aboutsummaryrefslogtreecommitdiff
path: root/util/interval-tree.c
blob: 4c0baf108f0efe18b51094a0cf160a20350ddb44 (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
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
/* SPDX-License-Identifier: GPL-2.0-or-later */

#include "qemu/osdep.h"
#include "qemu/interval-tree.h"
#include "qemu/atomic.h"

/*
 * Red Black Trees.
 *
 * For now, don't expose Linux Red-Black Trees separately, but retain the
 * separate type definitions to keep the implementation sane, and allow
 * the possibility of separating them later.
 *
 * Derived from include/linux/rbtree_augmented.h and its dependencies.
 */

/*
 * red-black trees properties:  https://en.wikipedia.org/wiki/Rbtree
 *
 *  1) A node is either red or black
 *  2) The root is black
 *  3) All leaves (NULL) are black
 *  4) Both children of every red node are black
 *  5) Every simple path from root to leaves contains the same number
 *     of black nodes.
 *
 *  4 and 5 give the O(log n) guarantee, since 4 implies you cannot have two
 *  consecutive red nodes in a path and every red node is therefore followed by
 *  a black. So if B is the number of black nodes on every simple path (as per
 *  5), then the longest possible path due to 4 is 2B.
 *
 *  We shall indicate color with case, where black nodes are uppercase and red
 *  nodes will be lowercase. Unknown color nodes shall be drawn as red within
 *  parentheses and have some accompanying text comment.
 *
 * Notes on lockless lookups:
 *
 * All stores to the tree structure (rb_left and rb_right) must be done using
 * WRITE_ONCE [qatomic_set for QEMU]. And we must not inadvertently cause
 * (temporary) loops in the tree structure as seen in program order.
 *
 * These two requirements will allow lockless iteration of the tree -- not
 * correct iteration mind you, tree rotations are not atomic so a lookup might
 * miss entire subtrees.
 *
 * But they do guarantee that any such traversal will only see valid elements
 * and that it will indeed complete -- does not get stuck in a loop.
 *
 * It also guarantees that if the lookup returns an element it is the 'correct'
 * one. But not returning an element does _NOT_ mean it's not present.
 *
 * NOTE:
 *
 * Stores to __rb_parent_color are not important for simple lookups so those
 * are left undone as of now. Nor did I check for loops involving parent
 * pointers.
 */

typedef enum RBColor
{
    RB_RED,
    RB_BLACK,
} RBColor;

typedef struct RBAugmentCallbacks {
    void (*propagate)(RBNode *node, RBNode *stop);
    void (*copy)(RBNode *old, RBNode *new);
    void (*rotate)(RBNode *old, RBNode *new);
} RBAugmentCallbacks;

static inline RBNode *rb_parent(const RBNode *n)
{
    return (RBNode *)(n->rb_parent_color & ~1);
}

static inline RBNode *rb_red_parent(const RBNode *n)
{
    return (RBNode *)n->rb_parent_color;
}

static inline RBColor pc_color(uintptr_t pc)
{
    return (RBColor)(pc & 1);
}

static inline bool pc_is_red(uintptr_t pc)
{
    return pc_color(pc) == RB_RED;
}

static inline bool pc_is_black(uintptr_t pc)
{
    return !pc_is_red(pc);
}

static inline RBColor rb_color(const RBNode *n)
{
    return pc_color(n->rb_parent_color);
}

static inline bool rb_is_red(const RBNode *n)
{
    return pc_is_red(n->rb_parent_color);
}

static inline bool rb_is_black(const RBNode *n)
{
    return pc_is_black(n->rb_parent_color);
}

static inline void rb_set_black(RBNode *n)
{
    n->rb_parent_color |= RB_BLACK;
}

static inline void rb_set_parent_color(RBNode *n, RBNode *p, RBColor color)
{
    n->rb_parent_color = (uintptr_t)p | color;
}

static inline void rb_set_parent(RBNode *n, RBNode *p)
{
    rb_set_parent_color(n, p, rb_color(n));
}

static inline void rb_link_node(RBNode *node, RBNode *parent, RBNode **rb_link)
{
    node->rb_parent_color = (uintptr_t)parent;
    node->rb_left = node->rb_right = NULL;

    qatomic_set(rb_link, node);
}

static RBNode *rb_next(RBNode *node)
{
    RBNode *parent;

    /* OMIT: if empty node, return null. */

    /*
     * If we have a right-hand child, go down and then left as far as we can.
     */
    if (node->rb_right) {
        node = node->rb_right;
        while (node->rb_left) {
            node = node->rb_left;
        }
        return node;
    }

    /*
     * No right-hand children. Everything down and left is smaller than us,
     * so any 'next' node must be in the general direction of our parent.
     * Go up the tree; any time the ancestor is a right-hand child of its
     * parent, keep going up. First time it's a left-hand child of its
     * parent, said parent is our 'next' node.
     */
    while ((parent = rb_parent(node)) && node == parent->rb_right) {
        node = parent;
    }

    return parent;
}

static inline void rb_change_child(RBNode *old, RBNode *new,
                                   RBNode *parent, RBRoot *root)
{
    if (!parent) {
        qatomic_set(&root->rb_node, new);
    } else if (parent->rb_left == old) {
        qatomic_set(&parent->rb_left, new);
    } else {
        qatomic_set(&parent->rb_right, new);
    }
}

static inline void rb_rotate_set_parents(RBNode *old, RBNode *new,
                                         RBRoot *root, RBColor color)
{
    RBNode *parent = rb_parent(old);

    new->rb_parent_color = old->rb_parent_color;
    rb_set_parent_color(old, new, color);
    rb_change_child(old, new, parent, root);
}

static void rb_insert_augmented(RBNode *node, RBRoot *root,
                                const RBAugmentCallbacks *augment)
{
    RBNode *parent = rb_red_parent(node), *gparent, *tmp;

    while (true) {
        /*
         * Loop invariant: node is red.
         */
        if (unlikely(!parent)) {
            /*
             * The inserted node is root. Either this is the first node, or
             * we recursed at Case 1 below and are no longer violating 4).
             */
            rb_set_parent_color(node, NULL, RB_BLACK);
            break;
        }

        /*
         * If there is a black parent, we are done.  Otherwise, take some
         * corrective action as, per 4), we don't want a red root or two
         * consecutive red nodes.
         */
        if (rb_is_black(parent)) {
            break;
        }

        gparent = rb_red_parent(parent);

        tmp = gparent->rb_right;
        if (parent != tmp) {    /* parent == gparent->rb_left */
            if (tmp && rb_is_red(tmp)) {
                /*
                 * Case 1 - node's uncle is red (color flips).
                 *
                 *       G            g
                 *      / \          / \
                 *     p   u  -->   P   U
                 *    /            /
                 *   n            n
                 *
                 * However, since g's parent might be red, and 4) does not
                 * allow this, we need to recurse at g.
                 */
                rb_set_parent_color(tmp, gparent, RB_BLACK);
                rb_set_parent_color(parent, gparent, RB_BLACK);
                node = gparent;
                parent = rb_parent(node);
                rb_set_parent_color(node, parent, RB_RED);
                continue;
            }

            tmp = parent->rb_right;
            if (node == tmp) {
                /*
                 * Case 2 - node's uncle is black and node is
                 * the parent's right child (left rotate at parent).
                 *
                 *      G             G
                 *     / \           / \
                 *    p   U  -->    n   U
                 *     \           /
                 *      n         p
                 *
                 * This still leaves us in violation of 4), the
                 * continuation into Case 3 will fix that.
                 */
                tmp = node->rb_left;
                qatomic_set(&parent->rb_right, tmp);
                qatomic_set(&node->rb_left, parent);
                if (tmp) {
                    rb_set_parent_color(tmp, parent, RB_BLACK);
                }
                rb_set_parent_color(parent, node, RB_RED);
                augment->rotate(parent, node);
                parent = node;
                tmp = node->rb_right;
            }

            /*
             * Case 3 - node's uncle is black and node is
             * the parent's left child (right rotate at gparent).
             *
             *        G           P
             *       / \         / \
             *      p   U  -->  n   g
             *     /                 \
             *    n                   U
             */
            qatomic_set(&gparent->rb_left, tmp); /* == parent->rb_right */
            qatomic_set(&parent->rb_right, gparent);
            if (tmp) {
                rb_set_parent_color(tmp, gparent, RB_BLACK);
            }
            rb_rotate_set_parents(gparent, parent, root, RB_RED);
            augment->rotate(gparent, parent);
            break;
        } else {
            tmp = gparent->rb_left;
            if (tmp && rb_is_red(tmp)) {
                /* Case 1 - color flips */
                rb_set_parent_color(tmp, gparent, RB_BLACK);
                rb_set_parent_color(parent, gparent, RB_BLACK);
                node = gparent;
                parent = rb_parent(node);
                rb_set_parent_color(node, parent, RB_RED);
                continue;
            }

            tmp = parent->rb_left;
            if (node == tmp) {
                /* Case 2 - right rotate at parent */
                tmp = node->rb_right;
                qatomic_set(&parent->rb_left, tmp);
                qatomic_set(&node->rb_right, parent);
                if (tmp) {
                    rb_set_parent_color(tmp, parent, RB_BLACK);
                }
                rb_set_parent_color(parent, node, RB_RED);
                augment->rotate(parent, node);
                parent = node;
                tmp = node->rb_left;
            }

            /* Case 3 - left rotate at gparent */
            qatomic_set(&gparent->rb_right, tmp); /* == parent->rb_left */
            qatomic_set(&parent->rb_left, gparent);
            if (tmp) {
                rb_set_parent_color(tmp, gparent, RB_BLACK);
            }
            rb_rotate_set_parents(gparent, parent, root, RB_RED);
            augment->rotate(gparent, parent);
            break;
        }
    }
}

static void rb_insert_augmented_cached(RBNode *node,
                                       RBRootLeftCached *root, bool newleft,
                                       const RBAugmentCallbacks *augment)
{
    if (newleft) {
        root->rb_leftmost = node;
    }
    rb_insert_augmented(node, &root->rb_root, augment);
}

static void rb_erase_color(RBNode *parent, RBRoot *root,
                           const RBAugmentCallbacks *augment)
{
    RBNode *node = NULL, *sibling, *tmp1, *tmp2;

    while (true) {
        /*
         * Loop invariants:
         * - node is black (or NULL on first iteration)
         * - node is not the root (parent is not NULL)
         * - All leaf paths going through parent and node have a
         *   black node count that is 1 lower than other leaf paths.
         */
        sibling = parent->rb_right;
        if (node != sibling) {  /* node == parent->rb_left */
            if (rb_is_red(sibling)) {
                /*
                 * Case 1 - left rotate at parent
                 *
                 *     P               S
                 *    / \             / \ 
                 *   N   s    -->    p   Sr
                 *      / \         / \ 
                 *     Sl  Sr      N   Sl
                 */
                tmp1 = sibling->rb_left;
                qatomic_set(&parent->rb_right, tmp1);
                qatomic_set(&sibling->rb_left, parent);
                rb_set_parent_color(tmp1, parent, RB_BLACK);
                rb_rotate_set_parents(parent, sibling, root, RB_RED);
                augment->rotate(parent, sibling);
                sibling = tmp1;
            }
            tmp1 = sibling->rb_right;
            if (!tmp1 || rb_is_black(tmp1)) {
                tmp2 = sibling->rb_left;
                if (!tmp2 || rb_is_black(tmp2)) {
                    /*
                     * Case 2 - sibling color flip
                     * (p could be either color here)
                     *
                     *    (p)           (p)
                     *    / \           / \ 
                     *   N   S    -->  N   s
                     *      / \           / \ 
                     *     Sl  Sr        Sl  Sr
                     *
                     * This leaves us violating 5) which
                     * can be fixed by flipping p to black
                     * if it was red, or by recursing at p.
                     * p is red when coming from Case 1.
                     */
                    rb_set_parent_color(sibling, parent, RB_RED);
                    if (rb_is_red(parent)) {
                        rb_set_black(parent);
                    } else {
                        node = parent;
                        parent = rb_parent(node);
                        if (parent) {
                            continue;
                        }
                    }
                    break;
                }
                /*
                 * Case 3 - right rotate at sibling
                 * (p could be either color here)
                 *
                 *   (p)           (p)
                 *   / \           / \
                 *  N   S    -->  N   sl
                 *     / \             \
                 *    sl  Sr            S
                 *                       \
                 *                        Sr
                 *
                 * Note: p might be red, and then bot
                 * p and sl are red after rotation (which
                 * breaks property 4). This is fixed in
                 * Case 4 (in rb_rotate_set_parents()
                 *         which set sl the color of p
                 *         and set p RB_BLACK)
                 *
                 *   (p)            (sl)
                 *   / \            /  \
                 *  N   sl   -->   P    S
                 *       \        /      \
                 *        S      N        Sr
                 *         \
                 *          Sr
                 */
                tmp1 = tmp2->rb_right;
                qatomic_set(&sibling->rb_left, tmp1);
                qatomic_set(&tmp2->rb_right, sibling);
                qatomic_set(&parent->rb_right, tmp2);
                if (tmp1) {
                    rb_set_parent_color(tmp1, sibling, RB_BLACK);
                }
                augment->rotate(sibling, tmp2);
                tmp1 = sibling;
                sibling = tmp2;
            }
            /*
             * Case 4 - left rotate at parent + color flips
             * (p and sl could be either color here.
             *  After rotation, p becomes black, s acquires
             *  p's color, and sl keeps its color)
             *
             *      (p)             (s)
             *      / \             / \
             *     N   S     -->   P   Sr
             *        / \         / \
             *      (sl) sr      N  (sl)
             */
            tmp2 = sibling->rb_left;
            qatomic_set(&parent->rb_right, tmp2);
            qatomic_set(&sibling->rb_left, parent);
            rb_set_parent_color(tmp1, sibling, RB_BLACK);
            if (tmp2) {
                rb_set_parent(tmp2, parent);
            }
            rb_rotate_set_parents(parent, sibling, root, RB_BLACK);
            augment->rotate(parent, sibling);
            break;
        } else {
            sibling = parent->rb_left;
            if (rb_is_red(sibling)) {
                /* Case 1 - right rotate at parent */
                tmp1 = sibling->rb_right;
                qatomic_set(&parent->rb_left, tmp1);
                qatomic_set(&sibling->rb_right, parent);
                rb_set_parent_color(tmp1, parent, RB_BLACK);
                rb_rotate_set_parents(parent, sibling, root, RB_RED);
                augment->rotate(parent, sibling);
                sibling = tmp1;
            }
            tmp1 = sibling->rb_left;
            if (!tmp1 || rb_is_black(tmp1)) {
                tmp2 = sibling->rb_right;
                if (!tmp2 || rb_is_black(tmp2)) {
                    /* Case 2 - sibling color flip */
                    rb_set_parent_color(sibling, parent, RB_RED);
                    if (rb_is_red(parent)) {
                        rb_set_black(parent);
                    } else {
                        node = parent;
                        parent = rb_parent(node);
                        if (parent) {
                            continue;
                        }
                    }
                    break;
                }
                /* Case 3 - left rotate at sibling */
                tmp1 = tmp2->rb_left;
                qatomic_set(&sibling->rb_right, tmp1);
                qatomic_set(&tmp2->rb_left, sibling);
                qatomic_set(&parent->rb_left, tmp2);
                if (tmp1) {
                    rb_set_parent_color(tmp1, sibling, RB_BLACK);
                }
                augment->rotate(sibling, tmp2);
                tmp1 = sibling;
                sibling = tmp2;
            }
            /* Case 4 - right rotate at parent + color flips */
            tmp2 = sibling->rb_right;
            qatomic_set(&parent->rb_left, tmp2);
            qatomic_set(&sibling->rb_right, parent);
            rb_set_parent_color(tmp1, sibling, RB_BLACK);
            if (tmp2) {
                rb_set_parent(tmp2, parent);
            }
            rb_rotate_set_parents(parent, sibling, root, RB_BLACK);
            augment->rotate(parent, sibling);
            break;
        }
    }
}

static void rb_erase_augmented(RBNode *node, RBRoot *root,
                               const RBAugmentCallbacks *augment)
{
    RBNode *child = node->rb_right;
    RBNode *tmp = node->rb_left;
    RBNode *parent, *rebalance;
    uintptr_t pc;

    if (!tmp) {
        /*
         * Case 1: node to erase has no more than 1 child (easy!)
         *
         * Note that if there is one child it must be red due to 5)
         * and node must be black due to 4). We adjust colors locally
         * so as to bypass rb_erase_color() later on.
         */
        pc = node->rb_parent_color;
        parent = rb_parent(node);
        rb_change_child(node, child, parent, root);
        if (child) {
            child->rb_parent_color = pc;
            rebalance = NULL;
        } else {
            rebalance = pc_is_black(pc) ? parent : NULL;
        }
        tmp = parent;
    } else if (!child) {
        /* Still case 1, but this time the child is node->rb_left */
        pc = node->rb_parent_color;
        parent = rb_parent(node);
        tmp->rb_parent_color = pc;
        rb_change_child(node, tmp, parent, root);
        rebalance = NULL;
        tmp = parent;
    } else {
        RBNode *successor = child, *child2;
        tmp = child->rb_left;
        if (!tmp) {
            /*
             * Case 2: node's successor is its right child
             *
             *    (n)          (s)
             *    / \          / \
             *  (x) (s)  ->  (x) (c)
             *        \
             *        (c)
             */
            parent = successor;
            child2 = successor->rb_right;

            augment->copy(node, successor);
        } else {
            /*
             * Case 3: node's successor is leftmost under
             * node's right child subtree
             *
             *    (n)          (s)
             *    / \          / \
             *  (x) (y)  ->  (x) (y)
             *      /            /
             *    (p)          (p)
             *    /            /
             *  (s)          (c)
             *    \
             *    (c)
             */
            do {
                parent = successor;
                successor = tmp;
                tmp = tmp->rb_left;
            } while (tmp);
            child2 = successor->rb_right;
            qatomic_set(&parent->rb_left, child2);
            qatomic_set(&successor->rb_right, child);
            rb_set_parent(child, successor);

            augment->copy(node, successor);
            augment->propagate(parent, successor);
        }

        tmp = node->rb_left;
        qatomic_set(&successor->rb_left, tmp);
        rb_set_parent(tmp, successor);

        pc = node->rb_parent_color;
        tmp = rb_parent(node);
        rb_change_child(node, successor, tmp, root);

        if (child2) {
            rb_set_parent_color(child2, parent, RB_BLACK);
            rebalance = NULL;
        } else {
            rebalance = rb_is_black(successor) ? parent : NULL;
        }
        successor->rb_parent_color = pc;
        tmp = successor;
    }

    augment->propagate(tmp, NULL);

    if (rebalance) {
        rb_erase_color(rebalance, root, augment);
    }
}

static void rb_erase_augmented_cached(RBNode *node, RBRootLeftCached *root,
                                      const RBAugmentCallbacks *augment)
{
    if (root->rb_leftmost == node) {
        root->rb_leftmost = rb_next(node);
    }
    rb_erase_augmented(node, &root->rb_root, augment);
}


/*
 * Interval trees.
 *
 * Derived from lib/interval_tree.c and its dependencies,
 * especially include/linux/interval_tree_generic.h.
 */

#define rb_to_itree(N)  container_of(N, IntervalTreeNode, rb)

static bool interval_tree_compute_max(IntervalTreeNode *node, bool exit)
{
    IntervalTreeNode *child;
    uint64_t max = node->last;

    if (node->rb.rb_left) {
        child = rb_to_itree(node->rb.rb_left);
        if (child->subtree_last > max) {
            max = child->subtree_last;
        }
    }
    if (node->rb.rb_right) {
        child = rb_to_itree(node->rb.rb_right);
        if (child->subtree_last > max) {
            max = child->subtree_last;
        }
    }
    if (exit && node->subtree_last == max) {
        return true;
    }
    node->subtree_last = max;
    return false;
}

static void interval_tree_propagate(RBNode *rb, RBNode *stop)
{
    while (rb != stop) {
        IntervalTreeNode *node = rb_to_itree(rb);
        if (interval_tree_compute_max(node, true)) {
            break;
        }
        rb = rb_parent(&node->rb);
    }
}

static void interval_tree_copy(RBNode *rb_old, RBNode *rb_new)
{
    IntervalTreeNode *old = rb_to_itree(rb_old);
    IntervalTreeNode *new = rb_to_itree(rb_new);

    new->subtree_last = old->subtree_last;
}

static void interval_tree_rotate(RBNode *rb_old, RBNode *rb_new)
{
    IntervalTreeNode *old = rb_to_itree(rb_old);
    IntervalTreeNode *new = rb_to_itree(rb_new);

    new->subtree_last = old->subtree_last;
    interval_tree_compute_max(old, false);
}

static const RBAugmentCallbacks interval_tree_augment = {
    .propagate = interval_tree_propagate,
    .copy = interval_tree_copy,
    .rotate = interval_tree_rotate,
};

/* Insert / remove interval nodes from the tree */
void interval_tree_insert(IntervalTreeNode *node, IntervalTreeRoot *root)
{
    RBNode **link = &root->rb_root.rb_node, *rb_parent = NULL;
    uint64_t start = node->start, last = node->last;
    IntervalTreeNode *parent;
    bool leftmost = true;

    while (*link) {
        rb_parent = *link;
        parent = rb_to_itree(rb_parent);

        if (parent->subtree_last < last) {
            parent->subtree_last = last;
        }
        if (start < parent->start) {
            link = &parent->rb.rb_left;
        } else {
            link = &parent->rb.rb_right;
            leftmost = false;
        }
    }

    node->subtree_last = last;
    rb_link_node(&node->rb, rb_parent, link);
    rb_insert_augmented_cached(&node->rb, root, leftmost,
                               &interval_tree_augment);
}

void interval_tree_remove(IntervalTreeNode *node, IntervalTreeRoot *root)
{
    rb_erase_augmented_cached(&node->rb, root, &interval_tree_augment);
}

/*
 * Iterate over intervals intersecting [start;last]
 *
 * Note that a node's interval intersects [start;last] iff:
 *   Cond1: node->start <= last
 * and
 *   Cond2: start <= node->last
 */

static IntervalTreeNode *interval_tree_subtree_search(IntervalTreeNode *node,
                                                      uint64_t start,
                                                      uint64_t last)
{
    while (true) {
        /*
         * Loop invariant: start <= node->subtree_last
         * (Cond2 is satisfied by one of the subtree nodes)
         */
        if (node->rb.rb_left) {
            IntervalTreeNode *left = rb_to_itree(node->rb.rb_left);

            if (start <= left->subtree_last) {
                /*
                 * Some nodes in left subtree satisfy Cond2.
                 * Iterate to find the leftmost such node N.
                 * If it also satisfies Cond1, that's the
                 * match we are looking for. Otherwise, there
                 * is no matching interval as nodes to the
                 * right of N can't satisfy Cond1 either.
                 */
                node = left;
                continue;
            }
        }
        if (node->start <= last) {         /* Cond1 */
            if (start <= node->last) {     /* Cond2 */
                return node; /* node is leftmost match */
            }
            if (node->rb.rb_right) {
                node = rb_to_itree(node->rb.rb_right);
                if (start <= node->subtree_last) {
                    continue;
                }
            }
        }
        return NULL; /* no match */
    }
}

IntervalTreeNode *interval_tree_iter_first(IntervalTreeRoot *root,
                                           uint64_t start, uint64_t last)
{
    IntervalTreeNode *node, *leftmost;

    if (!root->rb_root.rb_node) {
        return NULL;
    }

    /*
     * Fastpath range intersection/overlap between A: [a0, a1] and
     * B: [b0, b1] is given by:
     *
     *         a0 <= b1 && b0 <= a1
     *
     *  ... where A holds the lock range and B holds the smallest
     * 'start' and largest 'last' in the tree. For the later, we
     * rely on the root node, which by augmented interval tree
     * property, holds the largest value in its last-in-subtree.
     * This allows mitigating some of the tree walk overhead for
     * for non-intersecting ranges, maintained and consulted in O(1).
     */
    node = rb_to_itree(root->rb_root.rb_node);
    if (node->subtree_last < start) {
        return NULL;
    }

    leftmost = rb_to_itree(root->rb_leftmost);
    if (leftmost->start > last) {
        return NULL;
    }

    return interval_tree_subtree_search(node, start, last);
}

IntervalTreeNode *interval_tree_iter_next(IntervalTreeNode *node,
                                          uint64_t start, uint64_t last)
{
    RBNode *rb = node->rb.rb_right, *prev;

    while (true) {
        /*
         * Loop invariants:
         *   Cond1: node->start <= last
         *   rb == node->rb.rb_right
         *
         * First, search right subtree if suitable
         */
        if (rb) {
            IntervalTreeNode *right = rb_to_itree(rb);

            if (start <= right->subtree_last) {
                return interval_tree_subtree_search(right, start, last);
            }
        }

        /* Move up the tree until we come from a node's left child */
        do {
            rb = rb_parent(&node->rb);
            if (!rb) {
                return NULL;
            }
            prev = &node->rb;
            node = rb_to_itree(rb);
            rb = node->rb.rb_right;
        } while (prev == rb);

        /* Check if the node intersects [start;last] */
        if (last < node->start) {  /* !Cond1 */
            return NULL;
        }
        if (start <= node->last) { /* Cond2 */
            return node;
        }
    }
}

/* Occasionally useful for calling from within the debugger. */
#if 0
static void debug_interval_tree_int(IntervalTreeNode *node,
                                    const char *dir, int level)
{
    printf("%4d %*s %s [%" PRIu64 ",%" PRIu64 "] subtree_last:%" PRIu64 "\n",
           level, level + 1, dir, rb_is_red(&node->rb) ? "r" : "b",
           node->start, node->last, node->subtree_last);

    if (node->rb.rb_left) {
        debug_interval_tree_int(rb_to_itree(node->rb.rb_left), "<", level + 1);
    }
    if (node->rb.rb_right) {
        debug_interval_tree_int(rb_to_itree(node->rb.rb_right), ">", level + 1);
    }
}

void debug_interval_tree(IntervalTreeNode *node);
void debug_interval_tree(IntervalTreeNode *node)
{
    if (node) {
        debug_interval_tree_int(node, "*", 0);
    } else {
        printf("null\n");
    }
}
#endif