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yap-6.3/library/rbtrees.yap
Vitor Santos Costa 1a78144190 keys_to_list
2017-07-05 03:36:42 +01:00

1457 lines
38 KiB
Prolog

/**
* @file rbtrees.yap
* @author VITOR SANTOS COSTA <vsc@VITORs-MBP.lan>
* @author Jan Wielemaker
* @date Wed Nov 18 00:11:41 2015
*
* @brief Red-Black trees
*
*
*/
:- 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,
keys_to_rbtree/2,
ord_keys_to_rbtree/2,
is_rbtree/1,
rb_size/2,
rb_in/3
]).
/**
* @defgroup rbtrees Red-Black Trees
* @ingroup library
@{
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)).
*/
%% @pred rb_new(-T) is det.
% create an empty tree.
%
% 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('',_,_,'').
%% @pred rb_empty(?T) is semidet.
%
% Succeeds if T is an empty Red-Black tree.
rb_empty(t(Nil,Nil)) :- Nil = black('',_,_,'').
%% @pred 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).
%% @pred 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).
%% @pred 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).
%% @pred 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)
).
%% @pred 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)
).
%% @pred rb_update(+T, +Key, +NewVal, -TN) is semidet.
%% @pred 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)
).
%% @pred rb_rewrite(+T, +Key, +NewVal) is semidet.
%% @pred 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)
).
%% @pred 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.
keys_to_rbtree(List, T) :-
sort(List,Sorted),
ord_keys_to_rbtree(Sorted, T).
%% list_to_rbtree(+L, -T) is det.
%
% T is the red-black tree corresponding to the mapping in list L.
ord_keys_to_rbtree(List, T) :-
maplist(paux, List, Sorted),
ord_list_to_rbtree(Sorted, T).
paux(K, K-_).
%% 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 valid 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).
/** @pred rb_apply(+ _T_,+ _Key_,+ _G_,- _TN_)
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.
*/
/** @pred rb_clone(+ _T_,+ _NT_,+ _Nodes_)
=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_.
*/
/** @pred 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_.
*/
/** @pred 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_.
*/
/** @pred rb_delete(+ _T_,+ _Key_,- _TN_)
Delete element with key _Key_ from the tree _T_, returning a new
tree _TN_.
*/
/** @pred 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_.
*/
/** @pred rb_empty(? _T_)
Succeeds if tree _T_ is empty.
*/
/** @pred rb_fold(+ _T_,+ _G_,+ _Acc0_, - _AccF_)
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.
*/
/** @pred rb_insert(+ _T0_,+ _Key_,? _Value_,+ _TF_)
Add an element with key _Key_ and _Value_ to the tree
_T0_ creating a new red-black tree _TF_. Duplicated elements are not
allowed.
Add a new element with key _Key_ and _Value_ to the tree
_T0_ creating a new red-black tree _TF_. Fails is an element
with _Key_ exists in the tree.
*/
/** @pred rb_key_fold(+ _T_,+ _G_,+ _Acc0_, - _AccF_)
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,KeyL,VL,_,Acc0)` must hold, and if
_VR_ is the value of the next node in inorder,
`call(G,KeyR,VR,Acc1,_)` must hold.
*/
/** @pred rb_keys(+ _T_,+ _Keys_)
_Keys_ is an infix visit with all keys in tree _T_. Keys will be
sorted, but may be duplicate.
*/
/** @pred rb_lookup(+ _Key_,- _Value_,+ _T_)
Backtrack through all elements with key _Key_ in the red-black tree
_T_, returning for each the value _Value_.
*/
/** @pred rb_lookupall(+ _Key_,- _Value_,+ _T_)
Lookup all elements with key _Key_ in the red-black tree
_T_, returning the value _Value_.
*/
/** @pred rb_map(+ _T_,+ _G_,- _TN_)
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 or if `call(G,Val0,ValF)` is not
satisfiable for all _Var0_.
*/
/** @pred rb_max(+ _T_,- _Key_,- _Value_)
_Key_ is the maximal key in _T_, and is associated with _Val_.
*/
/** @pred rb_min(+ _T_,- _Key_,- _Value_)
_Key_ is the minimum key in _T_, and is associated with _Val_.
*/
/** @pred rb_new(? _T_)
Create a new tree.
*/
/** @pred rb_next(+ _T_, + _Key_,- _Next_,- _Value_)
_Next_ is the next element after _Key_ in _T_, and is
associated with _Val_.
*/
/** @pred 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.
*/
/** @pred rb_previous(+ _T_, + _Key_,- _Previous_,- _Value_)
_Previous_ is the previous element after _Key_ in _T_, and is
associated with _Val_.
*/
/** @pred rb_size(+ _T_,- _Size_)
_Size_ is the number of elements in _T_.
*/
/** @pred rb_update(+ _T_,+ _Key_,+ _NewVal_,- _TN_)
Tree _TN_ is tree _T_, but with value for _Key_ associated
with _NewVal_. Fails if it cannot find _Key_ in _T_.
*/
/** @pred rb_visit(+ _T_,- _Pairs_)
_Pairs_ is an infix visit of tree _T_, where each element of
_Pairs_ is of the form _K_- _Val_.
*/
%%! @}