跳转至

Size Balanced Tree

Size Balanced Tree (SBT) 是由中国 OI 选手陈启峰在 2007 年提出的一种自平衡二叉搜索树 (Self-Balanced Binary Search Tree, SBBST), 通过检查子树的节点数量进行自身的平衡维护。相比于红黑树,AVL 等主流自平衡二叉搜索树而言,Size Balanced Tree 支持在 \(O(\log n)\) 的时间复杂度内查询某个键值在树中的排名 (rank).

节点定义

相比与普通二叉搜索树,SBT 的每个节点 \(N\) 仅需要多维护一个整数字段 size, 用于储存以 \(N\) 为根的子树中节点的个数。节点类型 Node 的具体定义如下:

Identifier Type Description
left Node* 左子节点引用
right Node* 右子节点引用
size int 以该节点为根的子树中节点的个数

性质

Size Balanced Tree 中任意节点 \(N\) 满足如下几条性质:

Text Only
1
2
3
4
size(N.left) >= size(N.right.left)
size(N.left) >= size(N.right.right)
size(N.right) >= size(N.left.left)
size(N.right) >= size(N.left.right)

使用自然语言可描述为:任意节点的 size 不小于其兄弟节点(Sibling)的所有子节点(Nephew)的 size.

平衡维护

旋转

SBT 主要通过旋转操作改变自身高度从而进行平衡维护。其旋转操作与绝大部分自平衡二叉搜索树类似,唯一区别在于在完成旋转之后需要对旋转过程中左右子节点发生改变的节点更新 size. 示例代码如下:

C++
 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
inline void updateSize() {
  USize leftSize = this->left != nullptr ? this->left->size : 0;
  USize rightSize = this->right != nullptr ? this->right->size : 0;
  this->size = leftSize + rightSize + 1;
}

static void rotateLeft(NodePtr& node) {
  assert(node != nullptr);
  // clang-format off
  //     |                       |
  //     N                       S
  //    / \     l-rotate(N)     / \
  //   L   S    ==========>    N   R
  //      / \                 / \
  //     M   R               L   M
  // clang-format on
  NodePtr successor = node->right;
  node->right = successor->left;
  successor->left = node;

  node->updateSize();
  successor->updateSize();

  node = successor;
}

static void rotateRight(NodePtr& node) {
  assert(node != nullptr);
  // clang-format off
  //       |                   |
  //       N                   S
  //      / \   r-rotate(N)   / \
  //     S   R  ==========>  L   N
  //    / \                     / \
  //   L   M                   M   R
  // clang-format on
  NodePtr successor = node->left;
  node->left = successor->right;
  successor->right = node;

  node->updateSize();
  successor->updateSize();

  node = successor;
}

维护

Case 1

size(N.left) < size(N.right.left)

C++
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
if (size(node->right->left) > size(node->left)) {
  // clang-format off
  //     |                     |                      |
  //     N                     N                     [M]
  //    / \    r-rotate(R)    / \     l-rotate(N)    / \
  //  <L>  R   ==========>  <L> [M]   ==========>   N   R
  //      /                       \                /
  //    [M]                        R             <L>
  // clang-format on
  rotateRight(node->right);
  rotateLeft(node);
  fixBalance(node->left);
  fixBalance(node->right);
  fixBalance(node);
  return;
}

Case 2

size(N.left) < size(N.right.right)

C++
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
if (size(node->right->right) > size(node->left)) {
  // clang-format off
  //     |                       |
  //     N                       R
  //    / \     l-rotate(N)     / \
  //  <L>  R    ==========>    N  [M]
  //        \                 /
  //        [M]             <L>
  // clang-format on
  rotateLeft(node);
  fixBalance(node->left);
  fixBalance(node);
  return;
}

Case 3

size(N.right) < size(N.left.left)

C++
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
if (size(node->left->left) > size(node->right)) {
  // clang-format off
  //       |                       |
  //       N                       L
  //      / \     r-rotate(N)     / \
  //     L  <R>   ==========>   [M]  N
  //    /                             \
  //  [M]                             <R>
  // clang-format on
  rotateRight(node);
  fixBalance(node->right);
  fixBalance(node);
  return;
}

Case 4

size(N.right) < size(N.left.right)

C++
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
if (size(node->left->right) > size(node->right)) {
  // clang-format off
  //     |                     |                      |
  //     N                     N                     [M]
  //    / \    l-rotate(L)    / \     r-rotate(N)    / \
  //   L  <R>  ==========>  [M] <R>   ==========>   L   N
  //    \                   /                            \
  //    [M]                L                             <R>
  // clang-format on
  rotateLeft(node->left);
  rotateRight(node);
  fixBalance(node->left);
  fixBalance(node->right);
  fixBalance(node);
  return;
}

操作

插入

SBT 的插入操作需要在完成普通二叉搜索树的插入操作的基础上递归地进行节点 size 字段的更新及平衡维护。示例代码如下:

C++
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
if (compare(key, node->key)) {
  /* key < node->key */
  if (node->left == nullptr) {
    node->left = Node::from(key, value);
    node->updateSize();
  } else {
    insert(node->left, key, value, replace);
    node->updateSize();
    fixBalance(node);
  }
} else {
  /* key > node->key */
  if (node->right == nullptr) {
    node->right = Node::from(key, value);
    node->updateSize();
  } else {
    insert(node->right, key, value, replace);
    node->updateSize();
    fixBalance(node);
  }
}

删除

根据 Size Balanced Tree 的提出者陈启峰在其论文中对于删除操作的描述:

It can result in a destroyed SBT. But with the insertion above, a BST is still kept at the height of \(O(\log n)\) where \(n\) is the total number of insertions, not the current size.

删除操作虽然有可能使得 SBT 的性质被打破,但并不会使树的高度增高,因此不会影响后续操作的效率。但在实际情况下,如果在一次批量插入操作后只进行大量的删除和查询操作,依然有可能由于树的失衡影响整体效率,因此本文在实现 SBT 的删除操作时依然选择加入平衡维护。参考代码如下:

C++
  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
bool remove(NodePtr& node, K key, NodeConsumer action) {
  assert(node != nullptr);

  if (key != node->key) {
    if (compare(key, node->key)) {
      /* key < node->key */
      NodePtr& left = node->left;
      if (left != nullptr && remove(left, key, action)) {
        node->updateSize();
        fixBalance(node);
        return true;
      } else {
        return false;
      }
    } else {
      /* key > node->key */
      NodePtr& right = node->right;
      if (right != nullptr && remove(right, key, action)) {
        node->updateSize();
        fixBalance(node);
        return true;
      } else {
        return false;
      }
    }
  }

  assert(key == node->key);
  action(node);

  if (node->isLeaf()) {
    // Case 1: no child
    node = nullptr;
  } else if (node->right == nullptr) {
    // Case 2: left child only
    // clang-format off
    //     P
    //     |  remove(N)  P
    //     N  ========>  |
    //    /              L
    //   L
    // clang-format on
    node = node->left;
  } else if (node->left == nullptr) {
    // Case 3: right child only
    // clang-format off
    //   P
    //   |    remove(N)  P
    //   N    ========>  |
    //    \              R
    //     R
    // clang-format on
    node = node->right;
  } else if (node->right->left == nullptr) {
    // Case 4: both left and right child, right child has no left child
    // clang-format off
    //    |                 |
    //    N    remove(N)    R
    //   / \   ========>   /
    //  L   R             L
    // clang-format on
    NodePtr right = node->right;
    swapNode(node, right);
    right->right = node->right;
    node = right;
    node->updateSize();
    fixBalance(node);
  } else {
    // Case 5: both left and right child, right child is not a leaf
    // clang-format off
    //   Step 1. find the node N with the smallest key
    //           and its parent P on the right subtree
    //   Step 2. swap S and N
    //   Step 3. remove node N like Case 1 or Case 3
    //   Step 4. update size for all nodes on the path
    //           from S to P
    //     |                  |
    //     N                  S                 |
    //    / \                / \                S
    //   L  ..  swap(N, S)  L  ..  remove(N)   / \
    //       |  =========>      |  ========>  L  ..
    //       P                  P                 |
    //      / \                / \                P
    //     S  ..              N  ..              / \
    //      \                  \                R  ..
    //       R                  R
    //
    // clang-format on

    std::stack<NodePtr> path;

    // Step 1
    NodePtr successor = node->right;
    NodePtr parent = node;
    path.push(node);

    while (successor->left != nullptr) {
      path.push(successor);
      parent = successor;
      successor = parent->left;
    }

    // Step 2
    swapNode(node, successor);

    // Step 3
    parent->left = node->right;
    // Restore node
    node = successor;

    // Step 4
    while (!path.empty()) {
      path.top()->updateSize();
      path.pop();
    }
  }

  return true;
}

值得注意的是,在上述代码的 Case 5 中使用后继节点 \(S\)(也可以选择前驱节点)替换待删除节点 \(N\) 并删除替换后的 \(N\) 以后,需要更新替换前 \(S\) 节点的父节点 \(P\) 到替换后的 \(S\) 节点这条路径(如代码中注释所示)上的所有节点的 size 字段。本文的实现选择使用栈依次记录路径上的节点,最后再按遍历的相反顺序出栈进行更新。

查询排名

由于 SBT 节点中储存了子树节点个数的信息,因此可以在 \(O(\log n)\) 的时间复杂度下查询某个 key 的排名(或者大于/小于某个 key 的节点个数)。示例代码如下:

C++
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
USize countLess(ConstNodePtr node, K key, bool countEqual = false) const {
  if (node == nullptr) {
    return 0;
  } else if (key < node->key) {
    return countLess(node->left, key, countEqual);
  } else if (key > node->key) {
    return size(node->left) + 1 + countLess(node->right, key, countEqual);
  } else {
    return size(node->left) + (countEqual ? 1 : 0);
  }
}

USize countGreater(ConstNodePtr node, K key, bool countEqual = false) const {
  if (node == nullptr) {
    return 0;
  } else if (key < node->key) {
    return size(node->right) + 1 + countGreater(node->left, key, countEqual);
  } else if (key > node->key) {
    return countGreater(node->right, key, countEqual);
  } else {
    return size(node->right) + (countEqual ? 1 : 0);
  }
}

参考代码

下面的代码是用 SBT 实现的 Map,即有序不可重映射:

完整代码
C++
  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
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
/**
 * @file SizeBalancedTreeMap.hpp
 * @brief An SizeBalancedTree-based map implementation
 * @details The map is sorted according to the natural ordering of its
 *  keys or by a {@code Compare} function provided; This implementation
 *  provides guaranteed log(n) time cost for the contains, get, insert
 *  and remove operations.
 * @author [r.ivance](https://github.com/RIvance)
 */

#ifndef SIZE_BALANCED_TREE_MAP_HPP
#define SIZE_BALANCED_TREE_MAP_HPP

#include <cassert>
#include <cstddef>
#include <cstdint>
#include <functional>
#include <memory>
#include <stack>
#include <utility>
#include <vector>

/**
 * An SizeBalancedTree-based map implementation
 * http://wcipeg.com/wiki/Size_Balanced_Tree
 * @tparam Key the type of keys maintained by this map
 * @tparam Value the type of mapped values
 * @tparam Compare the compare function
 */
template <typename Key, typename Value, typename Compare = std::less<Key> >
class SizeBalancedTreeMap {
 private:
  using USize = size_t;

  Compare compare = Compare();

 public:
  struct Entry {
    Key key;
    Value value;

    bool operator==(const Entry &rhs) const noexcept {
      return this->key == rhs.key && this->value == rhs.value;
    }

    bool operator!=(const Entry &rhs) const noexcept {
      return this->key != rhs.key || this->value != rhs.value;
    }
  };

 private:
  struct Node {
    using Ptr = std::shared_ptr<Node>;
    using Provider = const std::function<Ptr(void)> &;
    using Consumer = const std::function<void(const Ptr &)> &;

    Key key;
    Value value{};

    Ptr left = nullptr;
    Ptr right = nullptr;

    USize size = 1;

    explicit Node(Key k) : key(std::move(k)) {}

    explicit Node(Key k, Value v) : key(std::move(k)), value(std::move(v)) {}

    ~Node() = default;

    inline bool isLeaf() const noexcept {
      return this->left == nullptr && this->right == nullptr;
    }

    inline void updateSize() {
      USize leftSize = this->left != nullptr ? this->left->size : 0;
      USize rightSize = this->right != nullptr ? this->right->size : 0;
      this->size = leftSize + rightSize + 1;
    }

    inline Entry entry() const { return Entry{key, value}; }

    static Ptr from(const Key &k) { return std::make_shared<Node>(Node(k)); }

    static Ptr from(const Key &k, const Value &v) {
      return std::make_shared<Node>(Node(k, v));
    }
  };

  using NodePtr = typename Node::Ptr;
  using ConstNodePtr = const NodePtr &;
  using NodeProvider = typename Node::Provider;
  using NodeConsumer = typename Node::Consumer;

  NodePtr root = nullptr;

  using K = const Key &;
  using V = const Value &;

 public:
  using EntryList = std::vector<Entry>;
  using KeyValueConsumer = const std::function<void(K, V)> &;
  using MutKeyValueConsumer = const std::function<void(K, Value &)> &;
  using KeyValueFilter = const std::function<bool(K, V)> &;

  class NoSuchMappingException : protected std::exception {
   private:
    const char *message;

   public:
    explicit NoSuchMappingException(const char *msg) : message(msg) {}

    const char *what() const noexcept override { return message; }
  };

  SizeBalancedTreeMap() noexcept = default;

  /**
   * Returns the number of entries in this map.
   * @return size_t
   */
  inline USize size() const noexcept {
    if (this->root != nullptr) {
      return this->root->size;
    } else {
      return 0;
    }
  }

  /**
   * Returns true if this collection contains no elements.
   * @return bool
   */
  inline bool empty() const noexcept { return this->root == nullptr; }

  /**
   * Removes all of the elements from this map.
   */
  void clear() noexcept { this->root = nullptr; }

  /**
   * Returns the value to which the specified key is mapped; If this map
   * contains no mapping for the key, a {@code NoSuchMappingException} will
   * be thrown.
   * @param key
   * @return SizeBalancedTreeMap<Key, Value>::Value
   * @throws NoSuchMappingException
   */
  Value get(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("Invalid key");
    } else {
      NodePtr node = this->getNode(this->root, key);
      if (node != nullptr) {
        return node->value;
      } else {
        throw NoSuchMappingException("Invalid key");
      }
    }
  }

  /**
   * Returns the value to which the specified key is mapped; If this map
   * contains no mapping for the key, a new mapping with a default value
   * will be inserted.
   * @param key
   * @return SizeBalancedTreeMap<Key, Value>::Value &
   */
  Value &getOrDefault(K key) {
    if (this->root == nullptr) {
      this->root = Node::from(key);
      return this->root->value;
    } else {
      return this
          ->getNodeOrProvide(this->root, key,
                             [&key]() { return Node::from(key); })
          ->value;
    }
  }

  /**
   * Returns true if this map contains a mapping for the specified key.
   * @param key
   * @return bool
   */
  bool contains(K key) const {
    return this->getNode(this->root, key) != nullptr;
  }

  /**
   * Associates the specified value with the specified key in this map.
   * @param key
   * @param value
   */
  void insert(K key, V value) {
    if (this->root == nullptr) {
      this->root = Node::from(key, value);
    } else {
      this->insert(this->root, key, value);
    }
  }

  /**
   * If the specified key is not already associated with a value, associates
   * it with the given value and returns true, else returns false.
   * @param key
   * @param value
   * @return bool
   */
  bool insertIfAbsent(K key, V value) {
    USize sizeBeforeInsertion = this->size();
    if (this->root == nullptr) {
      this->root = Node::from(key, value);
    } else {
      this->insert(this->root, key, value, false);
    }
    return this->size() > sizeBeforeInsertion;
  }

  /**
   * If the specified key is not already associated with a value, associates
   * it with the given value and returns the value, else returns the associated
   * value.
   * @param key
   * @param value
   * @return SizeBalancedTreeMap<Key, Value>::Value &
   */
  Value &getOrInsert(K key, V value) {
    if (this->root == nullptr) {
      this->root = Node::from(key, value);
      return root->value;
    } else {
      NodePtr node = getNodeOrProvide(this->root, key,
                                      [&]() { return Node::from(key, value); });
      return node->value;
    }
  }

  Value operator[](K key) const { return this->get(key); }

  Value &operator[](K key) { return this->getOrDefault(key); }

  /**
   * Removes the mapping for a key from this map if it is present;
   * Returns true if the mapping is present else returns false
   * @param key the key of the mapping
   * @return bool
   */
  bool remove(K key) {
    if (this->root == nullptr) {
      return false;
    } else {
      return this->remove(this->root, key, [](ConstNodePtr) {});
    }
  }

  /**
   * Removes the mapping for a key from this map if it is present and returns
   * the value which is mapped to the key; If this map contains no mapping for
   * the key, a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return SizeBalancedTreeMap<Key, Value>::Value
   * @throws NoSuchMappingException
   */
  Value getAndRemove(K key) {
    Value result;
    NodeConsumer action = [&](ConstNodePtr node) { result = node->value; };

    if (root == nullptr) {
      throw NoSuchMappingException("Invalid key");
    } else {
      if (remove(this->root, key, action)) {
        return result;
      } else {
        throw NoSuchMappingException("Invalid key");
      }
    }
  }

  /**
   * Gets the entry corresponding to the specified key; if no such entry
   * exists, returns the entry for the least key greater than the specified
   * key; if no such entry exists (i.e., the greatest key in the Tree is less
   * than the specified key), a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return SizeBalancedTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getCeilingEntry(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No ceiling entry in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (key == node->key) {
        return node->entry();
      }

      if (compare(key, node->key)) {
        /* key < node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          return node->entry();
        }
      } else {
        /* key > node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No ceiling entry in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->right) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException("No ceiling entry in this map");
            }
          }

          return parent->entry();
        }
      }
    }

    throw NoSuchMappingException("No ceiling entry in this map");
  }

  /**
   * Gets the entry corresponding to the specified key; if no such entry exists,
   * returns the entry for the greatest key less than the specified key;
   * if no such entry exists, a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return SizeBalancedTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getFloorEntry(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No floor entry exists in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (key == node->key) {
        return node->entry();
      }

      if (compare(key, node->key)) {
        /* key < node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No floor entry exists in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->left) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException("No floor entry exists in this map");
            }
          }

          return parent->entry();
        }
      } else {
        /* key > node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          return node->entry();
        }
      }
    }

    throw NoSuchMappingException("No floor entry exists in this map");
  }

  /**
   * Gets the entry for the least key greater than the specified
   * key; if no such entry exists, returns the entry for the least
   * key greater than the specified key; if no such entry exists,
   * a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return SizeBalancedTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getHigherEntry(K key) {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No higher entry exists in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (compare(key, node->key)) {
        /* key < node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          return node->entry();
        }
      } else {
        /* key >= node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No higher entry exists in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->right) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException(
                  "No higher entry exists in this map");
            }
          }

          return parent->entry();
        }
      }
    }

    throw NoSuchMappingException("No higher entry exists in this map");
  }

  /**
   * Returns the entry for the greatest key less than the specified key; if
   * no such entry exists (i.e., the least key in the Tree is greater than
   * the specified key), a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return SizeBalancedTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getLowerEntry(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No lower entry exists in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (compare(key, node->key) || key == node->key) {
        /* key <= node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No lower entry exists in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->left) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException("No lower entry exists in this map");
            }
          }

          return parent->entry();
        }
      } else {
        /* key > node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          return node->entry();
        }
      }
    }

    throw NoSuchMappingException("No lower entry exists in this map");
  }

  /**
   * Count the number of entries that are less than the given key
   * @param key
   * @return USize
   */
  USize countLessThan(K key) { return this->countLess(this->root, key); }

  /**
   * Count the number of entries that are less or equal to the given key
   * @param key
   * @return USize
   */
  USize countLessOrEqualTo(K key) {
    return this->countLess(this->root, key, true);
  }

  /**
   * Count the number of entries that are greater than the given key
   * @param key
   * @return USize
   */
  USize countGreaterThan(K key) { return this->countGreater(this->root, key); }

  /**
   * Count the number of entries that are greater or equal to the given key
   * @param key
   * @return USize
   */
  USize countGreaterOrEqualTo(K key) {
    return this->countGreater(this->root, key, true);
  }

  /**
   * Remove all entries that satisfy the filter condition.
   * @param filter
   */
  void removeAll(KeyValueFilter filter) {
    std::vector<Key> keys;
    this->inorderTraversal([&](ConstNodePtr node) {
      if (filter(node->key, node->value)) {
        keys.push_back(node->key);
      }
    });
    for (const Key &key : keys) {
      this->remove(key);
    }
  }

  /**
   * Performs the given action for each key and value entry in this map.
   * The value is immutable for the action.
   * @param action
   */
  void forEach(KeyValueConsumer action) const {
    this->inorderTraversal(
        [&](ConstNodePtr node) { action(node->key, node->value); });
  }

  /**
   * Performs the given action for each key and value entry in this map.
   * The value is mutable for the action.
   * @param action
   */
  void forEachMut(MutKeyValueConsumer action) {
    this->inorderTraversal(
        [&](ConstNodePtr node) { action(node->key, node->value); });
  }

  /**
   * Returns a list containing all of the entries in this map.
   * @return SizeBalancedTreeMap<Key, Value>::EntryList
   */
  EntryList toEntryList() const {
    EntryList entryList;
    this->inorderTraversal(
        [&](ConstNodePtr node) { entryList.push_back(node->entry()); });
    return entryList;
  }

 private:
  static USize size(ConstNodePtr node) {
    return node != nullptr ? node->size : 0;
  }

  static void rotateLeft(NodePtr &node) {
    assert(node != nullptr);
    // clang-format off
    //     |                       |
    //     N                       S
    //    / \     l-rotate(N)     / \
    //   L   S    ==========>    N   R
    //      / \                 / \
    //     M   R               L   M
    // clang-format on
    NodePtr successor = node->right;
    node->right = successor->left;
    successor->left = node;

    node->updateSize();
    successor->updateSize();

    node = successor;
  }

  static void rotateRight(NodePtr &node) {
    assert(node != nullptr);
    // clang-format off
    //       |                   |
    //       N                   S
    //      / \   r-rotate(N)   / \
    //     S   R  ==========>  L   N
    //    / \                     / \
    //   L   M                   M   R
    // clang-format on
    NodePtr successor = node->left;
    node->left = successor->right;
    successor->right = node;

    node->updateSize();
    successor->updateSize();

    node = successor;
  }

  static void swapNode(NodePtr &lhs, NodePtr &rhs) {
    std::swap(lhs->key, rhs->key);
    std::swap(lhs->value, rhs->value);
    std::swap(lhs, rhs);
  }

  static void fixBalance(NodePtr &node) {
    if (node == nullptr) {
      return;
    }

    if (node->left != nullptr) {
      if (size(node->left->left) > size(node->right)) {
        // clang-format off
        //       |                       |
        //       N                       L
        //      / \     r-rotate(N)     / \
        //     L  <R>   ==========>   [M]  N
        //    /                             \
        //  [M]                             <R>
        // clang-format on
        rotateRight(node);
        fixBalance(node->right);
        fixBalance(node);
        return;
      } else if (size(node->left->right) > size(node->right)) {
        // clang-format off
        //     |                     |                      |
        //     N                     N                     [M]
        //    / \    l-rotate(L)    / \     r-rotate(N)    / \
        //   L  <R>  ==========>  [M] <R>   ==========>   L   N
        //    \                   /                            \
        //    [M]                L                             <R>
        // clang-format on
        rotateLeft(node->left);
        rotateRight(node);
        fixBalance(node->left);
        fixBalance(node->right);
        fixBalance(node);
        return;
      }
    }

    if (node->right != nullptr) {
      if (size(node->right->right) > size(node->left)) {
        // clang-format off
        //     |                       |
        //     N                       R
        //    / \     l-rotate(N)     / \
        //  <L>  R    ==========>    N  [M]
        //        \                 /
        //        [M]             <L>
        // clang-format on
        rotateLeft(node);
        fixBalance(node->left);
        fixBalance(node);
        return;
      } else if (size(node->right->left) > size(node->left)) {
        // clang-format off
        //     |                     |                      |
        //     N                     N                     [M]
        //    / \    r-rotate(R)    / \     l-rotate(N)    / \
        //  <L>  R   ==========>  <L> [M]   ==========>   N   R
        //      /                       \                /
        //    [M]                        R             <L>
        // clang-format on
        rotateRight(node->right);
        rotateLeft(node);
        fixBalance(node->left);
        fixBalance(node->right);
        fixBalance(node);
        return;
      }
    }
  }

  NodePtr getNodeOrProvide(NodePtr &node, K key, NodeProvider provide) {
    assert(node != nullptr);

    if (key == node->key) {
      return node;
    }

    assert(key != node->key);

    NodePtr result;

    if (compare(key, node->key)) {
      /* key < node->key */
      if (node->left == nullptr) {
        result = node->left = provide();
        node->updateSize();
      } else {
        result = getNodeOrProvide(node->left, key, provide);
        node->updateSize();
        fixBalance(node);
      }
    } else {
      /* key > node->key */
      if (node->right == nullptr) {
        result = node->right = provide();
        node->updateSize();
      } else {
        result = getNodeOrProvide(node->right, key, provide);
        node->updateSize();
        fixBalance(node);
      }
    }

    return result;
  }

  NodePtr getNode(ConstNodePtr node, K key) const {
    assert(node != nullptr);

    if (key == node->key) {
      return node;
    }

    if (compare(key, node->key)) {
      /* key < node->key */
      return node->left == nullptr ? nullptr : getNode(node->left, key);
    } else {
      /* key > node->key */
      return node->right == nullptr ? nullptr : getNode(node->right, key);
    }
  }

  void insert(NodePtr &node, K key, V value, bool replace = true) {
    assert(node != nullptr);

    if (key == node->key) {
      if (replace) {
        node->value = value;
      }
      return;
    }

    assert(key != node->key);

    if (compare(key, node->key)) {
      /* key < node->key */
      if (node->left == nullptr) {
        node->left = Node::from(key, value);
        node->updateSize();
      } else {
        insert(node->left, key, value, replace);
        node->updateSize();
        fixBalance(node);
      }
    } else {
      /* key > node->key */
      if (node->right == nullptr) {
        node->right = Node::from(key, value);
        node->updateSize();
      } else {
        insert(node->right, key, value, replace);
        node->updateSize();
        fixBalance(node);
      }
    }
  }

  bool remove(NodePtr &node, K key, NodeConsumer action) {
    assert(node != nullptr);

    if (key != node->key) {
      if (compare(key, node->key)) {
        /* key < node->key */
        NodePtr &left = node->left;
        if (left != nullptr && remove(left, key, action)) {
          node->updateSize();
          fixBalance(node);
          return true;
        } else {
          return false;
        }
      } else {
        /* key > node->key */
        NodePtr &right = node->right;
        if (right != nullptr && remove(right, key, action)) {
          node->updateSize();
          fixBalance(node);
          return true;
        } else {
          return false;
        }
      }
    }

    assert(key == node->key);
    action(node);

    if (node->isLeaf()) {
      // Case 1: no child
      node = nullptr;
    } else if (node->right == nullptr) {
      // clang-format off
      // Case 2: left child only
      //     P
      //     |  remove(N)  P
      //     N  ========>  |
      //    /              L
      //   L
      // clang-format on
      node = node->left;
    } else if (node->left == nullptr) {
      // clang-format off
      // Case 3: right child only
      //   P
      //   |    remove(N)  P
      //   N    ========>  |
      //    \              R
      //     R
      // clang-format on
      node = node->right;
    } else if (node->right->left == nullptr) {
      // clang-format off
      // Case 4: both left and right child, right child has no left child
      //    |                 |
      //    N    remove(N)    R
      //   / \   ========>   /
      //  L   R             L
      // clang-format on
      NodePtr right = node->right;
      swapNode(node, right);
      right->right = node->right;
      node = right;
      node->updateSize();
      fixBalance(node);
    } else {
      // clang-format off
      // Case 5: both left and right child, right child is not a leaf
      //   Step 1. find the node N with the smallest key
      //           and its parent P on the right subtree
      //   Step 2. swap S and N
      //   Step 3. remove node N like Case 1 or Case 3
      //   Step 4. update size for all nodes on the path
      //           from S to P
      //     |                  |
      //     N                  S                 |
      //    / \                / \                S
      //   L  ..  swap(N, S)  L  ..  remove(N)   / \
      //       |  =========>      |  ========>  L  ..
      //       P                  P                 |
      //      / \                / \                P
      //     S  ..              N  ..              / \
      //      \                  \                R  ..
      //       R                  R
      // clang-format on

      std::stack<NodePtr> path;

      // Step 1
      NodePtr successor = node->right;
      NodePtr parent = node;
      path.push(node);

      while (successor->left != nullptr) {
        path.push(successor);
        parent = successor;
        successor = parent->left;
      }

      // Step 2
      swapNode(node, successor);

      // Step 3
      parent->left = node->right;
      // Restore node
      node = successor;

      // Step 4
      while (!path.empty()) {
        path.top()->updateSize();
        path.pop();
      }
    }

    return true;
  }

  USize countLess(ConstNodePtr node, K key, bool countEqual = false) const {
    if (node == nullptr) {
      return 0;
    } else if (key < node->key) {
      return countLess(node->left, key, countEqual);
    } else if (key > node->key) {
      return size(node->left) + 1 + countLess(node->right, key, countEqual);
    } else {
      return size(node->left) + (countEqual ? 1 : 0);
    }
  }

  USize countGreater(ConstNodePtr node, K key, bool countEqual = false) const {
    if (node == nullptr) {
      return 0;
    } else if (key < node->key) {
      return size(node->right) + 1 + countGreater(node->left, key, countEqual);
    } else if (key > node->key) {
      return countGreater(node->right, key, countEqual);
    } else {
      return size(node->right) + (countEqual ? 1 : 0);
    }
  }

  void inorderTraversal(NodeConsumer action) const {
    if (this->root == nullptr) {
      return;
    }

    std::stack<NodePtr> stack;
    NodePtr node = this->root;

    while (node != nullptr || !stack.empty()) {
      while (node != nullptr) {
        stack.push(node);
        node = node->left;
      }
      if (!stack.empty()) {
        node = stack.top();
        stack.pop();
        action(node);
        node = node->right;
      }
    }
  }
};

#endif  // SIZE_BALANCED_TREE_MAP_HPP