sort.rs

//! 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,
{
    // Limit the amount of iterations and fall back to heapsort, similarly to `slice::sort_unstable`.
    // This lowers the worst case running time from O(n^2) to O(n log n).
    // FIXME: Investigate whether it would be better to use something like Median of Medians
    // or Fast Deterministic Selection to guarantee O(n) worst case.
    let mut limit = usize::BITS - v.len().leading_zeros();

    // True if the last partitioning was reasonably balanced.
    let mut was_balanced = true;

    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;
        }

        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
        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);
        was_balanced = cmp::min(mid, v.len() - mid) >= v.len() / 8;

        // 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)
}