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
//! Slice sorting
//!
//! This module contains a sorting algorithm based on Orson Peters' pattern-defeating quicksort,
//! published at: <https://github.com/orlp/pdqsort>
//!
//! Unstable sorting is compatible with core because it doesn't allocate memory, unlike our
//! stable sorting implementation.

use crate::cmp;
use crate::mem::{self, MaybeUninit, SizedTypeProperties};
use crate::ptr;

/// When dropped, copies from `src` into `dest`.
struct CopyOnDrop<T> {
    src: *const T,
    dest: *mut T,
}

impl<T> Drop for CopyOnDrop<T> {
    fn drop(&mut self) {
        // SAFETY: This is a helper class.
        //         Please refer to its usage for correctness.
        //         Namely, one must be sure that `src` and `dst` does not overlap as required by `ptr::copy_nonoverlapping`.
        unsafe {
            ptr::copy_nonoverlapping(self.src, self.dest, 1);
        }
    }
}

/// Shifts the first element to the right until it encounters a greater or equal element.
fn shift_head<T, F>(v: &mut [T], is_less: &mut F)
where
    F: FnMut(&T, &T) -> bool,
{
    let len = v.len();
    // SAFETY: The unsafe operations below involves indexing without a bounds check (by offsetting a
    // pointer) and copying memory (`ptr::copy_nonoverlapping`).
    //
    // a. Indexing:
    //  1. We checked the size of the array to >=2.
    //  2. All the indexing that we will do is always between {0 <= index < len} at most.
    //
    // b. Memory copying
    //  1. We are obtaining pointers to references which are guaranteed to be valid.
    //  2. They cannot overlap because we obtain pointers to difference indices of the slice.
    //     Namely, `i` and `i-1`.
    //  3. If the slice is properly aligned, the elements are properly aligned.
    //     It is the caller's responsibility to make sure the slice is properly aligned.
    //
    // See comments below for further detail.
    unsafe {
        // If the first two elements are out-of-order...
        if len >= 2 && is_less(v.get_unchecked(1), v.get_unchecked(0)) {
            // Read the first element into a stack-allocated variable. If a following comparison
            // operation panics, `hole` will get dropped and automatically write the element back
            // into the slice.
            let tmp = mem::ManuallyDrop::new(ptr::read(v.get_unchecked(0)));
            let v = v.as_mut_ptr();
            let mut hole = CopyOnDrop { src: &*tmp, dest: v.add(1) };
            ptr::copy_nonoverlapping(v.add(1), v.add(0), 1);

            for i in 2..len {
                if !is_less(&*v.add(i), &*tmp) {
                    break;
                }

                // Move `i`-th element one place to the left, thus shifting the hole to the right.
                ptr::copy_nonoverlapping(v.add(i), v.add(i - 1), 1);
                hole.dest = v.add(i);
            }
            // `hole` gets dropped and thus copies `tmp` into the remaining hole in `v`.
        }
    }
}

/// Shifts the last element to the left until it encounters a smaller or equal element.
fn shift_tail<T, F>(v: &mut [T], is_less: &mut F)
where
    F: FnMut(&T, &T) -> bool,
{
    let len = v.len();
    // SAFETY: The unsafe operations below involves indexing without a bound check (by offsetting a
    // pointer) and copying memory (`ptr::copy_nonoverlapping`).
    //
    // a. Indexing:
    //  1. We checked the size of the array to >= 2.
    //  2. All the indexing that we will do is always between `0 <= index < len-1` at most.
    //
    // b. Memory copying
    //  1. We are obtaining pointers to references which are guaranteed to be valid.
    //  2. They cannot overlap because we obtain pointers to difference indices of the slice.
    //     Namely, `i` and `i+1`.
    //  3. If the slice is properly aligned, the elements are properly aligned.
    //     It is the caller's responsibility to make sure the slice is properly aligned.
    //
    // See comments below for further detail.
    unsafe {
        // If the last two elements are out-of-order...
        if len >= 2 && is_less(v.get_unchecked(len - 1), v.get_unchecked(len - 2)) {
            // Read the last element into a stack-allocated variable. If a following comparison
            // operation panics, `hole` will get dropped and automatically write the element back
            // into the slice.
            let tmp = mem::ManuallyDrop::new(ptr::read(v.get_unchecked(len - 1)));
            let v = v.as_mut_ptr();
            let mut hole = CopyOnDrop { src: &*tmp, dest: v.add(len - 2) };
            ptr::copy_nonoverlapping(v.add(len - 2), v.add(len - 1), 1);

            for i in (0..len - 2).rev() {
                if !is_less(&*tmp, &*v.add(i)) {
                    break;
                }

                // Move `i`-th element one place to the right, thus shifting the hole to the left.
                ptr::copy_nonoverlapping(v.add(i), v.add(i + 1), 1);
                hole.dest = v.add(i);
            }
            // `hole` gets dropped and thus copies `tmp` into the remaining hole in `v`.
        }
    }
}

/// Partially sorts a slice by shifting several out-of-order elements around.
///
/// Returns `true` if the slice is sorted at the end. This function is *O*(*n*) worst-case.
#[cold]
fn partial_insertion_sort<T, F>(v: &mut [T], is_less: &mut F) -> bool
where
    F: FnMut(&T, &T) -> bool,
{
    // Maximum number of adjacent out-of-order pairs that will get shifted.
    const MAX_STEPS: usize = 5;
    // If the slice is shorter than this, don't shift any elements.
    const SHORTEST_SHIFTING: usize = 50;

    let len = v.len();
    let mut i = 1;

    for _ in 0..MAX_STEPS {
        // SAFETY: We already explicitly did the bound checking with `i < len`.
        // All our subsequent indexing is only in the range `0 <= index < len`
        unsafe {
            // Find the next pair of adjacent out-of-order elements.
            while i < len && !is_less(v.get_unchecked(i), v.get_unchecked(i - 1)) {
                i += 1;
            }
        }

        // Are we done?
        if i == len {
            return true;
        }

        // Don't shift elements on short arrays, that has a performance cost.
        if len < SHORTEST_SHIFTING {
            return false;
        }

        // Swap the found pair of elements. This puts them in correct order.
        v.swap(i - 1, i);

        // Shift the smaller element to the left.
        shift_tail(&mut v[..i], is_less);
        // Shift the greater element to the right.
        shift_head(&mut v[i..], is_less);
    }

    // Didn't manage to sort the slice in the limited number of steps.
    false
}

/// Sorts a slice using insertion sort, which is *O*(*n*^2) worst-case.
fn insertion_sort<T, F>(v: &mut [T], is_less: &mut F)
where
    F: FnMut(&T, &T) -> bool,
{
    for i in 1..v.len() {
        shift_tail(&mut v[..i + 1], is_less);
    }
}

/// Sorts `v` using heapsort, which guarantees *O*(*n* \* log(*n*)) worst-case.
#[cold]
#[unstable(feature = "sort_internals", reason = "internal to sort module", issue = "none")]
pub fn heapsort<T, F>(v: &mut [T], mut is_less: F)
where
    F: FnMut(&T, &T) -> bool,
{
    // This binary heap respects the invariant `parent >= child`.
    let mut sift_down = |v: &mut [T], mut node| {
        loop {
            // Children of `node`.
            let mut child = 2 * node + 1;
            if child >= v.len() {
                break;
            }

            // Choose the greater child.
            if child + 1 < v.len() && is_less(&v[child], &v[child + 1]) {
                child += 1;
            }

            // Stop if the invariant holds at `node`.
            if !is_less(&v[node], &v[child]) {
                break;
            }

            // Swap `node` with the greater child, move one step down, and continue sifting.
            v.swap(node, child);
            node = child;
        }
    };

    // Build the heap in linear time.
    for i in (0..v.len() / 2).rev() {
        sift_down(v, i);
    }

    // Pop maximal elements from the heap.
    for i in (1..v.len()).rev() {
        v.swap(0, i);
        sift_down(&mut v[..i], 0);
    }
}

/// Partitions `v` into elements smaller than `pivot`, followed by elements greater than or equal
/// to `pivot`.
///
/// Returns the number of elements smaller than `pivot`.
///
/// Partitioning is performed block-by-block in order to minimize the cost of branching operations.
/// This idea is presented in the [BlockQuicksort][pdf] paper.
///
/// [pdf]: https://drops.dagstuhl.de/opus/volltexte/2016/6389/pdf/LIPIcs-ESA-2016-38.pdf
fn partition_in_blocks<T, F>(v: &mut [T], pivot: &T, is_less: &mut F) -> usize
where
    F: FnMut(&T, &T) -> bool,
{
    // Number of elements in a typical block.
    const BLOCK: usize = 128;

    // The partitioning algorithm repeats the following steps until completion:
    //
    // 1. Trace a block from the left side to identify elements greater than or equal to the pivot.
    // 2. Trace a block from the right side to identify elements smaller than the pivot.
    // 3. Exchange the identified elements between the left and right side.
    //
    // We keep the following variables for a block of elements:
    //
    // 1. `block` - Number of elements in the block.
    // 2. `start` - Start pointer into the `offsets` array.
    // 3. `end` - End pointer into the `offsets` array.
    // 4. `offsets - Indices of out-of-order elements within the block.

    // The current block on the left side (from `l` to `l.add(block_l)`).
    let mut l = v.as_mut_ptr();
    let mut block_l = BLOCK;
    let mut start_l = ptr::null_mut();
    let mut end_l = ptr::null_mut();
    let mut offsets_l = [MaybeUninit::<u8>::uninit(); BLOCK];

    // The current block on the right side (from `r.sub(block_r)` to `r`).
    // SAFETY: The documentation for .add() specifically mention that `vec.as_ptr().add(vec.len())` is always safe`
    let mut r = unsafe { l.add(v.len()) };
    let mut block_r = BLOCK;
    let mut start_r = ptr::null_mut();
    let mut end_r = ptr::null_mut();
    let mut offsets_r = [MaybeUninit::<u8>::uninit(); BLOCK];

    // FIXME: When we get VLAs, try creating one array of length `min(v.len(), 2 * BLOCK)` rather
    // than two fixed-size arrays of length `BLOCK`. VLAs might be more cache-efficient.

    // Returns the number of elements between pointers `l` (inclusive) and `r` (exclusive).
    fn width<T>(l: *mut T, r: *mut T) -> usize {
        assert!(mem::size_of::<T>() > 0);
        // FIXME: this should *likely* use `offset_from`, but more
        // investigation is needed (including running tests in miri).
        (r.addr() - l.addr()) / mem::size_of::<T>()
    }

    loop {
        // We are done with partitioning block-by-block when `l` and `r` get very close. Then we do
        // some patch-up work in order to partition the remaining elements in between.
        let is_done = width(l, r) <= 2 * BLOCK;

        if is_done {
            // Number of remaining elements (still not compared to the pivot).
            let mut rem = width(l, r);
            if start_l < end_l || start_r < end_r {
                rem -= BLOCK;
            }

            // Adjust block sizes so that the left and right block don't overlap, but get perfectly
            // aligned to cover the whole remaining gap.
            if start_l < end_l {
                block_r = rem;
            } else if start_r < end_r {
                block_l = rem;
            } else {
                // There were the same number of elements to switch on both blocks during the last
                // iteration, so there are no remaining elements on either block. Cover the remaining
                // items with roughly equally-sized blocks.
                block_l = rem / 2;
                block_r = rem - block_l;
            }
            debug_assert!(block_l <= BLOCK && block_r <= BLOCK);
            debug_assert!(width(l, r) == block_l + block_r);
        }

        if start_l == end_l {
            // Trace `block_l` elements from the left side.
            start_l = MaybeUninit::slice_as_mut_ptr(&mut offsets_l);
            end_l = start_l;
            let mut elem = l;

            for i in 0..block_l {
                // SAFETY: The unsafety operations below involve the usage of the `offset`.
                //         According to the conditions required by the function, we satisfy them because:
                //         1. `offsets_l` is stack-allocated, and thus considered separate allocated object.
                //         2. The function `is_less` returns a `bool`.
                //            Casting a `bool` will never overflow `isize`.
                //         3. We have guaranteed that `block_l` will be `<= BLOCK`.
                //            Plus, `end_l` was initially set to the begin pointer of `offsets_` which was declared on the stack.
                //            Thus, we know that even in the worst case (all invocations of `is_less` returns false) we will only be at most 1 byte pass the end.
                //        Another unsafety operation here is dereferencing `elem`.
                //        However, `elem` was initially the begin pointer to the slice which is always valid.
                unsafe {
                    // Branchless comparison.
                    *end_l = i as u8;
                    end_l = end_l.add(!is_less(&*elem, pivot) as usize);
                    elem = elem.add(1);
                }
            }
        }

        if start_r == end_r {
            // Trace `block_r` elements from the right side.
            start_r = MaybeUninit::slice_as_mut_ptr(&mut offsets_r);
            end_r = start_r;
            let mut elem = r;

            for i in 0..block_r {
                // SAFETY: The unsafety operations below involve the usage of the `offset`.
                //         According to the conditions required by the function, we satisfy them because:
                //         1. `offsets_r` is stack-allocated, and thus considered separate allocated object.
                //         2. The function `is_less` returns a `bool`.
                //            Casting a `bool` will never overflow `isize`.
                //         3. We have guaranteed that `block_r` will be `<= BLOCK`.
                //            Plus, `end_r` was initially set to the begin pointer of `offsets_` which was declared on the stack.
                //            Thus, we know that even in the worst case (all invocations of `is_less` returns true) we will only be at most 1 byte pass the end.
                //        Another unsafety operation here is dereferencing `elem`.
                //        However, `elem` was initially `1 * sizeof(T)` past the end and we decrement it by `1 * sizeof(T)` before accessing it.
                //        Plus, `block_r` was asserted to be less than `BLOCK` and `elem` will therefore at most be pointing to the beginning of the slice.
                unsafe {
                    // Branchless comparison.
                    elem = elem.sub(1);
                    *end_r = i as u8;
                    end_r = end_r.add(is_less(&*elem, pivot) as usize);
                }
            }
        }

        // Number of out-of-order elements to swap between the left and right side.
        let count = cmp::min(width(start_l, end_l), width(start_r, end_r));

        if count > 0 {
            macro_rules! left {
                () => {
                    l.add(usize::from(*start_l))
                };
            }
            macro_rules! right {
                () => {
                    r.sub(usize::from(*start_r) + 1)
                };
            }

            // Instead of swapping one pair at the time, it is more efficient to perform a cyclic
            // permutation. This is not strictly equivalent to swapping, but produces a similar
            // result using fewer memory operations.

            // SAFETY: The use of `ptr::read` is valid because there is at least one element in
            // both `offsets_l` and `offsets_r`, so `left!` is a valid pointer to read from.
            //
            // The uses of `left!` involve calls to `offset` on `l`, which points to the
            // beginning of `v`. All the offsets pointed-to by `start_l` are at most `block_l`, so
            // these `offset` calls are safe as all reads are within the block. The same argument
            // applies for the uses of `right!`.
            //
            // The calls to `start_l.offset` are valid because there are at most `count-1` of them,
            // plus the final one at the end of the unsafe block, where `count` is the minimum number
            // of collected offsets in `offsets_l` and `offsets_r`, so there is no risk of there not
            // being enough elements. The same reasoning applies to the calls to `start_r.offset`.
            //
            // The calls to `copy_nonoverlapping` are safe because `left!` and `right!` are guaranteed
            // not to overlap, and are valid because of the reasoning above.
            unsafe {
                let tmp = ptr::read(left!());
                ptr::copy_nonoverlapping(right!(), left!(), 1);

                for _ in 1..count {
                    start_l = start_l.add(1);
                    ptr::copy_nonoverlapping(left!(), right!(), 1);
                    start_r = start_r.add(1);
                    ptr::copy_nonoverlapping(right!(), left!(), 1);
                }

                ptr::copy_nonoverlapping(&tmp, right!(), 1);
                mem::forget(tmp);
                start_l = start_l.add(1);
                start_r = start_r.add(1);
            }
        }

        if start_l == end_l {
            // All out-of-order elements in the left block were moved. Move to the next block.

            // block-width-guarantee
            // SAFETY: if `!is_done` then the slice width is guaranteed to be at least `2*BLOCK` wide. There
            // are at most `BLOCK` elements in `offsets_l` because of its size, so the `offset` operation is
            // safe. Otherwise, the debug assertions in the `is_done` case guarantee that
            // `width(l, r) == block_l + block_r`, namely, that the block sizes have been adjusted to account
            // for the smaller number of remaining elements.
            l = unsafe { l.add(block_l) };
        }

        if start_r == end_r {
            // All out-of-order elements in the right block were moved. Move to the previous block.

            // SAFETY: Same argument as [block-width-guarantee]. Either this is a full block `2*BLOCK`-wide,
            // or `block_r` has been adjusted for the last handful of elements.
            r = unsafe { r.sub(block_r) };
        }

        if is_done {
            break;
        }
    }

    // All that remains now is at most one block (either the left or the right) with out-of-order
    // elements that need to be moved. Such remaining elements can be simply shifted to the end
    // within their block.

    if start_l < end_l {
        // The left block remains.
        // Move its remaining out-of-order elements to the far right.
        debug_assert_eq!(width(l, r), block_l);
        while start_l < end_l {
            // remaining-elements-safety
            // SAFETY: while the loop condition holds there are still elements in `offsets_l`, so it
            // is safe to point `end_l` to the previous element.
            //
            // The `ptr::swap` is safe if both its arguments are valid for reads and writes:
            //  - Per the debug assert above, the distance between `l` and `r` is `block_l`
            //    elements, so there can be at most `block_l` remaining offsets between `start_l`
            //    and `end_l`. This means `r` will be moved at most `block_l` steps back, which
            //    makes the `r.offset` calls valid (at that point `l == r`).
            //  - `offsets_l` contains valid offsets into `v` collected during the partitioning of
            //    the last block, so the `l.offset` calls are valid.
            unsafe {
                end_l = end_l.sub(1);
                ptr::swap(l.add(usize::from(*end_l)), r.sub(1));
                r = r.sub(1);
            }
        }
        width(v.as_mut_ptr(), r)
    } else if start_r < end_r {
        // The right block remains.
        // Move its remaining out-of-order elements to the far left.
        debug_assert_eq!(width(l, r), block_r);
        while start_r < end_r {
            // SAFETY: See the reasoning in [remaining-elements-safety].
            unsafe {
                end_r = end_r.sub(1);
                ptr::swap(l, r.sub(usize::from(*end_r) + 1));
                l = l.add(1);
            }
        }
        width(v.as_mut_ptr(), l)
    } else {
        // Nothing else to do, we're done.
        width(v.as_mut_ptr(), l)
    }
}

/// Partitions `v` into elements smaller than `v[pivot]`, followed by elements greater than or
/// equal to `v[pivot]`.
///
/// Returns a tuple of:
///
/// 1. Number of elements smaller than `v[pivot]`.
/// 2. True if `v` was already partitioned.
fn partition<T, F>(v: &mut [T], pivot: usize, is_less: &mut F) -> (usize, bool)
where
    F: FnMut(&T, &T) -> bool,
{
    let (mid, was_partitioned) = {
        // Place the pivot at the beginning of slice.
        v.swap(0, pivot);
        let (pivot, v) = v.split_at_mut(1);
        let pivot = &mut pivot[0];

        // Read the pivot into a stack-allocated variable for efficiency. If a following comparison
        // operation panics, the pivot will be automatically written back into the slice.

        // SAFETY: `pivot` is a reference to the first element of `v`, so `ptr::read` is safe.
        let tmp = mem::ManuallyDrop::new(unsafe { ptr::read(pivot) });
        let _pivot_guard = CopyOnDrop { src: &*tmp, dest: pivot };
        let pivot = &*tmp;

        // Find the first pair of out-of-order elements.
        let mut l = 0;
        let mut r = v.len();

        // SAFETY: The unsafety below involves indexing an array.
        // For the first one: We already do the bounds checking here with `l < r`.
        // For the second one: We initially have `l == 0` and `r == v.len()` and we checked that `l < r` at every indexing operation.
        //                     From here we know that `r` must be at least `r == l` which was shown to be valid from the first one.
        unsafe {
            // Find the first element greater than or equal to the pivot.
            while l < r && is_less(v.get_unchecked(l), pivot) {
                l += 1;
            }

            // Find the last element smaller that the pivot.
            while l < r && !is_less(v.get_unchecked(r - 1), pivot) {
                r -= 1;
            }
        }

        (l + partition_in_blocks(&mut v[l..r], pivot, is_less), l >= r)

        // `_pivot_guard` goes out of scope and writes the pivot (which is a stack-allocated
        // variable) back into the slice where it originally was. This step is critical in ensuring
        // safety!
    };

    // Place the pivot between the two partitions.
    v.swap(0, mid);

    (mid, was_partitioned)
}

/// Partitions `v` into elements equal to `v[pivot]` followed by elements greater than `v[pivot]`.
///
/// Returns the number of elements equal to the pivot. It is assumed that `v` does not contain
/// elements smaller than the pivot.
fn partition_equal<T, F>(v: &mut [T], pivot: usize, is_less: &mut F) -> usize
where
    F: FnMut(&T, &T) -> bool,
{
    // Place the pivot at the beginning of slice.
    v.swap(0, pivot);
    let (pivot, v) = v.split_at_mut(1);
    let pivot = &mut pivot[0];

    // Read the pivot into a stack-allocated variable for efficiency. If a following comparison
    // operation panics, the pivot will be automatically written back into the slice.
    // SAFETY: The pointer here is valid because it is obtained from a reference to a slice.
    let tmp = mem::ManuallyDrop::new(unsafe { ptr::read(pivot) });
    let _pivot_guard = CopyOnDrop { src: &*tmp, dest: pivot };
    let pivot = &*tmp;

    // Now partition the slice.
    let mut l = 0;
    let mut r = v.len();
    loop {
        // SAFETY: The unsafety below involves indexing an array.
        // For the first one: We already do the bounds checking here with `l < r`.
        // For the second one: We initially have `l == 0` and `r == v.len()` and we checked that `l < r` at every indexing operation.
        //                     From here we know that `r` must be at least `r == l` which was shown to be valid from the first one.
        unsafe {
            // Find the first element greater than the pivot.
            while l < r && !is_less(pivot, v.get_unchecked(l)) {
                l += 1;
            }

            // Find the last element equal to the pivot.
            while l < r && is_less(pivot, v.get_unchecked(r - 1)) {
                r -= 1;
            }

            // Are we done?
            if l >= r {
                break;
            }

            // Swap the found pair of out-of-order elements.
            r -= 1;
            let ptr = v.as_mut_ptr();
            ptr::swap(ptr.add(l), ptr.add(r));
            l += 1;
        }
    }

    // We found `l` elements equal to the pivot. Add 1 to account for the pivot itself.
    l + 1

    // `_pivot_guard` goes out of scope and writes the pivot (which is a stack-allocated variable)
    // back into the slice where it originally was. This step is critical in ensuring safety!
}

/// Scatters some elements around in an attempt to break patterns that might cause imbalanced
/// partitions in quicksort.
#[cold]
fn break_patterns<T>(v: &mut [T]) {
    let len = v.len();
    if len >= 8 {
        // Pseudorandom number generator from the "Xorshift RNGs" paper by George Marsaglia.
        let mut random = len as u32;
        let mut gen_u32 = || {
            random ^= random << 13;
            random ^= random >> 17;
            random ^= random << 5;
            random
        };
        let mut gen_usize = || {
            if usize::BITS <= 32 {
                gen_u32() as usize
            } else {
                (((gen_u32() as u64) << 32) | (gen_u32() as u64)) as usize
            }
        };

        // Take random numbers modulo this number.
        // The number fits into `usize` because `len` is not greater than `isize::MAX`.
        let modulus = len.next_power_of_two();

        // Some pivot candidates will be in the nearby of this index. Let's randomize them.
        let pos = len / 4 * 2;

        for i in 0..3 {
            // Generate a random number modulo `len`. However, in order to avoid costly operations
            // we first take it modulo a power of two, and then decrease by `len` until it fits
            // into the range `[0, len - 1]`.
            let mut other = gen_usize() & (modulus - 1);

            // `other` is guaranteed to be less than `2 * len`.
            if other >= len {
                other -= len;
            }

            v.swap(pos - 1 + i, other);
        }
    }
}

/// Chooses a pivot in `v` and returns the index and `true` if the slice is likely already sorted.
///
/// Elements in `v` might be reordered in the process.
fn choose_pivot<T, F>(v: &mut [T], is_less: &mut F) -> (usize, bool)
where
    F: FnMut(&T, &T) -> bool,
{
    // Minimum length to choose the median-of-medians method.
    // Shorter slices use the simple median-of-three method.
    const SHORTEST_MEDIAN_OF_MEDIANS: usize = 50;
    // Maximum number of swaps that can be performed in this function.
    const MAX_SWAPS: usize = 4 * 3;

    let len = v.len();

    // Three indices near which we are going to choose a pivot.
    let mut a = len / 4 * 1;
    let mut b = len / 4 * 2;
    let mut c = len / 4 * 3;

    // Counts the total number of swaps we are about to perform while sorting indices.
    let mut swaps = 0;

    if len >= 8 {
        // Swaps indices so that `v[a] <= v[b]`.
        // SAFETY: `len >= 8` so there are at least two elements in the neighborhoods of
        // `a`, `b` and `c`. This means the three calls to `sort_adjacent` result in
        // corresponding calls to `sort3` with valid 3-item neighborhoods around each
        // pointer, which in turn means the calls to `sort2` are done with valid
        // references. Thus the `v.get_unchecked` calls are safe, as is the `ptr::swap`
        // call.
        let mut sort2 = |a: &mut usize, b: &mut usize| unsafe {
            if is_less(v.get_unchecked(*b), v.get_unchecked(*a)) {
                ptr::swap(a, b);
                swaps += 1;
            }
        };

        // Swaps indices so that `v[a] <= v[b] <= v[c]`.
        let mut sort3 = |a: &mut usize, b: &mut usize, c: &mut usize| {
            sort2(a, b);
            sort2(b, c);
            sort2(a, b);
        };

        if len >= SHORTEST_MEDIAN_OF_MEDIANS {
            // Finds the median of `v[a - 1], v[a], v[a + 1]` and stores the index into `a`.
            let mut sort_adjacent = |a: &mut usize| {
                let tmp = *a;
                sort3(&mut (tmp - 1), a, &mut (tmp + 1));
            };

            // Find medians in the neighborhoods of `a`, `b`, and `c`.
            sort_adjacent(&mut a);
            sort_adjacent(&mut b);
            sort_adjacent(&mut c);
        }

        // Find the median among `a`, `b`, and `c`.
        sort3(&mut a, &mut b, &mut c);
    }

    if swaps < MAX_SWAPS {
        (b, swaps == 0)
    } else {
        // The maximum number of swaps was performed. Chances are the slice is descending or mostly
        // descending, so reversing will probably help sort it faster.
        v.reverse();
        (len - 1 - b, true)
    }
}

/// Sorts `v` recursively.
///
/// If the slice had a predecessor in the original array, it is specified as `pred`.
///
/// `limit` is the number of allowed imbalanced partitions before switching to `heapsort`. If zero,
/// this function will immediately switch to heapsort.
fn recurse<'a, T, F>(mut v: &'a mut [T], is_less: &mut F, mut pred: Option<&'a T>, mut limit: u32)
where
    F: FnMut(&T, &T) -> bool,
{
    // Slices of up to this length get sorted using insertion sort.
    const MAX_INSERTION: usize = 20;

    // True if the last partitioning was reasonably balanced.
    let mut was_balanced = true;
    // True if the last partitioning didn't shuffle elements (the slice was already partitioned).
    let mut was_partitioned = true;

    loop {
        let len = v.len();

        // Very short slices get sorted using insertion sort.
        if len <= MAX_INSERTION {
            insertion_sort(v, is_less);
            return;
        }

        // If too many bad pivot choices were made, simply fall back to heapsort in order to
        // guarantee `O(n * log(n))` worst-case.
        if limit == 0 {
            heapsort(v, is_less);
            return;
        }

        // If the last partitioning was imbalanced, try breaking patterns in the slice by shuffling
        // some elements around. Hopefully we'll choose a better pivot this time.
        if !was_balanced {
            break_patterns(v);
            limit -= 1;
        }

        // Choose a pivot and try guessing whether the slice is already sorted.
        let (pivot, likely_sorted) = choose_pivot(v, is_less);

        // If the last partitioning was decently balanced and didn't shuffle elements, and if pivot
        // selection predicts the slice is likely already sorted...
        if was_balanced && was_partitioned && likely_sorted {
            // Try identifying several out-of-order elements and shifting them to correct
            // positions. If the slice ends up being completely sorted, we're done.
            if partial_insertion_sort(v, is_less) {
                return;
            }
        }

        // If the chosen pivot is equal to the predecessor, then it's the smallest element in the
        // slice. Partition the slice into elements equal to and elements greater than the pivot.
        // This case is usually hit when the slice contains many duplicate elements.
        if let Some(p) = pred {
            if !is_less(p, &v[pivot]) {
                let mid = partition_equal(v, pivot, is_less);

                // Continue sorting elements greater than the pivot.
                v = &mut v[mid..];
                continue;
            }
        }

        // Partition the slice.
        let (mid, was_p) = partition(v, pivot, is_less);
        was_balanced = cmp::min(mid, len - mid) >= len / 8;
        was_partitioned = was_p;

        // Split the slice into `left`, `pivot`, and `right`.
        let (left, right) = v.split_at_mut(mid);
        let (pivot, right) = right.split_at_mut(1);
        let pivot = &pivot[0];

        // Recurse into the shorter side only in order to minimize the total number of recursive
        // calls and consume less stack space. Then just continue with the longer side (this is
        // akin to tail recursion).
        if left.len() < right.len() {
            recurse(left, is_less, pred, limit);
            v = right;
            pred = Some(pivot);
        } else {
            recurse(right, is_less, Some(pivot), limit);
            v = left;
        }
    }
}

/// Sorts `v` using pattern-defeating quicksort, which is *O*(*n* \* log(*n*)) worst-case.
pub fn quicksort<T, F>(v: &mut [T], mut is_less: F)
where
    F: FnMut(&T, &T) -> bool,
{
    // Sorting has no meaningful behavior on zero-sized types.
    if T::IS_ZST {
        return;
    }

    // Limit the number of imbalanced partitions to `floor(log2(len)) + 1`.
    let limit = usize::BITS - v.len().leading_zeros();

    recurse(v, &mut is_less, None, limit);
}

fn partition_at_index_loop<'a, T, F>(
    mut v: &'a mut [T],
    mut index: usize,
    is_less: &mut F,
    mut pred: Option<&'a T>,
) where
    F: FnMut(&T, &T) -> bool,
{
    loop {
        // For slices of up to this length it's probably faster to simply sort them.
        const MAX_INSERTION: usize = 10;
        if v.len() <= MAX_INSERTION {
            insertion_sort(v, is_less);
            return;
        }

        // Choose a pivot
        let (pivot, _) = choose_pivot(v, is_less);

        // If the chosen pivot is equal to the predecessor, then it's the smallest element in the
        // slice. Partition the slice into elements equal to and elements greater than the pivot.
        // This case is usually hit when the slice contains many duplicate elements.
        if let Some(p) = pred {
            if !is_less(p, &v[pivot]) {
                let mid = partition_equal(v, pivot, is_less);

                // If we've passed our index, then we're good.
                if mid > index {
                    return;
                }

                // Otherwise, continue sorting elements greater than the pivot.
                v = &mut v[mid..];
                index = index - mid;
                pred = None;
                continue;
            }
        }

        let (mid, _) = partition(v, pivot, is_less);

        // Split the slice into `left`, `pivot`, and `right`.
        let (left, right) = v.split_at_mut(mid);
        let (pivot, right) = right.split_at_mut(1);
        let pivot = &pivot[0];

        if mid < index {
            v = right;
            index = index - mid - 1;
            pred = Some(pivot);
        } else if mid > index {
            v = left;
        } else {
            // If mid == index, then we're done, since partition() guaranteed that all elements
            // after mid are greater than or equal to mid.
            return;
        }
    }
}

pub fn partition_at_index<T, F>(
    v: &mut [T],
    index: usize,
    mut is_less: F,
) -> (&mut [T], &mut T, &mut [T])
where
    F: FnMut(&T, &T) -> bool,
{
    use cmp::Ordering::Greater;
    use cmp::Ordering::Less;

    if index >= v.len() {
        panic!("partition_at_index index {} greater than length of slice {}", index, v.len());
    }

    if T::IS_ZST {
        // Sorting has no meaningful behavior on zero-sized types. Do nothing.
    } else if index == v.len() - 1 {
        // Find max element and place it in the last position of the array. We're free to use
        // `unwrap()` here because we know v must not be empty.
        let (max_index, _) = v
            .iter()
            .enumerate()
            .max_by(|&(_, x), &(_, y)| if is_less(x, y) { Less } else { Greater })
            .unwrap();
        v.swap(max_index, index);
    } else if index == 0 {
        // Find min element and place it in the first position of the array. We're free to use
        // `unwrap()` here because we know v must not be empty.
        let (min_index, _) = v
            .iter()
            .enumerate()
            .min_by(|&(_, x), &(_, y)| if is_less(x, y) { Less } else { Greater })
            .unwrap();
        v.swap(min_index, index);
    } else {
        partition_at_index_loop(v, index, &mut is_less, None);
    }

    let (left, right) = v.split_at_mut(index);
    let (pivot, right) = right.split_at_mut(1);
    let pivot = &mut pivot[0];
    (left, pivot, right)
}