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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 core because it doesn't allocate memory, unlike our
//! stable sorting implementation.
//!
//! In addition it also contains the core logic of the stable sort used by `slice::sort` based on
//! TimSort.
use crate::cmp;
use crate::mem::{self, MaybeUninit, SizedTypeProperties};
use crate::ptr;
// When dropped, copies from `src` into `dest`.
struct InsertionHole<T> {
src: *const T,
dest: *mut T,
}
impl<T> Drop for InsertionHole<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` and are both valid for writes.
unsafe {
ptr::copy_nonoverlapping(self.src, self.dest, 1);
}
}
}
/// Inserts `v[v.len() - 1]` into pre-sorted sequence `v[..v.len() - 1]` so that whole `v[..]`
/// becomes sorted.
unsafe fn insert_tail<T, F>(v: &mut [T], is_less: &mut F)
where
F: FnMut(&T, &T) -> bool,
{
debug_assert!(v.len() >= 2);
let arr_ptr = v.as_mut_ptr();
let i = v.len() - 1;
// SAFETY: caller must ensure v is at least len 2.
unsafe {
// See insert_head which talks about why this approach is beneficial.
let i_ptr = arr_ptr.add(i);
// It's important that we use i_ptr here. If this check is positive and we continue,
// We want to make sure that no other copy of the value was seen by is_less.
// Otherwise we would have to copy it back.
if is_less(&*i_ptr, &*i_ptr.sub(1)) {
// It's important, that we use tmp for comparison from now on. As it is the value that
// will be copied back. And notionally we could have created a divergence if we copy
// back the wrong value.
let tmp = mem::ManuallyDrop::new(ptr::read(i_ptr));
// Intermediate state of the insertion process is always tracked by `hole`, which
// serves two purposes:
// 1. Protects integrity of `v` from panics in `is_less`.
// 2. Fills the remaining hole in `v` in the end.
//
// Panic safety:
//
// If `is_less` panics at any point during the process, `hole` will get dropped and
// fill the hole in `v` with `tmp`, thus ensuring that `v` still holds every object it
// initially held exactly once.
let mut hole = InsertionHole { src: &*tmp, dest: i_ptr.sub(1) };
ptr::copy_nonoverlapping(hole.dest, i_ptr, 1);
// SAFETY: We know i is at least 1.
for j in (0..(i - 1)).rev() {
let j_ptr = arr_ptr.add(j);
if !is_less(&*tmp, &*j_ptr) {
break;
}
ptr::copy_nonoverlapping(j_ptr, hole.dest, 1);
hole.dest = j_ptr;
}
// `hole` gets dropped and thus copies `tmp` into the remaining hole in `v`.
}
}
}
/// Inserts `v[0]` into pre-sorted sequence `v[1..]` so that whole `v[..]` becomes sorted.
///
/// This is the integral subroutine of insertion sort.
unsafe fn insert_head<T, F>(v: &mut [T], is_less: &mut F)
where
F: FnMut(&T, &T) -> bool,
{
debug_assert!(v.len() >= 2);
// SAFETY: caller must ensure v is at least len 2.
unsafe {
if is_less(v.get_unchecked(1), v.get_unchecked(0)) {
let arr_ptr = v.as_mut_ptr();
// There are three ways to implement insertion here:
//
// 1. Swap adjacent elements until the first one gets to its final destination.
// However, this way we copy data around more than is necessary. If elements are big
// structures (costly to copy), this method will be slow.
//
// 2. Iterate until the right place for the first element is found. Then shift the
// elements succeeding it to make room for it and finally place it into the
// remaining hole. This is a good method.
//
// 3. Copy the first element into a temporary variable. Iterate until the right place
// for it is found. As we go along, copy every traversed element into the slot
// preceding it. Finally, copy data from the temporary variable into the remaining
// hole. This method is very good. Benchmarks demonstrated slightly better
// performance than with the 2nd method.
//
// All methods were benchmarked, and the 3rd showed best results. So we chose that one.
let tmp = mem::ManuallyDrop::new(ptr::read(arr_ptr));
// Intermediate state of the insertion process is always tracked by `hole`, which
// serves two purposes:
// 1. Protects integrity of `v` from panics in `is_less`.
// 2. Fills the remaining hole in `v` in the end.
//
// Panic safety:
//
// If `is_less` panics at any point during the process, `hole` will get dropped and
// fill the hole in `v` with `tmp`, thus ensuring that `v` still holds every object it
// initially held exactly once.
let mut hole = InsertionHole { src: &*tmp, dest: arr_ptr.add(1) };
ptr::copy_nonoverlapping(arr_ptr.add(1), arr_ptr.add(0), 1);
for i in 2..v.len() {
if !is_less(&v.get_unchecked(i), &*tmp) {
break;
}
ptr::copy_nonoverlapping(arr_ptr.add(i), arr_ptr.add(i - 1), 1);
hole.dest = arr_ptr.add(i);
}
// `hole` gets dropped and thus copies `tmp` into the remaining hole in `v`.
}
}
}
/// Sort `v` assuming `v[..offset]` is already sorted.
///
/// Never inline this function to avoid code bloat. It still optimizes nicely and has practically no
/// performance impact. Even improving performance in some cases.
#[inline(never)]
pub(super) fn insertion_sort_shift_left<T, F>(v: &mut [T], offset: usize, is_less: &mut F)
where
F: FnMut(&T, &T) -> bool,
{
let len = v.len();
// Using assert here improves performance.
assert!(offset != 0 && offset <= len);
// Shift each element of the unsorted region v[i..] as far left as is needed to make v sorted.
for i in offset..len {
// SAFETY: we tested that `offset` must be at least 1, so this loop is only entered if len
// >= 2. The range is exclusive and we know `i` must be at least 1 so this slice has at
// >least len 2.
unsafe {
insert_tail(&mut v[..=i], is_less);
}
}
}
/// Sort `v` assuming `v[offset..]` is already sorted.
///
/// Never inline this function to avoid code bloat. It still optimizes nicely and has practically no
/// performance impact. Even improving performance in some cases.
#[inline(never)]
fn insertion_sort_shift_right<T, F>(v: &mut [T], offset: usize, is_less: &mut F)
where
F: FnMut(&T, &T) -> bool,
{
let len = v.len();
// Using assert here improves performance.
assert!(offset != 0 && offset <= len && len >= 2);
// Shift each element of the unsorted region v[..i] as far left as is needed to make v sorted.
for i in (0..offset).rev() {
// SAFETY: we tested that `offset` must be at least 1, so this loop is only entered if len
// >= 2.We ensured that the slice length is always at least 2 long. We know that start_found
// will be at least one less than end, and the range is exclusive. Which gives us i always
// <= (end - 2).
unsafe {
insert_head(&mut v[i..len], is_less);
}
}
}
/// 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);
if i >= 2 {
// Shift the smaller element to the left.
insertion_sort_shift_left(&mut v[..i], i - 1, is_less);
// Shift the greater element to the right.
insertion_sort_shift_right(&mut v[..i], 1, is_less);
}
}
// Didn't manage to sort the slice in the limited number of steps.
false
}
/// 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() {
// We need a branch to be sure not to out-of-bounds index,
// but it's highly predictable. The comparison, however,
// is better done branchless, especially for primitives.
child += is_less(&v[child], &v[child + 1]) as usize;
}
// 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.
pub(super) 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 = InsertionHole { 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.
pub(super) 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 = InsertionHole { src: &*tmp, dest: pivot };
let pivot = &*tmp;
let len = v.len();
if len == 0 {
return 0;
}
// Now partition the slice.
let mut l = 0;
let mut r = 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.
loop {
r -= 1;
if l >= r || !is_less(pivot, v.get_unchecked(r)) {
break;
}
}
// Are we done?
if l >= r {
break;
}
// Swap the found pair of out-of-order elements.
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]
pub(super) fn break_patterns<T>(v: &mut [T]) {
let len = v.len();
if len >= 8 {
let mut seed = len;
let mut gen_usize = || {
// Pseudorandom number generator from the "Xorshift RNGs" paper by George Marsaglia.
if usize::BITS <= 32 {
let mut r = seed as u32;
r ^= r << 13;
r ^= r >> 17;
r ^= r << 5;
seed = r as usize;
seed
} else {
let mut r = seed as u64;
r ^= r << 13;
r ^= r >> 7;
r ^= r << 17;
seed = r as usize;
seed
}
};
// 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.
pub(super) 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 {
if len >= 2 {
insertion_sort_shift_left(v, 1, 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);
}
/// Merges non-decreasing runs `v[..mid]` and `v[mid..]` using `buf` as temporary storage, and
/// stores the result into `v[..]`.
///
/// # Safety
///
/// The two slices must be non-empty and `mid` must be in bounds. Buffer `buf` must be long enough
/// to hold a copy of the shorter slice. Also, `T` must not be a zero-sized type.
unsafe fn merge<T, F>(v: &mut [T], mid: usize, buf: *mut T, is_less: &mut F)
where
F: FnMut(&T, &T) -> bool,
{
let len = v.len();
let v = v.as_mut_ptr();
// SAFETY: mid and len must be in-bounds of v.
let (v_mid, v_end) = unsafe { (v.add(mid), v.add(len)) };
// The merge process first copies the shorter run into `buf`. Then it traces the newly copied
// run and the longer run forwards (or backwards), comparing their next unconsumed elements and
// copying the lesser (or greater) one into `v`.
//
// As soon as the shorter run is fully consumed, the process is done. If the longer run gets
// consumed first, then we must copy whatever is left of the shorter run into the remaining
// hole in `v`.
//
// Intermediate state of the process is always tracked by `hole`, which serves two purposes:
// 1. Protects integrity of `v` from panics in `is_less`.
// 2. Fills the remaining hole in `v` if the longer run gets consumed first.
//
// Panic safety:
//
// If `is_less` panics at any point during the process, `hole` will get dropped and fill the
// hole in `v` with the unconsumed range in `buf`, thus ensuring that `v` still holds every
// object it initially held exactly once.
let mut hole;
if mid <= len - mid {
// The left run is shorter.
// SAFETY: buf must have enough capacity for `v[..mid]`.
unsafe {
ptr::copy_nonoverlapping(v, buf, mid);
hole = MergeHole { start: buf, end: buf.add(mid), dest: v };
}
// Initially, these pointers point to the beginnings of their arrays.
let left = &mut hole.start;
let mut right = v_mid;
let out = &mut hole.dest;
while *left < hole.end && right < v_end {
// Consume the lesser side.
// If equal, prefer the left run to maintain stability.
// SAFETY: left and right must be valid and part of v same for out.
unsafe {
let is_l = is_less(&*right, &**left);
let to_copy = if is_l { right } else { *left };
ptr::copy_nonoverlapping(to_copy, *out, 1);
*out = out.add(1);
right = right.add(is_l as usize);
*left = left.add(!is_l as usize);
}
}
} else {
// The right run is shorter.
// SAFETY: buf must have enough capacity for `v[mid..]`.
unsafe {
ptr::copy_nonoverlapping(v_mid, buf, len - mid);
hole = MergeHole { start: buf, end: buf.add(len - mid), dest: v_mid };
}
// Initially, these pointers point past the ends of their arrays.
let left = &mut hole.dest;
let right = &mut hole.end;
let mut out = v_end;
while v < *left && buf < *right {
// Consume the greater side.
// If equal, prefer the right run to maintain stability.
// SAFETY: left and right must be valid and part of v same for out.
unsafe {
let is_l = is_less(&*right.sub(1), &*left.sub(1));
*left = left.sub(is_l as usize);
*right = right.sub(!is_l as usize);
let to_copy = if is_l { *left } else { *right };
out = out.sub(1);
ptr::copy_nonoverlapping(to_copy, out, 1);
}
}
}
// Finally, `hole` gets dropped. If the shorter run was not fully consumed, whatever remains of
// it will now be copied into the hole in `v`.
// When dropped, copies the range `start..end` into `dest..`.
struct MergeHole<T> {
start: *mut T,
end: *mut T,
dest: *mut T,
}
impl<T> Drop for MergeHole<T> {
fn drop(&mut self) {
// SAFETY: `T` is not a zero-sized type, and these are pointers into a slice's elements.
unsafe {
let len = self.end.sub_ptr(self.start);
ptr::copy_nonoverlapping(self.start, self.dest, len);
}
}
}
}
/// This merge sort borrows some (but not all) ideas from TimSort, which used to be described in
/// detail [here](https://github.com/python/cpython/blob/main/Objects/listsort.txt). However Python
/// has switched to a Powersort based implementation.
///
/// The algorithm identifies strictly descending and non-descending subsequences, which are called
/// natural runs. There is a stack of pending runs yet to be merged. Each newly found run is pushed
/// onto the stack, and then some pairs of adjacent runs are merged until these two invariants are
/// satisfied:
///
/// 1. for every `i` in `1..runs.len()`: `runs[i - 1].len > runs[i].len`
/// 2. for every `i` in `2..runs.len()`: `runs[i - 2].len > runs[i - 1].len + runs[i].len`
///
/// The invariants ensure that the total running time is *O*(*n* \* log(*n*)) worst-case.
pub fn merge_sort<T, CmpF, ElemAllocF, ElemDeallocF, RunAllocF, RunDeallocF>(
v: &mut [T],
is_less: &mut CmpF,
elem_alloc_fn: ElemAllocF,
elem_dealloc_fn: ElemDeallocF,
run_alloc_fn: RunAllocF,
run_dealloc_fn: RunDeallocF,
) where
CmpF: FnMut(&T, &T) -> bool,
ElemAllocF: Fn(usize) -> *mut T,
ElemDeallocF: Fn(*mut T, usize),
RunAllocF: Fn(usize) -> *mut TimSortRun,
RunDeallocF: Fn(*mut TimSortRun, usize),
{
// Slices of up to this length get sorted using insertion sort.
const MAX_INSERTION: usize = 20;
// The caller should have already checked that.
debug_assert!(!T::IS_ZST);
let len = v.len();
// Short arrays get sorted in-place via insertion sort to avoid allocations.
if len <= MAX_INSERTION {
if len >= 2 {
insertion_sort_shift_left(v, 1, is_less);
}
return;
}
// Allocate a buffer to use as scratch memory. We keep the length 0 so we can keep in it
// shallow copies of the contents of `v` without risking the dtors running on copies if
// `is_less` panics. When merging two sorted runs, this buffer holds a copy of the shorter run,
// which will always have length at most `len / 2`.
let buf = BufGuard::new(len / 2, elem_alloc_fn, elem_dealloc_fn);
let buf_ptr = buf.buf_ptr.as_ptr();
let mut runs = RunVec::new(run_alloc_fn, run_dealloc_fn);
let mut end = 0;
let mut start = 0;
// Scan forward. Memory pre-fetching prefers forward scanning vs backwards scanning, and the
// code-gen is usually better. For the most sensitive types such as integers, these are merged
// bidirectionally at once. So there is no benefit in scanning backwards.
while end < len {
let (streak_end, was_reversed) = find_streak(&v[start..], is_less);
end += streak_end;
if was_reversed {
v[start..end].reverse();
}
// Insert some more elements into the run if it's too short. Insertion sort is faster than
// merge sort on short sequences, so this significantly improves performance.
end = provide_sorted_batch(v, start, end, is_less);
// Push this run onto the stack.
runs.push(TimSortRun { start, len: end - start });
start = end;
// Merge some pairs of adjacent runs to satisfy the invariants.
while let Some(r) = collapse(runs.as_slice(), len) {
let left = runs[r];
let right = runs[r + 1];
let merge_slice = &mut v[left.start..right.start + right.len];
// SAFETY: `buf_ptr` must hold enough capacity for the shorter of the two sides, and
// neither side may be on length 0.
unsafe {
merge(merge_slice, left.len, buf_ptr, is_less);
}
runs[r + 1] = TimSortRun { start: left.start, len: left.len + right.len };
runs.remove(r);
}
}
// Finally, exactly one run must remain in the stack.
debug_assert!(runs.len() == 1 && runs[0].start == 0 && runs[0].len == len);
// Examines the stack of runs and identifies the next pair of runs to merge. More specifically,
// if `Some(r)` is returned, that means `runs[r]` and `runs[r + 1]` must be merged next. If the
// algorithm should continue building a new run instead, `None` is returned.
//
// TimSort is infamous for its buggy implementations, as described here:
// http://envisage-project.eu/timsort-specification-and-verification/
//
// The gist of the story is: we must enforce the invariants on the top four runs on the stack.
// Enforcing them on just top three is not sufficient to ensure that the invariants will still
// hold for *all* runs in the stack.
//
// This function correctly checks invariants for the top four runs. Additionally, if the top
// run starts at index 0, it will always demand a merge operation until the stack is fully
// collapsed, in order to complete the sort.
#[inline]
fn collapse(runs: &[TimSortRun], stop: usize) -> Option<usize> {
let n = runs.len();
if n >= 2
&& (runs[n - 1].start + runs[n - 1].len == stop
|| runs[n - 2].len <= runs[n - 1].len
|| (n >= 3 && runs[n - 3].len <= runs[n - 2].len + runs[n - 1].len)
|| (n >= 4 && runs[n - 4].len <= runs[n - 3].len + runs[n - 2].len))
{
if n >= 3 && runs[n - 3].len < runs[n - 1].len { Some(n - 3) } else { Some(n - 2) }
} else {
None
}
}
// Extremely basic versions of Vec.
// Their use is super limited and by having the code here, it allows reuse between the sort
// implementations.
struct BufGuard<T, ElemDeallocF>
where
ElemDeallocF: Fn(*mut T, usize),
{
buf_ptr: ptr::NonNull<T>,
capacity: usize,
elem_dealloc_fn: ElemDeallocF,
}
impl<T, ElemDeallocF> BufGuard<T, ElemDeallocF>
where
ElemDeallocF: Fn(*mut T, usize),
{
fn new<ElemAllocF>(
len: usize,
elem_alloc_fn: ElemAllocF,
elem_dealloc_fn: ElemDeallocF,
) -> Self
where
ElemAllocF: Fn(usize) -> *mut T,
{
Self {
buf_ptr: ptr::NonNull::new(elem_alloc_fn(len)).unwrap(),
capacity: len,
elem_dealloc_fn,
}
}
}
impl<T, ElemDeallocF> Drop for BufGuard<T, ElemDeallocF>
where
ElemDeallocF: Fn(*mut T, usize),
{
fn drop(&mut self) {
(self.elem_dealloc_fn)(self.buf_ptr.as_ptr(), self.capacity);
}
}
struct RunVec<RunAllocF, RunDeallocF>
where
RunAllocF: Fn(usize) -> *mut TimSortRun,
RunDeallocF: Fn(*mut TimSortRun, usize),
{
buf_ptr: ptr::NonNull<TimSortRun>,
capacity: usize,
len: usize,
run_alloc_fn: RunAllocF,
run_dealloc_fn: RunDeallocF,
}
impl<RunAllocF, RunDeallocF> RunVec<RunAllocF, RunDeallocF>
where
RunAllocF: Fn(usize) -> *mut TimSortRun,
RunDeallocF: Fn(*mut TimSortRun, usize),
{
fn new(run_alloc_fn: RunAllocF, run_dealloc_fn: RunDeallocF) -> Self {
// Most slices can be sorted with at most 16 runs in-flight.
const START_RUN_CAPACITY: usize = 16;
Self {
buf_ptr: ptr::NonNull::new(run_alloc_fn(START_RUN_CAPACITY)).unwrap(),
capacity: START_RUN_CAPACITY,
len: 0,
run_alloc_fn,
run_dealloc_fn,
}
}
fn push(&mut self, val: TimSortRun) {
if self.len == self.capacity {
let old_capacity = self.capacity;
let old_buf_ptr = self.buf_ptr.as_ptr();
self.capacity = self.capacity * 2;
self.buf_ptr = ptr::NonNull::new((self.run_alloc_fn)(self.capacity)).unwrap();
// SAFETY: buf_ptr new and old were correctly allocated and old_buf_ptr has
// old_capacity valid elements.
unsafe {
ptr::copy_nonoverlapping(old_buf_ptr, self.buf_ptr.as_ptr(), old_capacity);
}
(self.run_dealloc_fn)(old_buf_ptr, old_capacity);
}
// SAFETY: The invariant was just checked.
unsafe {
self.buf_ptr.as_ptr().add(self.len).write(val);
}
self.len += 1;
}
fn remove(&mut self, index: usize) {
if index >= self.len {
panic!("Index out of bounds");
}
// SAFETY: buf_ptr needs to be valid and len invariant upheld.
unsafe {
// the place we are taking from.
let ptr = self.buf_ptr.as_ptr().add(index);
// Shift everything down to fill in that spot.
ptr::copy(ptr.add(1), ptr, self.len - index - 1);
}
self.len -= 1;
}
fn as_slice(&self) -> &[TimSortRun] {
// SAFETY: Safe as long as buf_ptr is valid and len invariant was upheld.
unsafe { &*ptr::slice_from_raw_parts(self.buf_ptr.as_ptr(), self.len) }
}
fn len(&self) -> usize {
self.len
}
}
impl<RunAllocF, RunDeallocF> core::ops::Index<usize> for RunVec<RunAllocF, RunDeallocF>
where
RunAllocF: Fn(usize) -> *mut TimSortRun,
RunDeallocF: Fn(*mut TimSortRun, usize),
{
type Output = TimSortRun;
fn index(&self, index: usize) -> &Self::Output {
if index < self.len {
// SAFETY: buf_ptr and len invariant must be upheld.
unsafe {
return &*(self.buf_ptr.as_ptr().add(index));
}
}
panic!("Index out of bounds");
}
}
impl<RunAllocF, RunDeallocF> core::ops::IndexMut<usize> for RunVec<RunAllocF, RunDeallocF>
where
RunAllocF: Fn(usize) -> *mut TimSortRun,
RunDeallocF: Fn(*mut TimSortRun, usize),
{
fn index_mut(&mut self, index: usize) -> &mut Self::Output {
if index < self.len {
// SAFETY: buf_ptr and len invariant must be upheld.
unsafe {
return &mut *(self.buf_ptr.as_ptr().add(index));
}
}
panic!("Index out of bounds");
}
}
impl<RunAllocF, RunDeallocF> Drop for RunVec<RunAllocF, RunDeallocF>
where
RunAllocF: Fn(usize) -> *mut TimSortRun,
RunDeallocF: Fn(*mut TimSortRun, usize),
{
fn drop(&mut self) {
// As long as TimSortRun is Copy we don't need to drop them individually but just the
// whole allocation.
(self.run_dealloc_fn)(self.buf_ptr.as_ptr(), self.capacity);
}
}
}
/// Internal type used by merge_sort.
#[derive(Clone, Copy, Debug)]
pub struct TimSortRun {
len: usize,
start: usize,
}
/// Takes a range as denoted by start and end, that is already sorted and extends it to the right if
/// necessary with sorts optimized for smaller ranges such as insertion sort.
fn provide_sorted_batch<T, F>(v: &mut [T], start: usize, mut end: usize, is_less: &mut F) -> usize
where
F: FnMut(&T, &T) -> bool,
{
let len = v.len();
assert!(end >= start && end <= len);
// This value is a balance between least comparisons and best performance, as
// influenced by for example cache locality.
const MIN_INSERTION_RUN: usize = 10;
// Insert some more elements into the run if it's too short. Insertion sort is faster than
// merge sort on short sequences, so this significantly improves performance.
let start_end_diff = end - start;
if start_end_diff < MIN_INSERTION_RUN && end < len {
// v[start_found..end] are elements that are already sorted in the input. We want to extend
// the sorted region to the left, so we push up MIN_INSERTION_RUN - 1 to the right. Which is
// more efficient that trying to push those already sorted elements to the left.
end = cmp::min(start + MIN_INSERTION_RUN, len);
let presorted_start = cmp::max(start_end_diff, 1);
insertion_sort_shift_left(&mut v[start..end], presorted_start, is_less);
}
end
}
/// Finds a streak of presorted elements starting at the beginning of the slice. Returns the first
/// value that is not part of said streak, and a bool denoting whether the streak was reversed.
/// Streaks can be increasing or decreasing.
fn find_streak<T, F>(v: &[T], is_less: &mut F) -> (usize, bool)
where
F: FnMut(&T, &T) -> bool,
{
let len = v.len();
if len < 2 {
return (len, false);
}
let mut end = 2;
// SAFETY: See below specific.
unsafe {
// SAFETY: We checked that len >= 2, so 0 and 1 are valid indices.
let assume_reverse = is_less(v.get_unchecked(1), v.get_unchecked(0));
// SAFETY: We know end >= 2 and check end < len.
// From that follows that accessing v at end and end - 1 is safe.
if assume_reverse {
while end < len && is_less(v.get_unchecked(end), v.get_unchecked(end - 1)) {
end += 1;
}
(end, true)
} else {
while end < len && !is_less(v.get_unchecked(end), v.get_unchecked(end - 1)) {
end += 1;
}
(end, false)
}
}
}