sort.rs.html -- source
Description: Source to the Rust file `src/libcore/slice/sort.rs`.
Keywords: rust, rustlang, rust-lang

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    //! 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 libcore because it doesn't allocate memory, unlike our
    //! stable sorting implementation.

    // ignore-tidy-undocumented-unsafe

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

    /// When dropped, copies from `src` into `dest`.
    struct CopyOnDrop<T> {
        src: *mut 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 bound check (`get_unchecked` and `get_unchecked_mut`)
        // 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 mut tmp = mem::ManuallyDrop::new(ptr::read(v.get_unchecked(0)));
                let mut hole = CopyOnDrop { src: &mut *tmp, dest: v.get_unchecked_mut(1) };
                ptr::copy_nonoverlapping(v.get_unchecked(1), v.get_unchecked_mut(0), 1);

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

                    // Move `i`-th element one place to the left, thus shifting the hole to the right.
                    ptr::copy_nonoverlapping(v.get_unchecked(i), v.get_unchecked_mut(i - 1), 1);
                    hole.dest = v.get_unchecked_mut(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 (`get_unchecked` and `get_unchecked_mut`)
        // 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 mut tmp = mem::ManuallyDrop::new(ptr::read(v.get_unchecked(len - 1)));
                let mut hole = CopyOnDrop { src: &mut *tmp, dest: v.get_unchecked_mut(len - 2) };
                ptr::copy_nonoverlapping(v.get_unchecked(len - 2), v.get_unchecked_mut(len - 1), 1);

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

                    // Move `i`-th element one place to the right, thus shifting the hole to the left.
                    ptr::copy_nonoverlapping(v.get_unchecked(i), v.get_unchecked_mut(i + 1), 1);
                    hole.dest = v.get_unchecked_mut(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]
    pub fn heapsort<T, F>(v: &mut [T], is_less: &mut 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 left = 2 * node + 1;
                let right = 2 * node + 2;

                // Choose the greater child.
                let greater =
                    if right < v.len() && is_less(&v[left], &v[right]) { right } else { left };

                // Stop if the invariant holds at `node`.
                if greater >= v.len() || !is_less(&v[node], &v[greater]) {
                    break;
                }

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

        // 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]: http://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);
            (r as usize - l as usize) / 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 {
                    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::first_ptr_mut(&mut offsets_l);
                end_l = MaybeUninit::first_ptr_mut(&mut offsets_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.offset(!is_less(&*elem, pivot) as isize);
                        elem = elem.offset(1);
                    }
                }
            }

            if start_r == end_r {
                // Trace `block_r` elements from the right side.
                start_r = MaybeUninit::first_ptr_mut(&mut offsets_r);
                end_r = MaybeUninit::first_ptr_mut(&mut offsets_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.offset(-1);
                        *end_r = i as u8;
                        end_r = end_r.offset(is_less(&*elem, pivot) as isize);
                    }
                }
            }

            // 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.offset(*start_l as isize)
                    };
                }
                macro_rules! right {
                    () => {
                        r.offset(-(*start_r as isize) - 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.
                unsafe {
                    let tmp = ptr::read(left!());
                    ptr::copy_nonoverlapping(right!(), left!(), 1);

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

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

            if start_l == end_l {
                // All out-of-order elements in the left block were moved. Move to the next block.
                l = unsafe { l.offset(block_l as isize) };
            }

            if start_r == end_r {
                // All out-of-order elements in the right block were moved. Move to the previous block.
                r = unsafe { r.offset(-(block_r as isize)) };
            }

            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 {
                unsafe {
                    end_l = end_l.offset(-1);
                    ptr::swap(l.offset(*end_l as isize), r.offset(-1));
                    r = r.offset(-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 {
                unsafe {
                    end_r = end_r.offset(-1);
                    ptr::swap(l, r.offset(-(*end_r as isize) - 1));
                    l = l.offset(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.
            let mut tmp = mem::ManuallyDrop::new(unsafe { ptr::read(pivot) });
            let _pivot_guard = CopyOnDrop { src: &mut *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 mut tmp = mem::ManuallyDrop::new(unsafe { ptr::read(pivot) });
        let _pivot_guard = CopyOnDrop { src: &mut *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;
                ptr::swap(v.get_unchecked_mut(l), v.get_unchecked_mut(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 mem::size_of::<usize>() <= 4 {
                    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]`.
            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: usize)
    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 mem::size_of::<T>() == 0 {
            return;
        }

        // Limit the number of imbalanced partitions to `floor(log2(len)) + 1`.
        let limit = mem::size_of::<usize>() * 8 - v.len().leading_zeros() as usize;

        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 mem::size_of::<T>() == 0 {
            // 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)
    }