
% value associated with Key in TN is ValF. Fails if or if
% call(G,Val0,ValF) is not satisfiable for all Var0. Assumes keys
% are not repeated.
rb_partial_map(t(Nil,T0), Map, Goal, t(Nil,TF)) :-
partial_map(T0, Map, [], Nil, Goal, TF).
rb_partial_map(t(Nil,T0), Map, Map0, Goal, t(Nil,TF)) :-
partial_map(T0, Map, Map0, Nil, Goal, TF).
partial_map(T,[],[],_,_,T) :- !.
partial_map(black('',_,_,_),Map,Map,Nil,_,Nil) :- !.
partial_map(red(L,K,V,R),Map,MapF,Nil,Goal,red(NL,K,NV,NR)) :-
partial_map(L,Map,MapI,Nil,Goal,NL),
(
MapI == [] ->
NR = R, NV = V, MapF = []
;
MapI = [K1|MapR],
(
K == K1
->
( call(Goal,V,NV) -> true ; NV = V ),
MapN = MapR
;
NV = V,
MapN = MapI
),
partial_map(R,MapN,MapF,Nil,Goal,NR)
).
partial_map(black(L,K,V,R),Map,MapF,Nil,Goal,black(NL,K,NV,NR)) :-
partial_map(L,Map,MapI,Nil,Goal,NL),
(
MapI == [] ->
NR = R, NV = V, MapF = []
;
MapI = [K1|MapR],
(
K == K1
->
( call(Goal,V,NV) -> true ; NV = V ),
MapN = MapR
;
NV = V,
MapN = MapI
),
partial_map(R,MapN,MapF,Nil,Goal,NR)
).
%
% whole keys
%
%% rb_keys(+T, -Keys)
%
% Keys is unified with an ordered list of all keys in the
% Red-Black tree T.
rb_keys(t(_,T),Lf) :-
keys(T,[],Lf).
rb_keys(t(_,T),L0,Lf) :-
keys(T,L0,Lf).
keys(black('',_,_,''),L,L) :- !.
keys(red(L,K,_,R),L0,Lf) :-
keys(L,[K|L1],Lf),
keys(R,L0,L1).
keys(black(L,K,_,R),L0,Lf) :-
keys(L,[K|L1],Lf),
keys(R,L0,L1).
%% list_to_rbtree(+L, -T) is det.
%
% T is the red-black tree corresponding to the mapping in list L.
list_to_rbtree(List, T) :-
sort(List,Sorted),
ord_list_to_rbtree(Sorted, T).
%% ord_list_to_rbtree(+L, -T) is det.
%
% T is the red-black tree corresponding to the mapping in ordered
% list L.
ord_list_to_rbtree([], t(Nil,Nil)) :- !,
Nil = black('', _, _, '').
ord_list_to_rbtree([K-V], t(Nil,black(Nil,K,V,Nil))) :- !,
Nil = black('', _, _, '').
ord_list_to_rbtree(List, t(Nil,Tree)) :-
Nil = black('', _, _, ''),
Ar =.. [seq|List],
functor(Ar,_,L),
Height is truncate(log(L)/log(2)),
construct_rbtree(1, L, Ar, Height, Nil, Tree).
construct_rbtree(L, M, _, _, Nil, Nil) :- M < L, !.
construct_rbtree(L, L, Ar, Depth, Nil, Node) :- !,
arg(L, Ar, K-Val),
build_node(Depth, Nil, K, Val, Nil, Node).
construct_rbtree(I0, Max, Ar, Depth, Nil, Node) :-
I is (I0+Max)//2,
arg(I, Ar, K-Val),
build_node(Depth, Left, K, Val, Right, Node),
I1 is I-1,
NewDepth is Depth-1,
construct_rbtree(I0, I1, Ar, NewDepth, Nil, Left),
I2 is I+1,
construct_rbtree(I2, Max, Ar, NewDepth, Nil, Right).
build_node( 0, Left, K, Val, Right, red(Left, K, Val, Right)) :- !.
build_node( _, Left, K, Val, Right, black(Left, K, Val, Right)).
%% rb_size(+T, -Size) is det.
%
% Size is the number of elements in T.
rb_size(t(_,T),Size) :-
size(T,0,Size).
size(black('',_,_,_),Sz,Sz) :- !.
size(red(L,_,_,R),Sz0,Szf) :-
Sz1 is Sz0+1,
size(L,Sz1,Sz2),
size(R,Sz2,Szf).
size(black(L,_,_,R),Sz0,Szf) :-
Sz1 is Sz0+1,
size(L,Sz1,Sz2),
size(R,Sz2,Szf).
%% is_rbtree(?Term) is semidet.
%
% True if Term is a valide Red-Black tree.
%
% @tbd Catch variables.
is_rbtree(X) :-
var(X), !, fail.
is_rbtree(t(Nil,Nil)) :- !.
is_rbtree(t(_,T)) :-
catch(rbtree1(T), msg(_,_), fail).
is_rbtree(X,_) :-
var(X), !, fail.
is_rbtree(T,Goal) :-
catch(rbtree1(T), msg(S,Args), (once(Goal),format(S,Args))).
%
% This code checks if a tree is ordered and a rbtree
%
%
rbtree(t(_,black('',_,_,''))) :- !.
rbtree(t(_,T)) :-
catch(rbtree1(T),msg(S,Args),format(S,Args)).
rbtree1(black(L,K,_,R)) :-
find_path_blacks(L, 0, Bls),
check_rbtree(L,-inf,K,Bls),
check_rbtree(R,K,+inf,Bls).
rbtree1(red(_,_,_,_)) :-
throw(msg("root should be black",[])).
find_path_blacks(black('',_,_,''), Bls, Bls) :- !.
find_path_blacks(black(L,_,_,_), Bls0, Bls) :-
Bls1 is Bls0+1,
find_path_blacks(L, Bls1, Bls).
find_path_blacks(red(L,_,_,_), Bls0, Bls) :-
find_path_blacks(L, Bls0, Bls).
check_rbtree(black('',_,_,''),Min,Max,Bls0) :- !,
check_height(Bls0,Min,Max).
check_rbtree(red(L,K,_,R),Min,Max,Bls) :-
check_val(K,Min,Max),
check_red_child(L),
check_red_child(R),
check_rbtree(L,Min,K,Bls),
check_rbtree(R,K,Max,Bls).
check_rbtree(black(L,K,_,R),Min,Max,Bls0) :-
check_val(K,Min,Max),
Bls is Bls0-1,
check_rbtree(L,Min,K,Bls),
check_rbtree(R,K,Max,Bls).
check_height(0,_,_) :- !.
check_height(Bls0,Min,Max) :-
throw(msg("Unbalance ~d between ~w and ~w~n",[Bls0,Min,Max])).
check_val(K, Min, Max) :- ( K @> Min ; Min == -inf), (K @< Max ; Max == +inf), !.
check_val(K, Min, Max) :-
throw(msg("not ordered: ~w not between ~w and ~w~n",[K,Min,Max])).
check_red_child(black(_,_,_,_)).
check_red_child(red(_,K,_,_)) :-
throw(msg("must be red: ~w~n",[K])).
%count(1,16,X), format("deleting ~d~n",[X]), new(1,a,T0), insert(T0,2,b,T1), insert(T1,3,c,T2), insert(T2,4,c,T3), insert(T3,5,c,T4), insert(T4,6,c,T5), insert(T5,7,c,T6), insert(T6,8,c,T7), insert(T7,9,c,T8), insert(T8,10,c,T9),insert(T9,11,c,T10), insert(T10,12,c,T11),insert(T11,13,c,T12),insert(T12,14,c,T13),insert(T13,15,c,T14), insert(T14,16,c,T15),delete(T15,X,T16),pretty_print(T16),rbtree(T16),fail.
% count(1,16,X0), X is -X0, format("deleting ~d~n",[X]), new(-1,a,T0), insert(T0,-2,b,T1), insert(T1,-3,c,T2), insert(T2,-4,c,T3), insert(T3,-5,c,T4), insert(T4,-6,c,T5), insert(T5,-7,c,T6), insert(T6,-8,c,T7), insert(T7,-9,c,T8), insert(T8,-10,c,T9),insert(T9,-11,c,T10), insert(T10,-12,c,T11),insert(T11,-13,c,T12),insert(T12,-14,c,T13),insert(T13,-15,c,T14), insert(T14,-16,c,T15),delete(T15,X,T16),pretty_print(T16),rbtree(T16),fail.
count(I,_,I).
count(I,M,L) :-
I < M, I1 is I+1, count(I1,M,L).
test_pos :-
rb_new(1,a,T0),
N = 10000,
build_ptree(2,N,T0,T),
% pretty_print(T),
rbtree(T),
clean_tree(1,N,T,_),
bclean_tree(N,1,T,_),
count(1,N,X), ( rb_delete(T,X,TF) -> true ; abort ),
% pretty_print(TF),
rbtree(TF),
% format("done ~d~n",[X]),
fail.
test_pos.
build_ptree(X,X,T0,TF) :- !,
rb_insert(T0,X,X,TF).
build_ptree(X1,X,T0,TF) :-
rb_insert(T0,X1,X1,TI),
X2 is X1+1,
build_ptree(X2,X,TI,TF).
clean_tree(X,X,T0,TF) :- !,
rb_delete(T0,X,TF),
( rbtree(TF) -> true ; abort).
clean_tree(X1,X,T0,TF) :-
rb_delete(T0,X1,TI),
X2 is X1+1,
( rbtree(TI) -> true ; abort),
clean_tree(X2,X,TI,TF).
bclean_tree(X,X,T0,TF) :- !,
format("cleaning ~d~n", [X]),
rb_delete(T0,X,TF),
( rbtree(TF) -> true ; abort).
bclean_tree(X1,X,T0,TF) :-
format("cleaning ~d~n", [X1]),
rb_delete(T0,X1,TI),
X2 is X1-1,
( rbtree(TI) -> true ; abort),
bclean_tree(X2,X,TI,TF).
test_neg :-
Size = 10000,
rb_new(-1,a,T0),
build_ntree(2,Size,T0,T),
% pretty_print(T),
rbtree(T),
MSize is -Size,
clean_tree(MSize,-1,T,_),
bclean_tree(-1,MSize,T,_),
count(1,Size,X), NX is -X, ( rb_delete(T,NX,TF) -> true ; abort ),
% pretty_print(TF),
rbtree(TF),
% format("done ~d~n",[X]),
fail.
test_neg.
build_ntree(X,X,T0,TF) :- !,
X1 is -X,
rb_insert(T0,X1,X1,TF).
build_ntree(X1,X,T0,TF) :-
NX1 is -X1,
rb_insert(T0,NX1,NX1,TI),
X2 is X1+1,
build_ntree(X2,X,TI,TF).