summaryrefslogtreecommitdiff
path: root/lib/stdlib/src/gb_trees.erl
blob: c0cdde012efa320b79ec9a76f56e7ec4051bb0a4 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
%% Licensed under the Apache License, Version 2.0 (the "License");
%% you may not use this file except in compliance with the License.
%% You may obtain a copy of the License at
%%
%%     http://www.apache.org/licenses/LICENSE-2.0
%%
%% Unless required by applicable law or agreed to in writing, software
%% distributed under the License is distributed on an "AS IS" BASIS,
%% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
%% See the License for the specific language governing permissions and
%% limitations under the License.
%%
%% =====================================================================
%% General Balanced Trees - highly efficient dictionaries.
%%
%% Copyright (C) 1999-2001 Sven-Olof Nyström, Richard Carlsson
%%
%% An efficient implementation of Prof. Arne Andersson's General
%% Balanced Trees. These have no storage overhead compared to plain
%% unbalanced binary trees, and their performance is in general better
%% than AVL trees.
%% ---------------------------------------------------------------------
%% Operations:
%%
%% - empty(): returns empty tree.
%%
%% - is_empty(T): returns 'true' if T is an empty tree, and 'false'
%%   otherwise.
%%
%% - size(T): returns the number of nodes in the tree as an integer.
%%   Returns 0 (zero) if the tree is empty.
%%
%% - lookup(X, T): looks up key X in tree T; returns {value, V}, or
%%   `none' if the key is not present.
%%
%% - get(X, T): retreives the value stored with key X in tree T. Assumes
%%   that the key is present in the tree.
%%
%% - insert(X, V, T): inserts key X with value V into tree T; returns
%%   the new tree. Assumes that the key is *not* present in the tree.
%%
%% - update(X, V, T): updates key X to value V in tree T; returns the
%%   new tree. Assumes that the key is present in the tree.
%%
%% - enter(X, V, T): inserts key X with value V into tree T if the key
%%   is not present in the tree, otherwise updates key X to value V in
%%   T. Returns the new tree.
%%
%% - delete(X, T): removes key X from tree T; returns new tree. Assumes
%%   that the key is present in the tree.
%%
%% - delete_any(X, T): removes key X from tree T if the key is present
%%   in the tree, otherwise does nothing; returns new tree.
%%
%% - take(X, T): removes element with key X from tree T; returns new tree
%%   without removed element. Assumes that the key is present in the tree.
%%
%% - take_any(X, T): removes element with key X from tree T and returns
%%   a new tree if the key is present; otherwise does nothing and returns
%%   'error'.
%%
%% - balance(T): rebalances tree T. Note that this is rarely necessary,
%%   but may be motivated when a large number of entries have been
%%   deleted from the tree without further insertions. Rebalancing could
%%   then be forced in order to minimise lookup times, since deletion
%%   only does not rebalance the tree.
%%
%% - is_defined(X, T): returns `true' if key X is present in tree T, and
%%   `false' otherwise.
%%
%% - keys(T): returns an ordered list of all keys in tree T.
%%
%% - values(T): returns the list of values for all keys in tree T,
%%   sorted by their corresponding keys. Duplicates are not removed.
%%
%% - to_list(T): returns an ordered list of {Key, Value} pairs for all
%%   keys in tree T.
%%
%% - from_orddict(L): turns an ordered list L of {Key, Value} pairs into
%%   a tree. The list must not contain duplicate keys.
%%
%% - smallest(T): returns {X, V}, where X is the smallest key in tree T,
%%   and V is the value associated with X in T. Assumes that the tree T
%%   is nonempty.
%%
%% - largest(T): returns {X, V}, where X is the largest key in tree T,
%%   and V is the value associated with X in T. Assumes that the tree T
%%   is nonempty.
%%
%% - take_smallest(T): returns {X, V, T1}, where X is the smallest key
%%   in tree T, V is the value associated with X in T, and T1 is the
%%   tree T with key X deleted. Assumes that the tree T is nonempty.
%%
%% - take_largest(T): returns {X, V, T1}, where X is the largest key
%%   in tree T, V is the value associated with X in T, and T1 is the
%%   tree T with key X deleted. Assumes that the tree T is nonempty.
%%
%% - iterator(T): returns an iterator that can be used for traversing
%%   the entries of tree T; see `next'. The implementation of this is
%%   very efficient; traversing the whole tree using `next' is only
%%   slightly slower than getting the list of all elements using
%%   `to_list' and traversing that. The main advantage of the iterator
%%   approach is that it does not require the complete list of all
%%   elements to be built in memory at one time.
%%
%% - iterator_from(K, T): returns an iterator that can be used for
%%   traversing the entries of tree T with key greater than or
%%   equal to K; see `next'.
%%
%% - next(S): returns {X, V, S1} where X is the smallest key referred to
%%   by the iterator S, and S1 is the new iterator to be used for
%%   traversing the remaining entries, or the atom `none' if no entries
%%   remain.
%%
%% - map(F, T): maps the function F(K, V) -> V' to all key-value pairs
%%   of the tree T and returns a new tree T' with the same set of keys
%%   as T and the new set of values V'.

-module(gb_trees).

-export([empty/0, is_empty/1, size/1, lookup/2, get/2, insert/3,
	 update/3, enter/3, delete/2, delete_any/2, balance/1,
	 is_defined/2, keys/1, values/1, to_list/1, from_orddict/1,
	 smallest/1, largest/1, take/2, take_any/2,
         take_smallest/1, take_largest/1,
	 iterator/1, iterator_from/2, next/1, map/2]).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Data structure:
%% - {Size, Tree}, where `Tree' is composed of nodes of the form:
%%   - {Key, Value, Smaller, Bigger}, and the "empty tree" node:
%%   - nil.
%%
%% I make no attempt to balance trees after deletions. Since deletions
%% don't increase the height of a tree, I figure this is OK.
%%
%% Original balance condition h(T) <= ceil(c * log(|T|)) has been
%% changed to the similar (but not quite equivalent) condition 2 ^ h(T)
%% <= |T| ^ c. I figure this should also be OK.
%%
%% Performance is comparable to the AVL trees in the Erlang book (and
%% faster in general due to less overhead); the difference is that
%% deletion works for my trees, but not for the book's trees. Behaviour
%% is logaritmic (as it should be).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Some macros.

-define(p, 2). % It seems that p = 2 is optimal for sorted keys

-define(pow(A, _), A * A). % correct with exponent as defined above.

-define(div2(X), X bsr 1). 

-define(mul2(X), X bsl 1).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Some types.

-export_type([tree/0, tree/2, iter/0, iter/2]).

-type gb_tree_node(K, V) :: 'nil'
                          | {K, V, gb_tree_node(K, V), gb_tree_node(K, V)}.
-opaque tree(Key, Value) :: {non_neg_integer(), gb_tree_node(Key, Value)}.
-type tree() :: tree(_, _).
-opaque iter(Key, Value) :: [gb_tree_node(Key, Value)].
-type iter() :: iter(_, _).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec empty() -> tree().

empty() ->
    {0, nil}.

-spec is_empty(Tree) -> boolean() when
      Tree :: tree().

is_empty({0, nil}) ->
    true;
is_empty(_) ->
    false.

-spec size(Tree) -> non_neg_integer() when
      Tree :: tree().

size({Size, _}) when is_integer(Size), Size >= 0 ->
    Size.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec lookup(Key, Tree) -> 'none' | {'value', Value} when
      Tree :: tree(Key, Value).

lookup(Key, {_, T}) ->
    lookup_1(Key, T).

%% The term order is an arithmetic total order, so we should not
%% test exact equality for the keys. (If we do, then it becomes
%% possible that neither `>', `<', nor `=:=' matches.) Testing '<'
%% and '>' first is statistically better than testing for
%% equality, and also allows us to skip the test completely in the
%% remaining case.

lookup_1(Key, {Key1, _, Smaller, _}) when Key < Key1 ->
    lookup_1(Key, Smaller);
lookup_1(Key, {Key1, _, _, Bigger}) when Key > Key1 ->
    lookup_1(Key, Bigger);
lookup_1(_, {_, Value, _, _}) ->
    {value, Value};
lookup_1(_, nil) ->
    none.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% This is a specialized version of `lookup'.

-spec is_defined(Key, Tree) -> boolean() when
      Tree :: tree(Key, Value :: term()).

is_defined(Key, {_, T}) ->
    is_defined_1(Key, T).

is_defined_1(Key, {Key1, _, Smaller, _}) when Key < Key1 ->
    is_defined_1(Key, Smaller);
is_defined_1(Key, {Key1, _, _, Bigger}) when Key > Key1 ->
    is_defined_1(Key, Bigger);
is_defined_1(_, {_, _, _, _}) ->
    true;
is_defined_1(_, nil) ->
    false.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% This is a specialized version of `lookup'.

-spec get(Key, Tree) -> Value when
      Tree :: tree(Key, Value).

get(Key, {_, T}) ->
    get_1(Key, T).

get_1(Key, {Key1, _, Smaller, _}) when Key < Key1 ->
    get_1(Key, Smaller);
get_1(Key, {Key1, _, _, Bigger}) when Key > Key1 ->
    get_1(Key, Bigger);
get_1(_, {_, Value, _, _}) ->
    Value.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec update(Key, Value, Tree1) -> Tree2 when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

update(Key, Val, {S, T}) ->
    T1 = update_1(Key, Val, T),
    {S, T1}.

%% See `lookup' for notes on the term comparison order.

update_1(Key, Value, {Key1, V, Smaller, Bigger}) when Key < Key1 -> 
    {Key1, V, update_1(Key, Value, Smaller), Bigger};
update_1(Key, Value, {Key1, V, Smaller, Bigger}) when Key > Key1 ->
    {Key1, V, Smaller, update_1(Key, Value, Bigger)};
update_1(Key, Value, {_, _, Smaller, Bigger}) ->
    {Key, Value, Smaller, Bigger}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec insert(Key, Value, Tree1) -> Tree2 when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

insert(Key, Val, {S, T}) when is_integer(S) ->
    S1 = S+1,
    {S1, insert_1(Key, Val, T, ?pow(S1, ?p))}.

insert_1(Key, Value, {Key1, V, Smaller, Bigger}, S) when Key < Key1 -> 
    case insert_1(Key, Value, Smaller, ?div2(S)) of
	{T1, H1, S1} ->
	    T = {Key1, V, T1, Bigger},
	    {H2, S2} = count(Bigger),
	    H = ?mul2(erlang:max(H1, H2)),
	    SS = S1 + S2 + 1,
	    P = ?pow(SS, ?p),
	    if
		H > P -> 
		    balance(T, SS);
		true ->
		    {T, H, SS}
	    end;
	T1 ->
	    {Key1, V, T1, Bigger}
    end;
insert_1(Key, Value, {Key1, V, Smaller, Bigger}, S) when Key > Key1 -> 
    case insert_1(Key, Value, Bigger, ?div2(S)) of
	{T1, H1, S1} ->
	    T = {Key1, V, Smaller, T1},
	    {H2, S2} = count(Smaller),
	    H = ?mul2(erlang:max(H1, H2)),
	    SS = S1 + S2 + 1,
	    P = ?pow(SS, ?p),
	    if
		H > P -> 
		    balance(T, SS);
		true ->
		    {T, H, SS}
	    end;
	T1 ->
	    {Key1, V, Smaller, T1}
    end;
insert_1(Key, Value, nil, S) when S =:= 0 ->
    {{Key, Value, nil, nil}, 1, 1};
insert_1(Key, Value, nil, _S) ->
    {Key, Value, nil, nil};
insert_1(Key, _, _, _) ->
    erlang:error({key_exists, Key}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec enter(Key, Value, Tree1) -> Tree2 when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

enter(Key, Val, T) ->
    case is_defined(Key, T) of
	true ->
	    update(Key, Val, T);
	false ->
	    insert(Key, Val, T)
    end.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

count({_, _, nil, nil}) ->
    {1, 1};
count({_, _, Sm, Bi}) ->
    {H1, S1} = count(Sm),
    {H2, S2} = count(Bi),
    {?mul2(erlang:max(H1, H2)), S1 + S2 + 1};
count(nil) ->
    {1, 0}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec balance(Tree1) -> Tree2 when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

balance({S, T}) ->
    {S, balance(T, S)}.

balance(T, S) ->
    balance_list(to_list_1(T), S).

balance_list(L, S) ->
    {T, []} = balance_list_1(L, S),
    T.

balance_list_1(L, S) when S > 1 ->
    Sm = S - 1,
    S2 = Sm div 2,
    S1 = Sm - S2,
    {T1, [{K, V} | L1]} = balance_list_1(L, S1),
    {T2, L2} = balance_list_1(L1, S2),
    T = {K, V, T1, T2},
    {T, L2};
balance_list_1([{Key, Val} | L], 1) ->
    {{Key, Val, nil, nil}, L};
balance_list_1(L, 0) ->
    {nil, L}.

-spec from_orddict(List) -> Tree when
      List :: [{Key, Value}],
      Tree :: tree(Key, Value).

from_orddict(L) ->
    S = length(L),
    {S, balance_list(L, S)}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec delete_any(Key, Tree1) -> Tree2 when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

delete_any(Key, T) ->
    case is_defined(Key, T) of
	true ->
	    delete(Key, T);
	false ->
	    T
    end.

%%% delete. Assumes that key is present.

-spec delete(Key, Tree1) -> Tree2 when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

delete(Key, {S, T}) when is_integer(S), S >= 0 ->
    {S - 1, delete_1(Key, T)}.

%% See `lookup' for notes on the term comparison order.

delete_1(Key, {Key1, Value, Smaller, Larger}) when Key < Key1 ->
    Smaller1 = delete_1(Key, Smaller),
    {Key1, Value, Smaller1, Larger};
delete_1(Key, {Key1, Value, Smaller, Bigger}) when Key > Key1 ->
    Bigger1 = delete_1(Key, Bigger),
    {Key1, Value, Smaller, Bigger1};
delete_1(_, {_, _, Smaller, Larger}) ->
    merge(Smaller, Larger).

merge(Smaller, nil) ->
    Smaller;
merge(nil, Larger) ->
    Larger;
merge(Smaller, Larger) ->
    {Key, Value, Larger1} = take_smallest1(Larger),
    {Key, Value, Smaller, Larger1}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec take_any(Key, Tree1) -> {Value, Tree2} | 'error' when
      Tree1 :: tree(Key, _),
      Tree2 :: tree(Key, _),
      Key   :: term(),
      Value :: term().

take_any(Key, Tree) ->
    case is_defined(Key, Tree) of
        true -> take(Key, Tree);
        false -> error
    end.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec take(Key, Tree1) -> {Value, Tree2} when
      Tree1 :: tree(Key, _),
      Tree2 :: tree(Key, _),
      Key   :: term(),
      Value :: term().

take(Key, {S, T}) when is_integer(S), S >= 0 ->
    {Value, Res} = take_1(Key, T),
    {Value, {S - 1, Res}}.

take_1(Key, {Key1, Value, Smaller, Larger}) when Key < Key1 ->
    {Value2, Smaller1} = take_1(Key, Smaller),
    {Value2, {Key1, Value, Smaller1, Larger}};
take_1(Key, {Key1, Value, Smaller, Bigger}) when Key > Key1 ->
    {Value2, Bigger1} = take_1(Key, Bigger),
    {Value2, {Key1, Value, Smaller, Bigger1}};
take_1(_, {_Key, Value, Smaller, Larger}) ->
    {Value, merge(Smaller, Larger)}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec take_smallest(Tree1) -> {Key, Value, Tree2} when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

take_smallest({Size, Tree}) when is_integer(Size), Size >= 0 ->
    {Key, Value, Larger} = take_smallest1(Tree),
    {Key, Value, {Size - 1, Larger}}.

take_smallest1({Key, Value, nil, Larger}) ->
    {Key, Value, Larger};
take_smallest1({Key, Value, Smaller, Larger}) ->
    {Key1, Value1, Smaller1} = take_smallest1(Smaller),
    {Key1, Value1, {Key, Value, Smaller1, Larger}}.

-spec smallest(Tree) -> {Key, Value} when
      Tree :: tree(Key, Value).

smallest({_, Tree}) ->
    smallest_1(Tree).

smallest_1({Key, Value, nil, _Larger}) ->
    {Key, Value};
smallest_1({_Key, _Value, Smaller, _Larger}) ->
    smallest_1(Smaller).

-spec take_largest(Tree1) -> {Key, Value, Tree2} when
      Tree1 :: tree(Key, Value),
      Tree2 :: tree(Key, Value).

take_largest({Size, Tree}) when is_integer(Size), Size >= 0 ->
    {Key, Value, Smaller} = take_largest1(Tree),
    {Key, Value, {Size - 1, Smaller}}.

take_largest1({Key, Value, Smaller, nil}) ->
    {Key, Value, Smaller};
take_largest1({Key, Value, Smaller, Larger}) ->
    {Key1, Value1, Larger1} = take_largest1(Larger),
    {Key1, Value1, {Key, Value, Smaller, Larger1}}.

-spec largest(Tree) -> {Key, Value} when
      Tree :: tree(Key, Value).

largest({_, Tree}) ->
    largest_1(Tree).

largest_1({Key, Value, _Smaller, nil}) ->
    {Key, Value};
largest_1({_Key, _Value, _Smaller, Larger}) ->
    largest_1(Larger).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec to_list(Tree) -> [{Key, Value}] when
      Tree :: tree(Key, Value).
			   
to_list({_, T}) ->
    to_list(T, []).

to_list_1(T) -> to_list(T, []).

to_list({Key, Value, Small, Big}, L) ->
    to_list(Small, [{Key, Value} | to_list(Big, L)]);
to_list(nil, L) -> L.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec keys(Tree) -> [Key] when
      Tree :: tree(Key, Value :: term()).

keys({_, T}) ->
    keys(T, []).

keys({Key, _Value, Small, Big}, L) ->
    keys(Small, [Key | keys(Big, L)]);
keys(nil, L) -> L.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec values(Tree) -> [Value] when
      Tree :: tree(Key :: term(), Value).

values({_, T}) ->
    values(T, []).

values({_Key, Value, Small, Big}, L) ->
    values(Small, [Value | values(Big, L)]);
values(nil, L) -> L.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec iterator(Tree) -> Iter when
      Tree :: tree(Key, Value),
      Iter :: iter(Key, Value).

iterator({_, T}) ->
    iterator_1(T).

iterator_1(T) ->
    iterator(T, []).

%% The iterator structure is really just a list corresponding to
%% the call stack of an in-order traversal. This is quite fast.

iterator({_, _, nil, _} = T, As) ->
    [T | As];
iterator({_, _, L, _} = T, As) ->
    iterator(L, [T | As]);
iterator(nil, As) ->
    As.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec iterator_from(Key, Tree) -> Iter when
      Tree :: tree(Key, Value),
      Iter :: iter(Key, Value).

iterator_from(S, {_, T}) ->
    iterator_1_from(S, T).

iterator_1_from(S, T) ->
    iterator_from(S, T, []).

iterator_from(S, {K, _, _, T}, As) when K < S ->
    iterator_from(S, T, As);
iterator_from(_, {_, _, nil, _} = T, As) ->
    [T | As];
iterator_from(S, {_, _, L, _} = T, As) ->
    iterator_from(S, L, [T | As]);
iterator_from(_, nil, As) ->
    As.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec next(Iter1) -> 'none' | {Key, Value, Iter2} when
      Iter1 :: iter(Key, Value),
      Iter2 :: iter(Key, Value).

next([{X, V, _, T} | As]) ->
    {X, V, iterator(T, As)};
next([]) ->
    none.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-spec map(Function, Tree1) -> Tree2 when
      Function :: fun((K :: Key, V1 :: Value1) -> V2 :: Value2),
      Tree1 :: tree(Key, Value1),
      Tree2 :: tree(Key, Value2).

map(F, {Size, Tree}) when is_function(F, 2) ->
    {Size, map_1(F, Tree)}.

map_1(_, nil) -> nil;
map_1(F, {K, V, Smaller, Larger}) ->
    {K, F(K, V), map_1(F, Smaller), map_1(F, Larger)}.