rust/compiler/rustc_index/src/bit_set.rs

use std::fmt;
use std::iter;
use std::marker::PhantomData;
use std::mem;
use std::ops::{BitAnd, BitAndAssign, BitOrAssign, Bound, Not, Range, RangeBounds, Shl};
use std::rc::Rc;
use std::slice;

use arrayvec::ArrayVec;
use smallvec::{smallvec, SmallVec};

#[cfg(feature = "nightly")]
use rustc_macros::{Decodable_Generic, Encodable_Generic};

use crate::{Idx, IndexVec};

use Chunk::*;

#[cfg(test)]
mod tests;

type Word = u64;
const WORD_BYTES: usize = mem::size_of::<Word>();
const WORD_BITS: usize = WORD_BYTES * 8;

// The choice of chunk size has some trade-offs.
//
// A big chunk size tends to favour cases where many large `ChunkedBitSet`s are
// present, because they require fewer `Chunk`s, reducing the number of
// allocations and reducing peak memory usage. Also, fewer chunk operations are
// required, though more of them might be `Mixed`.
//
// A small chunk size tends to favour cases where many small `ChunkedBitSet`s
// are present, because less space is wasted at the end of the final chunk (if
// it's not full).
const CHUNK_WORDS: usize = 32;
const CHUNK_BITS: usize = CHUNK_WORDS * WORD_BITS; // 2048 bits

/// ChunkSize is small to keep `Chunk` small. The static assertion ensures it's
/// not too small.
type ChunkSize = u16;
const _: () = assert!(CHUNK_BITS <= ChunkSize::MAX as usize);

pub trait BitRelations<Rhs> {
    fn union(&mut self, other: &Rhs) -> bool;
    fn subtract(&mut self, other: &Rhs) -> bool;
    fn intersect(&mut self, other: &Rhs) -> bool;
}

#[inline]
fn inclusive_start_end<T: Idx>(
    range: impl RangeBounds<T>,
    domain: usize,
) -> Option<(usize, usize)> {
    // Both start and end are inclusive.
    let start = match range.start_bound().cloned() {
        Bound::Included(start) => start.index(),
        Bound::Excluded(start) => start.index() + 1,
        Bound::Unbounded => 0,
    };
    let end = match range.end_bound().cloned() {
        Bound::Included(end) => end.index(),
        Bound::Excluded(end) => end.index().checked_sub(1)?,
        Bound::Unbounded => domain - 1,
    };
    assert!(end < domain);
    if start > end {
        return None;
    }
    Some((start, end))
}

macro_rules! bit_relations_inherent_impls {
    () => {
        /// Sets `self = self | other` and returns `true` if `self` changed
        /// (i.e., if new bits were added).
        pub fn union<Rhs>(&mut self, other: &Rhs) -> bool
        where
            Self: BitRelations<Rhs>,
        {
            <Self as BitRelations<Rhs>>::union(self, other)
        }

        /// Sets `self = self - other` and returns `true` if `self` changed.
        /// (i.e., if any bits were removed).
        pub fn subtract<Rhs>(&mut self, other: &Rhs) -> bool
        where
            Self: BitRelations<Rhs>,
        {
            <Self as BitRelations<Rhs>>::subtract(self, other)
        }

        /// Sets `self = self & other` and return `true` if `self` changed.
        /// (i.e., if any bits were removed).
        pub fn intersect<Rhs>(&mut self, other: &Rhs) -> bool
        where
            Self: BitRelations<Rhs>,
        {
            <Self as BitRelations<Rhs>>::intersect(self, other)
        }
    };
}

/// A fixed-size bitset type with a dense representation.
///
/// NOTE: Use [`GrowableBitSet`] if you need support for resizing after creation.
///
/// `T` is an index type, typically a newtyped `usize` wrapper, but it can also
/// just be `usize`.
///
/// All operations that involve an element will panic if the element is equal
/// to or greater than the domain size. All operations that involve two bitsets
/// will panic if the bitsets have differing domain sizes.
///
#[cfg_attr(feature = "nightly", derive(Decodable_Generic, Encodable_Generic))]
#[derive(Eq, PartialEq, Hash)]
pub struct BitSet<T> {
    domain_size: usize,
    words: SmallVec<[Word; 2]>,
    marker: PhantomData<T>,
}

impl<T> BitSet<T> {
    /// Gets the domain size.
    pub fn domain_size(&self) -> usize {
        self.domain_size
    }
}

impl<T: Idx> BitSet<T> {
    /// Creates a new, empty bitset with a given `domain_size`.
    #[inline]
    pub fn new_empty(domain_size: usize) -> BitSet<T> {
        let num_words = num_words(domain_size);
        BitSet { domain_size, words: smallvec![0; num_words], marker: PhantomData }
    }

    /// Creates a new, filled bitset with a given `domain_size`.
    #[inline]
    pub fn new_filled(domain_size: usize) -> BitSet<T> {
        let num_words = num_words(domain_size);
        let mut result =
            BitSet { domain_size, words: smallvec![!0; num_words], marker: PhantomData };
        result.clear_excess_bits();
        result
    }

    /// Clear all elements.
    #[inline]
    pub fn clear(&mut self) {
        self.words.fill(0);
    }

    /// Clear excess bits in the final word.
    fn clear_excess_bits(&mut self) {
        clear_excess_bits_in_final_word(self.domain_size, &mut self.words);
    }

    /// Count the number of set bits in the set.
    pub fn count(&self) -> usize {
        self.words.iter().map(|e| e.count_ones() as usize).sum()
    }

    /// Returns `true` if `self` contains `elem`.
    #[inline]
    pub fn contains(&self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        let (word_index, mask) = word_index_and_mask(elem);
        (self.words[word_index] & mask) != 0
    }

    /// Is `self` is a (non-strict) superset of `other`?
    #[inline]
    pub fn superset(&self, other: &BitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size);
        self.words.iter().zip(&other.words).all(|(a, b)| (a & b) == *b)
    }

    /// Is the set empty?
    #[inline]
    pub fn is_empty(&self) -> bool {
        self.words.iter().all(|a| *a == 0)
    }

    /// Insert `elem`. Returns whether the set has changed.
    #[inline]
    pub fn insert(&mut self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        let (word_index, mask) = word_index_and_mask(elem);
        let word_ref = &mut self.words[word_index];
        let word = *word_ref;
        let new_word = word | mask;
        *word_ref = new_word;
        new_word != word
    }

    #[inline]
    pub fn insert_range(&mut self, elems: impl RangeBounds<T>) {
        let Some((start, end)) = inclusive_start_end(elems, self.domain_size) else {
            return;
        };

        let (start_word_index, start_mask) = word_index_and_mask(start);
        let (end_word_index, end_mask) = word_index_and_mask(end);

        // Set all words in between start and end (exclusively of both).
        for word_index in (start_word_index + 1)..end_word_index {
            self.words[word_index] = !0;
        }

        if start_word_index != end_word_index {
            // Start and end are in different words, so we handle each in turn.
            //
            // We set all leading bits. This includes the start_mask bit.
            self.words[start_word_index] |= !(start_mask - 1);
            // And all trailing bits (i.e. from 0..=end) in the end word,
            // including the end.
            self.words[end_word_index] |= end_mask | (end_mask - 1);
        } else {
            self.words[start_word_index] |= end_mask | (end_mask - start_mask);
        }
    }

    /// Sets all bits to true.
    pub fn insert_all(&mut self) {
        self.words.fill(!0);
        self.clear_excess_bits();
    }

    /// Returns `true` if the set has changed.
    #[inline]
    pub fn remove(&mut self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        let (word_index, mask) = word_index_and_mask(elem);
        let word_ref = &mut self.words[word_index];
        let word = *word_ref;
        let new_word = word & !mask;
        *word_ref = new_word;
        new_word != word
    }

    /// Iterates over the indices of set bits in a sorted order.
    #[inline]
    pub fn iter(&self) -> BitIter<'_, T> {
        BitIter::new(&self.words)
    }

    /// Set `self = self | other`. In contrast to `union` returns `true` if the set contains at
    /// least one bit that is not in `other` (i.e. `other` is not a superset of `self`).
    ///
    /// This is an optimization for union of a hybrid bitset.
    fn reverse_union_sparse(&mut self, sparse: &SparseBitSet<T>) -> bool {
        assert!(sparse.domain_size == self.domain_size);
        self.clear_excess_bits();

        let mut not_already = false;
        // Index of the current word not yet merged.
        let mut current_index = 0;
        // Mask of bits that came from the sparse set in the current word.
        let mut new_bit_mask = 0;
        for (word_index, mask) in sparse.iter().map(|x| word_index_and_mask(*x)) {
            // Next bit is in a word not inspected yet.
            if word_index > current_index {
                self.words[current_index] |= new_bit_mask;
                // Were there any bits in the old word that did not occur in the sparse set?
                not_already |= (self.words[current_index] ^ new_bit_mask) != 0;
                // Check all words we skipped for any set bit.
                not_already |= self.words[current_index + 1..word_index].iter().any(|&x| x != 0);
                // Update next word.
                current_index = word_index;
                // Reset bit mask, no bits have been merged yet.
                new_bit_mask = 0;
            }
            // Add bit and mark it as coming from the sparse set.
            // self.words[word_index] |= mask;
            new_bit_mask |= mask;
        }
        self.words[current_index] |= new_bit_mask;
        // Any bits in the last inspected word that were not in the sparse set?
        not_already |= (self.words[current_index] ^ new_bit_mask) != 0;
        // Any bits in the tail? Note `clear_excess_bits` before.
        not_already |= self.words[current_index + 1..].iter().any(|&x| x != 0);

        not_already
    }

    pub fn last_set_in(&self, range: impl RangeBounds<T>) -> Option<T> {
        let (start, end) = inclusive_start_end(range, self.domain_size)?;
        let (start_word_index, _) = word_index_and_mask(start);
        let (end_word_index, end_mask) = word_index_and_mask(end);

        let end_word = self.words[end_word_index] & (end_mask | (end_mask - 1));
        if end_word != 0 {
            let pos = max_bit(end_word) + WORD_BITS * end_word_index;
            if start <= pos {
                return Some(T::new(pos));
            }
        }

        // We exclude end_word_index from the range here, because we don't want
        // to limit ourselves to *just* the last word: the bits set it in may be
        // after `end`, so it may not work out.
        if let Some(offset) =
            self.words[start_word_index..end_word_index].iter().rposition(|&w| w != 0)
        {
            let word_idx = start_word_index + offset;
            let start_word = self.words[word_idx];
            let pos = max_bit(start_word) + WORD_BITS * word_idx;
            if start <= pos {
                return Some(T::new(pos));
            }
        }

        None
    }

    bit_relations_inherent_impls! {}
}

// dense REL dense
impl<T: Idx> BitRelations<BitSet<T>> for BitSet<T> {
    fn union(&mut self, other: &BitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size);
        bitwise(&mut self.words, &other.words, |a, b| a | b)
    }

    fn subtract(&mut self, other: &BitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size);
        bitwise(&mut self.words, &other.words, |a, b| a & !b)
    }

    fn intersect(&mut self, other: &BitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size);
        bitwise(&mut self.words, &other.words, |a, b| a & b)
    }
}

impl<T: Idx> From<GrowableBitSet<T>> for BitSet<T> {
    fn from(bit_set: GrowableBitSet<T>) -> Self {
        bit_set.bit_set
    }
}

/// A fixed-size bitset type with a partially dense, partially sparse
/// representation. The bitset is broken into chunks, and chunks that are all
/// zeros or all ones are represented and handled very efficiently.
///
/// This type is especially efficient for sets that typically have a large
/// `domain_size` with significant stretches of all zeros or all ones, and also
/// some stretches with lots of 0s and 1s mixed in a way that causes trouble
/// for `IntervalSet`.
///
/// `T` is an index type, typically a newtyped `usize` wrapper, but it can also
/// just be `usize`.
///
/// All operations that involve an element will panic if the element is equal
/// to or greater than the domain size. All operations that involve two bitsets
/// will panic if the bitsets have differing domain sizes.
#[derive(PartialEq, Eq)]
pub struct ChunkedBitSet<T> {
    domain_size: usize,

    /// The chunks. Each one contains exactly CHUNK_BITS values, except the
    /// last one which contains 1..=CHUNK_BITS values.
    chunks: Box<[Chunk]>,

    marker: PhantomData<T>,
}

// Note: the chunk domain size is duplicated in each variant. This is a bit
// inconvenient, but it allows the type size to be smaller than if we had an
// outer struct containing a chunk domain size plus the `Chunk`, because the
// compiler can place the chunk domain size after the tag.
#[derive(Clone, Debug, PartialEq, Eq)]
enum Chunk {
    /// A chunk that is all zeros; we don't represent the zeros explicitly.
    Zeros(ChunkSize),

    /// A chunk that is all ones; we don't represent the ones explicitly.
    Ones(ChunkSize),

    /// A chunk that has a mix of zeros and ones, which are represented
    /// explicitly and densely. It never has all zeros or all ones.
    ///
    /// If this is the final chunk there may be excess, unused words. This
    /// turns out to be both simpler and have better performance than
    /// allocating the minimum number of words, largely because we avoid having
    /// to store the length, which would make this type larger. These excess
    /// words are always be zero, as are any excess bits in the final in-use
    /// word.
    ///
    /// The second field is the count of 1s set in the chunk, and must satisfy
    /// `0 < count < chunk_domain_size`.
    ///
    /// The words are within an `Rc` because it's surprisingly common to
    /// duplicate an entire chunk, e.g. in `ChunkedBitSet::clone_from()`, or
    /// when a `Mixed` chunk is union'd into a `Zeros` chunk. When we do need
    /// to modify a chunk we use `Rc::make_mut`.
    Mixed(ChunkSize, ChunkSize, Rc<[Word; CHUNK_WORDS]>),
}

// This type is used a lot. Make sure it doesn't unintentionally get bigger.
#[cfg(target_pointer_width = "64")]
crate::static_assert_size!(Chunk, 16);

impl<T> ChunkedBitSet<T> {
    pub fn domain_size(&self) -> usize {
        self.domain_size
    }

    #[cfg(test)]
    fn assert_valid(&self) {
        if self.domain_size == 0 {
            assert!(self.chunks.is_empty());
            return;
        }

        assert!((self.chunks.len() - 1) * CHUNK_BITS <= self.domain_size);
        assert!(self.chunks.len() * CHUNK_BITS >= self.domain_size);
        for chunk in self.chunks.iter() {
            chunk.assert_valid();
        }
    }
}

impl<T: Idx> ChunkedBitSet<T> {
    /// Creates a new bitset with a given `domain_size` and chunk kind.
    fn new(domain_size: usize, is_empty: bool) -> Self {
        let chunks = if domain_size == 0 {
            Box::new([])
        } else {
            // All the chunks have a chunk_domain_size of `CHUNK_BITS` except
            // the final one.
            let final_chunk_domain_size = {
                let n = domain_size % CHUNK_BITS;
                if n == 0 { CHUNK_BITS } else { n }
            };
            let mut chunks =
                vec![Chunk::new(CHUNK_BITS, is_empty); num_chunks(domain_size)].into_boxed_slice();
            *chunks.last_mut().unwrap() = Chunk::new(final_chunk_domain_size, is_empty);
            chunks
        };
        ChunkedBitSet { domain_size, chunks, marker: PhantomData }
    }

    /// Creates a new, empty bitset with a given `domain_size`.
    #[inline]
    pub fn new_empty(domain_size: usize) -> Self {
        ChunkedBitSet::new(domain_size, /* is_empty */ true)
    }

    /// Creates a new, filled bitset with a given `domain_size`.
    #[inline]
    pub fn new_filled(domain_size: usize) -> Self {
        ChunkedBitSet::new(domain_size, /* is_empty */ false)
    }

    #[cfg(test)]
    fn chunks(&self) -> &[Chunk] {
        &self.chunks
    }

    /// Count the number of bits in the set.
    pub fn count(&self) -> usize {
        self.chunks.iter().map(|chunk| chunk.count()).sum()
    }

    /// Returns `true` if `self` contains `elem`.
    #[inline]
    pub fn contains(&self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        let chunk = &self.chunks[chunk_index(elem)];
        match &chunk {
            Zeros(_) => false,
            Ones(_) => true,
            Mixed(_, _, words) => {
                let (word_index, mask) = chunk_word_index_and_mask(elem);
                (words[word_index] & mask) != 0
            }
        }
    }

    #[inline]
    pub fn iter(&self) -> ChunkedBitIter<'_, T> {
        ChunkedBitIter::new(self)
    }

    /// Insert `elem`. Returns whether the set has changed.
    pub fn insert(&mut self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        let chunk_index = chunk_index(elem);
        let chunk = &mut self.chunks[chunk_index];
        match *chunk {
            Zeros(chunk_domain_size) => {
                if chunk_domain_size > 1 {
                    #[cfg(feature = "nightly")]
                    let mut words = {
                        // We take some effort to avoid copying the words.
                        let words = Rc::<[Word; CHUNK_WORDS]>::new_zeroed();
                        // SAFETY: `words` can safely be all zeroes.
                        unsafe { words.assume_init() }
                    };
                    #[cfg(not(feature = "nightly"))]
                    let mut words = {
                        // FIXME: unconditionally use `Rc::new_zeroed` once it is stable (#63291).
                        let words = mem::MaybeUninit::<[Word; CHUNK_WORDS]>::zeroed();
                        // SAFETY: `words` can safely be all zeroes.
                        let words = unsafe { words.assume_init() };
                        // Unfortunate possibly-large copy
                        Rc::new(words)
                    };
                    let words_ref = Rc::get_mut(&mut words).unwrap();

                    let (word_index, mask) = chunk_word_index_and_mask(elem);
                    words_ref[word_index] |= mask;
                    *chunk = Mixed(chunk_domain_size, 1, words);
                } else {
                    *chunk = Ones(chunk_domain_size);
                }
                true
            }
            Ones(_) => false,
            Mixed(chunk_domain_size, ref mut count, ref mut words) => {
                // We skip all the work if the bit is already set.
                let (word_index, mask) = chunk_word_index_and_mask(elem);
                if (words[word_index] & mask) == 0 {
                    *count += 1;
                    if *count < chunk_domain_size {
                        let words = Rc::make_mut(words);
                        words[word_index] |= mask;
                    } else {
                        *chunk = Ones(chunk_domain_size);
                    }
                    true
                } else {
                    false
                }
            }
        }
    }

    /// Sets all bits to true.
    pub fn insert_all(&mut self) {
        for chunk in self.chunks.iter_mut() {
            *chunk = match *chunk {
                Zeros(chunk_domain_size)
                | Ones(chunk_domain_size)
                | Mixed(chunk_domain_size, ..) => Ones(chunk_domain_size),
            }
        }
    }

    /// Returns `true` if the set has changed.
    pub fn remove(&mut self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        let chunk_index = chunk_index(elem);
        let chunk = &mut self.chunks[chunk_index];
        match *chunk {
            Zeros(_) => false,
            Ones(chunk_domain_size) => {
                if chunk_domain_size > 1 {
                    #[cfg(feature = "nightly")]
                    let mut words = {
                        // We take some effort to avoid copying the words.
                        let words = Rc::<[Word; CHUNK_WORDS]>::new_zeroed();
                        // SAFETY: `words` can safely be all zeroes.
                        unsafe { words.assume_init() }
                    };
                    #[cfg(not(feature = "nightly"))]
                    let mut words = {
                        // FIXME: unconditionally use `Rc::new_zeroed` once it is stable (#63291).
                        let words = mem::MaybeUninit::<[Word; CHUNK_WORDS]>::zeroed();
                        // SAFETY: `words` can safely be all zeroes.
                        let words = unsafe { words.assume_init() };
                        // Unfortunate possibly-large copy
                        Rc::new(words)
                    };
                    let words_ref = Rc::get_mut(&mut words).unwrap();

                    // Set only the bits in use.
                    let num_words = num_words(chunk_domain_size as usize);
                    words_ref[..num_words].fill(!0);
                    clear_excess_bits_in_final_word(
                        chunk_domain_size as usize,
                        &mut words_ref[..num_words],
                    );
                    let (word_index, mask) = chunk_word_index_and_mask(elem);
                    words_ref[word_index] &= !mask;
                    *chunk = Mixed(chunk_domain_size, chunk_domain_size - 1, words);
                } else {
                    *chunk = Zeros(chunk_domain_size);
                }
                true
            }
            Mixed(chunk_domain_size, ref mut count, ref mut words) => {
                // We skip all the work if the bit is already clear.
                let (word_index, mask) = chunk_word_index_and_mask(elem);
                if (words[word_index] & mask) != 0 {
                    *count -= 1;
                    if *count > 0 {
                        let words = Rc::make_mut(words);
                        words[word_index] &= !mask;
                    } else {
                        *chunk = Zeros(chunk_domain_size);
                    }
                    true
                } else {
                    false
                }
            }
        }
    }

    bit_relations_inherent_impls! {}
}

impl<T: Idx> BitRelations<ChunkedBitSet<T>> for ChunkedBitSet<T> {
    fn union(&mut self, other: &ChunkedBitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size);
        debug_assert_eq!(self.chunks.len(), other.chunks.len());

        let mut changed = false;
        for (mut self_chunk, other_chunk) in self.chunks.iter_mut().zip(other.chunks.iter()) {
            match (&mut self_chunk, &other_chunk) {
                (_, Zeros(_)) | (Ones(_), _) => {}
                (Zeros(self_chunk_domain_size), Ones(other_chunk_domain_size))
                | (Mixed(self_chunk_domain_size, ..), Ones(other_chunk_domain_size))
                | (Zeros(self_chunk_domain_size), Mixed(other_chunk_domain_size, ..)) => {
                    // `other_chunk` fully overwrites `self_chunk`
                    debug_assert_eq!(self_chunk_domain_size, other_chunk_domain_size);
                    *self_chunk = other_chunk.clone();
                    changed = true;
                }
                (
                    Mixed(
                        self_chunk_domain_size,
                        ref mut self_chunk_count,
                        ref mut self_chunk_words,
                    ),
                    Mixed(_other_chunk_domain_size, _other_chunk_count, other_chunk_words),
                ) => {
                    // First check if the operation would change
                    // `self_chunk.words`. If not, we can avoid allocating some
                    // words, and this happens often enough that it's a
                    // performance win. Also, we only need to operate on the
                    // in-use words, hence the slicing.
                    let op = |a, b| a | b;
                    let num_words = num_words(*self_chunk_domain_size as usize);
                    if bitwise_changes(
                        &self_chunk_words[0..num_words],
                        &other_chunk_words[0..num_words],
                        op,
                    ) {
                        let self_chunk_words = Rc::make_mut(self_chunk_words);
                        let has_changed = bitwise(
                            &mut self_chunk_words[0..num_words],
                            &other_chunk_words[0..num_words],
                            op,
                        );
                        debug_assert!(has_changed);
                        *self_chunk_count = self_chunk_words[0..num_words]
                            .iter()
                            .map(|w| w.count_ones() as ChunkSize)
                            .sum();
                        if *self_chunk_count == *self_chunk_domain_size {
                            *self_chunk = Ones(*self_chunk_domain_size);
                        }
                        changed = true;
                    }
                }
            }
        }
        changed
    }

    fn subtract(&mut self, _other: &ChunkedBitSet<T>) -> bool {
        unimplemented!("implement if/when necessary");
    }

    fn intersect(&mut self, _other: &ChunkedBitSet<T>) -> bool {
        unimplemented!("implement if/when necessary");
    }
}

impl<T: Idx> BitRelations<HybridBitSet<T>> for ChunkedBitSet<T> {
    fn union(&mut self, other: &HybridBitSet<T>) -> bool {
        // FIXME: This is slow if `other` is dense, but it hasn't been a problem
        // in practice so far.
        // If a faster implementation of this operation is required, consider
        // reopening https://github.com/rust-lang/rust/pull/94625
        assert_eq!(self.domain_size, other.domain_size());
        sequential_update(|elem| self.insert(elem), other.iter())
    }

    fn subtract(&mut self, other: &HybridBitSet<T>) -> bool {
        // FIXME: This is slow if `other` is dense, but it hasn't been a problem
        // in practice so far.
        // If a faster implementation of this operation is required, consider
        // reopening https://github.com/rust-lang/rust/pull/94625
        assert_eq!(self.domain_size, other.domain_size());
        sequential_update(|elem| self.remove(elem), other.iter())
    }

    fn intersect(&mut self, _other: &HybridBitSet<T>) -> bool {
        unimplemented!("implement if/when necessary");
    }
}

impl<T: Idx> BitRelations<ChunkedBitSet<T>> for BitSet<T> {
    fn union(&mut self, other: &ChunkedBitSet<T>) -> bool {
        sequential_update(|elem| self.insert(elem), other.iter())
    }

    fn subtract(&mut self, _other: &ChunkedBitSet<T>) -> bool {
        unimplemented!("implement if/when necessary");
    }

    fn intersect(&mut self, other: &ChunkedBitSet<T>) -> bool {
        assert_eq!(self.domain_size(), other.domain_size);
        let mut changed = false;
        for (i, chunk) in other.chunks.iter().enumerate() {
            let mut words = &mut self.words[i * CHUNK_WORDS..];
            if words.len() > CHUNK_WORDS {
                words = &mut words[..CHUNK_WORDS];
            }
            match chunk {
                Chunk::Zeros(..) => {
                    for word in words {
                        if *word != 0 {
                            changed = true;
                            *word = 0;
                        }
                    }
                }
                Chunk::Ones(..) => (),
                Chunk::Mixed(_, _, data) => {
                    for (i, word) in words.iter_mut().enumerate() {
                        let new_val = *word & data[i];
                        if new_val != *word {
                            changed = true;
                            *word = new_val;
                        }
                    }
                }
            }
        }
        changed
    }
}

impl<T> Clone for ChunkedBitSet<T> {
    fn clone(&self) -> Self {
        ChunkedBitSet {
            domain_size: self.domain_size,
            chunks: self.chunks.clone(),
            marker: PhantomData,
        }
    }

    /// WARNING: this implementation of clone_from will panic if the two
    /// bitsets have different domain sizes. This constraint is not inherent to
    /// `clone_from`, but it works with the existing call sites and allows a
    /// faster implementation, which is important because this function is hot.
    fn clone_from(&mut self, from: &Self) {
        assert_eq!(self.domain_size, from.domain_size);
        debug_assert_eq!(self.chunks.len(), from.chunks.len());

        self.chunks.clone_from(&from.chunks)
    }
}

pub struct ChunkedBitIter<'a, T: Idx> {
    index: usize,
    bitset: &'a ChunkedBitSet<T>,
}

impl<'a, T: Idx> ChunkedBitIter<'a, T> {
    #[inline]
    fn new(bitset: &'a ChunkedBitSet<T>) -> ChunkedBitIter<'a, T> {
        ChunkedBitIter { index: 0, bitset }
    }
}

impl<'a, T: Idx> Iterator for ChunkedBitIter<'a, T> {
    type Item = T;
    fn next(&mut self) -> Option<T> {
        while self.index < self.bitset.domain_size() {
            let elem = T::new(self.index);
            let chunk = &self.bitset.chunks[chunk_index(elem)];
            match &chunk {
                Zeros(chunk_domain_size) => {
                    self.index += *chunk_domain_size as usize;
                }
                Ones(_chunk_domain_size) => {
                    self.index += 1;
                    return Some(elem);
                }
                Mixed(_chunk_domain_size, _, words) => loop {
                    let elem = T::new(self.index);
                    self.index += 1;
                    let (word_index, mask) = chunk_word_index_and_mask(elem);
                    if (words[word_index] & mask) != 0 {
                        return Some(elem);
                    }
                    if self.index % CHUNK_BITS == 0 {
                        break;
                    }
                },
            }
        }
        None
    }

    fn fold<B, F>(mut self, mut init: B, mut f: F) -> B
    where
        F: FnMut(B, Self::Item) -> B,
    {
        // If `next` has already been called, we may not be at the start of a chunk, so we first
        // advance the iterator to the start of the next chunk, before proceeding in chunk sized
        // steps.
        while self.index % CHUNK_BITS != 0 {
            let Some(item) = self.next() else { return init };
            init = f(init, item);
        }
        let start_chunk = self.index / CHUNK_BITS;
        let chunks = &self.bitset.chunks[start_chunk..];
        for (i, chunk) in chunks.iter().enumerate() {
            let base = (start_chunk + i) * CHUNK_BITS;
            match chunk {
                Chunk::Zeros(_) => (),
                Chunk::Ones(limit) => {
                    for j in 0..(*limit as usize) {
                        init = f(init, T::new(base + j));
                    }
                }
                Chunk::Mixed(_, _, words) => {
                    init = BitIter::new(&**words).fold(init, |val, mut item: T| {
                        item.increment_by(base);
                        f(val, item)
                    });
                }
            }
        }
        init
    }
}

impl Chunk {
    #[cfg(test)]
    fn assert_valid(&self) {
        match *self {
            Zeros(chunk_domain_size) | Ones(chunk_domain_size) => {
                assert!(chunk_domain_size as usize <= CHUNK_BITS);
            }
            Mixed(chunk_domain_size, count, ref words) => {
                assert!(chunk_domain_size as usize <= CHUNK_BITS);
                assert!(0 < count && count < chunk_domain_size);

                // Check the number of set bits matches `count`.
                assert_eq!(
                    words.iter().map(|w| w.count_ones() as ChunkSize).sum::<ChunkSize>(),
                    count
                );

                // Check the not-in-use words are all zeroed.
                let num_words = num_words(chunk_domain_size as usize);
                if num_words < CHUNK_WORDS {
                    assert_eq!(
                        words[num_words..]
                            .iter()
                            .map(|w| w.count_ones() as ChunkSize)
                            .sum::<ChunkSize>(),
                        0
                    );
                }
            }
        }
    }

    fn new(chunk_domain_size: usize, is_empty: bool) -> Self {
        debug_assert!(chunk_domain_size <= CHUNK_BITS);
        let chunk_domain_size = chunk_domain_size as ChunkSize;
        if is_empty { Zeros(chunk_domain_size) } else { Ones(chunk_domain_size) }
    }

    /// Count the number of 1s in the chunk.
    fn count(&self) -> usize {
        match *self {
            Zeros(_) => 0,
            Ones(chunk_domain_size) => chunk_domain_size as usize,
            Mixed(_, count, _) => count as usize,
        }
    }
}

// Applies a function to mutate a bitset, and returns true if any
// of the applications return true
fn sequential_update<T: Idx>(
    mut self_update: impl FnMut(T) -> bool,
    it: impl Iterator<Item = T>,
) -> bool {
    it.fold(false, |changed, elem| self_update(elem) | changed)
}

// Optimization of intersection for SparseBitSet that's generic
// over the RHS
fn sparse_intersect<T: Idx>(
    set: &mut SparseBitSet<T>,
    other_contains: impl Fn(&T) -> bool,
) -> bool {
    let size = set.elems.len();
    set.elems.retain(|elem| other_contains(elem));
    set.elems.len() != size
}

// Optimization of dense/sparse intersection. The resulting set is
// guaranteed to be at most the size of the sparse set, and hence can be
// represented as a sparse set. Therefore the sparse set is copied and filtered,
// then returned as the new set.
fn dense_sparse_intersect<T: Idx>(
    dense: &BitSet<T>,
    sparse: &SparseBitSet<T>,
) -> (SparseBitSet<T>, bool) {
    let mut sparse_copy = sparse.clone();
    sparse_intersect(&mut sparse_copy, |el| dense.contains(*el));
    let n = sparse_copy.len();
    (sparse_copy, n != dense.count())
}

// hybrid REL dense
impl<T: Idx> BitRelations<BitSet<T>> for HybridBitSet<T> {
    fn union(&mut self, other: &BitSet<T>) -> bool {
        assert_eq!(self.domain_size(), other.domain_size);
        match self {
            HybridBitSet::Sparse(sparse) => {
                // `self` is sparse and `other` is dense. To
                // merge them, we have two available strategies:
                // * Densify `self` then merge other
                // * Clone other then integrate bits from `self`
                // The second strategy requires dedicated method
                // since the usual `union` returns the wrong
                // result. In the dedicated case the computation
                // is slightly faster if the bits of the sparse
                // bitset map to only few words of the dense
                // representation, i.e. indices are near each
                // other.
                //
                // Benchmarking seems to suggest that the second
                // option is worth it.
                let mut new_dense = other.clone();
                let changed = new_dense.reverse_union_sparse(sparse);
                *self = HybridBitSet::Dense(new_dense);
                changed
            }

            HybridBitSet::Dense(dense) => dense.union(other),
        }
    }

    fn subtract(&mut self, other: &BitSet<T>) -> bool {
        assert_eq!(self.domain_size(), other.domain_size);
        match self {
            HybridBitSet::Sparse(sparse) => {
                sequential_update(|elem| sparse.remove(elem), other.iter())
            }
            HybridBitSet::Dense(dense) => dense.subtract(other),
        }
    }

    fn intersect(&mut self, other: &BitSet<T>) -> bool {
        assert_eq!(self.domain_size(), other.domain_size);
        match self {
            HybridBitSet::Sparse(sparse) => sparse_intersect(sparse, |elem| other.contains(*elem)),
            HybridBitSet::Dense(dense) => dense.intersect(other),
        }
    }
}

// dense REL hybrid
impl<T: Idx> BitRelations<HybridBitSet<T>> for BitSet<T> {
    fn union(&mut self, other: &HybridBitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size());
        match other {
            HybridBitSet::Sparse(sparse) => {
                sequential_update(|elem| self.insert(elem), sparse.iter().cloned())
            }
            HybridBitSet::Dense(dense) => self.union(dense),
        }
    }

    fn subtract(&mut self, other: &HybridBitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size());
        match other {
            HybridBitSet::Sparse(sparse) => {
                sequential_update(|elem| self.remove(elem), sparse.iter().cloned())
            }
            HybridBitSet::Dense(dense) => self.subtract(dense),
        }
    }

    fn intersect(&mut self, other: &HybridBitSet<T>) -> bool {
        assert_eq!(self.domain_size, other.domain_size());
        match other {
            HybridBitSet::Sparse(sparse) => {
                let (updated, changed) = dense_sparse_intersect(self, sparse);

                // We can't directly assign the SparseBitSet to the BitSet, and
                // doing `*self = updated.to_dense()` would cause a drop / reallocation. Instead,
                // the BitSet is cleared and `updated` is copied into `self`.
                self.clear();
                for elem in updated.iter() {
                    self.insert(*elem);
                }
                changed
            }
            HybridBitSet::Dense(dense) => self.intersect(dense),
        }
    }
}

// hybrid REL hybrid
impl<T: Idx> BitRelations<HybridBitSet<T>> for HybridBitSet<T> {
    fn union(&mut self, other: &HybridBitSet<T>) -> bool {
        assert_eq!(self.domain_size(), other.domain_size());
        match self {
            HybridBitSet::Sparse(_) => {
                match other {
                    HybridBitSet::Sparse(other_sparse) => {
                        // Both sets are sparse. Add the elements in
                        // `other_sparse` to `self` one at a time. This
                        // may or may not cause `self` to be densified.
                        let mut changed = false;
                        for elem in other_sparse.iter() {
                            changed |= self.insert(*elem);
                        }
                        changed
                    }

                    HybridBitSet::Dense(other_dense) => self.union(other_dense),
                }
            }

            HybridBitSet::Dense(self_dense) => self_dense.union(other),
        }
    }

    fn subtract(&mut self, other: &HybridBitSet<T>) -> bool {
        assert_eq!(self.domain_size(), other.domain_size());
        match self {
            HybridBitSet::Sparse(self_sparse) => {
                sequential_update(|elem| self_sparse.remove(elem), other.iter())
            }
            HybridBitSet::Dense(self_dense) => self_dense.subtract(other),
        }
    }

    fn intersect(&mut self, other: &HybridBitSet<T>) -> bool {
        assert_eq!(self.domain_size(), other.domain_size());
        match self {
            HybridBitSet::Sparse(self_sparse) => {
                sparse_intersect(self_sparse, |elem| other.contains(*elem))
            }
            HybridBitSet::Dense(self_dense) => match other {
                HybridBitSet::Sparse(other_sparse) => {
                    let (updated, changed) = dense_sparse_intersect(self_dense, other_sparse);
                    *self = HybridBitSet::Sparse(updated);
                    changed
                }
                HybridBitSet::Dense(other_dense) => self_dense.intersect(other_dense),
            },
        }
    }
}

impl<T> Clone for BitSet<T> {
    fn clone(&self) -> Self {
        BitSet { domain_size: self.domain_size, words: self.words.clone(), marker: PhantomData }
    }

    fn clone_from(&mut self, from: &Self) {
        self.domain_size = from.domain_size;
        self.words.clone_from(&from.words);
    }
}

impl<T: Idx> fmt::Debug for BitSet<T> {
    fn fmt(&self, w: &mut fmt::Formatter<'_>) -> fmt::Result {
        w.debug_list().entries(self.iter()).finish()
    }
}

impl<T: Idx> fmt::Debug for ChunkedBitSet<T> {
    fn fmt(&self, w: &mut fmt::Formatter<'_>) -> fmt::Result {
        w.debug_list().entries(self.iter()).finish()
    }
}

impl<T: Idx> ToString for BitSet<T> {
    fn to_string(&self) -> String {
        let mut result = String::new();
        let mut sep = '[';

        // Note: this is a little endian printout of bytes.

        // i tracks how many bits we have printed so far.
        let mut i = 0;
        for word in &self.words {
            let mut word = *word;
            for _ in 0..WORD_BYTES {
                // for each byte in `word`:
                let remain = self.domain_size - i;
                // If less than a byte remains, then mask just that many bits.
                let mask = if remain <= 8 { (1 << remain) - 1 } else { 0xFF };
                assert!(mask <= 0xFF);
                let byte = word & mask;

                result.push_str(&format!("{sep}{byte:02x}"));

                if remain <= 8 {
                    break;
                }
                word >>= 8;
                i += 8;
                sep = '-';
            }
            sep = '|';
        }
        result.push(']');

        result
    }
}

pub struct BitIter<'a, T: Idx> {
    /// A copy of the current word, but with any already-visited bits cleared.
    /// (This lets us use `trailing_zeros()` to find the next set bit.) When it
    /// is reduced to 0, we move onto the next word.
    word: Word,

    /// The offset (measured in bits) of the current word.
    offset: usize,

    /// Underlying iterator over the words.
    iter: slice::Iter<'a, Word>,

    marker: PhantomData<T>,
}

impl<'a, T: Idx> BitIter<'a, T> {
    #[inline]
    fn new(words: &'a [Word]) -> BitIter<'a, T> {
        // We initialize `word` and `offset` to degenerate values. On the first
        // call to `next()` we will fall through to getting the first word from
        // `iter`, which sets `word` to the first word (if there is one) and
        // `offset` to 0. Doing it this way saves us from having to maintain
        // additional state about whether we have started.
        BitIter {
            word: 0,
            offset: usize::MAX - (WORD_BITS - 1),
            iter: words.iter(),
            marker: PhantomData,
        }
    }
}

impl<'a, T: Idx> Iterator for BitIter<'a, T> {
    type Item = T;
    fn next(&mut self) -> Option<T> {
        loop {
            if self.word != 0 {
                // Get the position of the next set bit in the current word,
                // then clear the bit.
                let bit_pos = self.word.trailing_zeros() as usize;
                let bit = 1 << bit_pos;
                self.word ^= bit;
                return Some(T::new(bit_pos + self.offset));
            }

            // Move onto the next word. `wrapping_add()` is needed to handle
            // the degenerate initial value given to `offset` in `new()`.
            let word = self.iter.next()?;
            self.word = *word;
            self.offset = self.offset.wrapping_add(WORD_BITS);
        }
    }
}

#[inline]
fn bitwise<Op>(out_vec: &mut [Word], in_vec: &[Word], op: Op) -> bool
where
    Op: Fn(Word, Word) -> Word,
{
    assert_eq!(out_vec.len(), in_vec.len());
    let mut changed = 0;
    for (out_elem, in_elem) in iter::zip(out_vec, in_vec) {
        let old_val = *out_elem;
        let new_val = op(old_val, *in_elem);
        *out_elem = new_val;
        // This is essentially equivalent to a != with changed being a bool, but
        // in practice this code gets auto-vectorized by the compiler for most
        // operators. Using != here causes us to generate quite poor code as the
        // compiler tries to go back to a boolean on each loop iteration.
        changed |= old_val ^ new_val;
    }
    changed != 0
}

/// Does this bitwise operation change `out_vec`?
#[inline]
fn bitwise_changes<Op>(out_vec: &[Word], in_vec: &[Word], op: Op) -> bool
where
    Op: Fn(Word, Word) -> Word,
{
    assert_eq!(out_vec.len(), in_vec.len());
    for (out_elem, in_elem) in iter::zip(out_vec, in_vec) {
        let old_val = *out_elem;
        let new_val = op(old_val, *in_elem);
        if old_val != new_val {
            return true;
        }
    }
    false
}

const SPARSE_MAX: usize = 8;

/// A fixed-size bitset type with a sparse representation and a maximum of
/// `SPARSE_MAX` elements. The elements are stored as a sorted `ArrayVec` with
/// no duplicates.
///
/// This type is used by `HybridBitSet`; do not use directly.
#[derive(Clone, Debug)]
pub struct SparseBitSet<T> {
    domain_size: usize,
    elems: ArrayVec<T, SPARSE_MAX>,
}

impl<T: Idx> SparseBitSet<T> {
    fn new_empty(domain_size: usize) -> Self {
        SparseBitSet { domain_size, elems: ArrayVec::new() }
    }

    fn len(&self) -> usize {
        self.elems.len()
    }

    fn is_empty(&self) -> bool {
        self.elems.len() == 0
    }

    fn contains(&self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        self.elems.contains(&elem)
    }

    fn insert(&mut self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        let changed = if let Some(i) = self.elems.iter().position(|&e| e.index() >= elem.index()) {
            if self.elems[i] == elem {
                // `elem` is already in the set.
                false
            } else {
                // `elem` is smaller than one or more existing elements.
                self.elems.insert(i, elem);
                true
            }
        } else {
            // `elem` is larger than all existing elements.
            self.elems.push(elem);
            true
        };
        assert!(self.len() <= SPARSE_MAX);
        changed
    }

    fn remove(&mut self, elem: T) -> bool {
        assert!(elem.index() < self.domain_size);
        if let Some(i) = self.elems.iter().position(|&e| e == elem) {
            self.elems.remove(i);
            true
        } else {
            false
        }
    }

    fn to_dense(&self) -> BitSet<T> {
        let mut dense = BitSet::new_empty(self.domain_size);
        for elem in self.elems.iter() {
            dense.insert(*elem);
        }
        dense
    }

    fn iter(&self) -> slice::Iter<'_, T> {
        self.elems.iter()
    }

    bit_relations_inherent_impls! {}
}

impl<T: Idx + Ord> SparseBitSet<T> {
    pub fn last_set_in(&self, range: impl RangeBounds<T>) -> Option<T> {
        let mut last_leq = None;
        for e in self.iter() {
            if range.contains(e) {
                last_leq = Some(*e);
            }
        }
        last_leq
    }
}

/// A fixed-size bitset type with a hybrid representation: sparse when there
/// are up to a `SPARSE_MAX` elements in the set, but dense when there are more
/// than `SPARSE_MAX`.
///
/// This type is especially efficient for sets that typically have a small
/// number of elements, but a large `domain_size`, and are cleared frequently.
///
/// `T` is an index type, typically a newtyped `usize` wrapper, but it can also
/// just be `usize`.
///
/// All operations that involve an element will panic if the element is equal
/// to or greater than the domain size. All operations that involve two bitsets
/// will panic if the bitsets have differing domain sizes.
#[derive(Clone)]
pub enum HybridBitSet<T> {
    Sparse(SparseBitSet<T>),
    Dense(BitSet<T>),
}

impl<T: Idx> fmt::Debug for HybridBitSet<T> {
    fn fmt(&self, w: &mut fmt::Formatter<'_>) -> fmt::Result {
        match self {
            Self::Sparse(b) => b.fmt(w),
            Self::Dense(b) => b.fmt(w),
        }
    }
}

impl<T: Idx> HybridBitSet<T> {
    pub fn new_empty(domain_size: usize) -> Self {
        HybridBitSet::Sparse(SparseBitSet::new_empty(domain_size))
    }

    pub fn domain_size(&self) -> usize {
        match self {
            HybridBitSet::Sparse(sparse) => sparse.domain_size,
            HybridBitSet::Dense(dense) => dense.domain_size,
        }
    }

    pub fn clear(&mut self) {
        let domain_size = self.domain_size();
        *self = HybridBitSet::new_empty(domain_size);
    }

    pub fn contains(&self, elem: T) -> bool {
        match self {
            HybridBitSet::Sparse(sparse) => sparse.contains(elem),
            HybridBitSet::Dense(dense) => dense.contains(elem),
        }
    }

    pub fn superset(&self, other: &HybridBitSet<T>) -> bool {
        match (self, other) {
            (HybridBitSet::Dense(self_dense), HybridBitSet::Dense(other_dense)) => {
                self_dense.superset(other_dense)
            }
            _ => {
                assert!(self.domain_size() == other.domain_size());
                other.iter().all(|elem| self.contains(elem))
            }
        }
    }

    pub fn is_empty(&self) -> bool {
        match self {
            HybridBitSet::Sparse(sparse) => sparse.is_empty(),
            HybridBitSet::Dense(dense) => dense.is_empty(),
        }
    }

    /// Returns the previous element present in the bitset from `elem`,
    /// inclusively of elem. That is, will return `Some(elem)` if elem is in the
    /// bitset.
    pub fn last_set_in(&self, range: impl RangeBounds<T>) -> Option<T>
    where
        T: Ord,
    {
        match self {
            HybridBitSet::Sparse(sparse) => sparse.last_set_in(range),
            HybridBitSet::Dense(dense) => dense.last_set_in(range),
        }
    }

    pub fn insert(&mut self, elem: T) -> bool {
        // No need to check `elem` against `self.domain_size` here because all
        // the match cases check it, one way or another.
        match self {
            HybridBitSet::Sparse(sparse) if sparse.len() < SPARSE_MAX => {
                // The set is sparse and has space for `elem`.
                sparse.insert(elem)
            }
            HybridBitSet::Sparse(sparse) if sparse.contains(elem) => {
                // The set is sparse and does not have space for `elem`, but
                // that doesn't matter because `elem` is already present.
                false
            }
            HybridBitSet::Sparse(sparse) => {
                // The set is sparse and full. Convert to a dense set.
                let mut dense = sparse.to_dense();
                let changed = dense.insert(elem);
                assert!(changed);
                *self = HybridBitSet::Dense(dense);
                changed
            }
            HybridBitSet::Dense(dense) => dense.insert(elem),
        }
    }

    pub fn insert_range(&mut self, elems: impl RangeBounds<T>) {
        // No need to check `elem` against `self.domain_size` here because all
        // the match cases check it, one way or another.
        let start = match elems.start_bound().cloned() {
            Bound::Included(start) => start.index(),
            Bound::Excluded(start) => start.index() + 1,
            Bound::Unbounded => 0,
        };
        let end = match elems.end_bound().cloned() {
            Bound::Included(end) => end.index() + 1,
            Bound::Excluded(end) => end.index(),
            Bound::Unbounded => self.domain_size() - 1,
        };
        let Some(len) = end.checked_sub(start) else { return };
        match self {
            HybridBitSet::Sparse(sparse) if sparse.len() + len < SPARSE_MAX => {
                // The set is sparse and has space for `elems`.
                for elem in start..end {
                    sparse.insert(T::new(elem));
                }
            }
            HybridBitSet::Sparse(sparse) => {
                // The set is sparse and full. Convert to a dense set.
                let mut dense = sparse.to_dense();
                dense.insert_range(elems);
                *self = HybridBitSet::Dense(dense);
            }
            HybridBitSet::Dense(dense) => dense.insert_range(elems),
        }
    }

    pub fn insert_all(&mut self) {
        let domain_size = self.domain_size();
        match self {
            HybridBitSet::Sparse(_) => {
                *self = HybridBitSet::Dense(BitSet::new_filled(domain_size));
            }
            HybridBitSet::Dense(dense) => dense.insert_all(),
        }
    }

    pub fn remove(&mut self, elem: T) -> bool {
        // Note: we currently don't bother going from Dense back to Sparse.
        match self {
            HybridBitSet::Sparse(sparse) => sparse.remove(elem),
            HybridBitSet::Dense(dense) => dense.remove(elem),
        }
    }

    /// Converts to a dense set, consuming itself in the process.
    pub fn to_dense(self) -> BitSet<T> {
        match self {
            HybridBitSet::Sparse(sparse) => sparse.to_dense(),
            HybridBitSet::Dense(dense) => dense,
        }
    }

    pub fn iter(&self) -> HybridIter<'_, T> {
        match self {
            HybridBitSet::Sparse(sparse) => HybridIter::Sparse(sparse.iter()),
            HybridBitSet::Dense(dense) => HybridIter::Dense(dense.iter()),
        }
    }

    bit_relations_inherent_impls! {}
}

pub enum HybridIter<'a, T: Idx> {
    Sparse(slice::Iter<'a, T>),
    Dense(BitIter<'a, T>),
}

impl<'a, T: Idx> Iterator for HybridIter<'a, T> {
    type Item = T;

    fn next(&mut self) -> Option<T> {
        match self {
            HybridIter::Sparse(sparse) => sparse.next().copied(),
            HybridIter::Dense(dense) => dense.next(),
        }
    }
}

/// A resizable bitset type with a dense representation.
///
/// `T` is an index type, typically a newtyped `usize` wrapper, but it can also
/// just be `usize`.
///
/// All operations that involve an element will panic if the element is equal
/// to or greater than the domain size.
#[derive(Clone, Debug, PartialEq)]
pub struct GrowableBitSet<T: Idx> {
    bit_set: BitSet<T>,
}

impl<T: Idx> Default for GrowableBitSet<T> {
    fn default() -> Self {
        GrowableBitSet::new_empty()
    }
}

impl<T: Idx> GrowableBitSet<T> {
    /// Ensure that the set can hold at least `min_domain_size` elements.
    pub fn ensure(&mut self, min_domain_size: usize) {
        if self.bit_set.domain_size < min_domain_size {
            self.bit_set.domain_size = min_domain_size;
        }

        let min_num_words = num_words(min_domain_size);
        if self.bit_set.words.len() < min_num_words {
            self.bit_set.words.resize(min_num_words, 0)
        }
    }

    pub fn new_empty() -> GrowableBitSet<T> {
        GrowableBitSet { bit_set: BitSet::new_empty(0) }
    }

    pub fn with_capacity(capacity: usize) -> GrowableBitSet<T> {
        GrowableBitSet { bit_set: BitSet::new_empty(capacity) }
    }

    /// Returns `true` if the set has changed.
    #[inline]
    pub fn insert(&mut self, elem: T) -> bool {
        self.ensure(elem.index() + 1);
        self.bit_set.insert(elem)
    }

    /// Returns `true` if the set has changed.
    #[inline]
    pub fn remove(&mut self, elem: T) -> bool {
        self.ensure(elem.index() + 1);
        self.bit_set.remove(elem)
    }

    #[inline]
    pub fn is_empty(&self) -> bool {
        self.bit_set.is_empty()
    }

    #[inline]
    pub fn contains(&self, elem: T) -> bool {
        let (word_index, mask) = word_index_and_mask(elem);
        self.bit_set.words.get(word_index).is_some_and(|word| (word & mask) != 0)
    }

    #[inline]
    pub fn iter(&self) -> BitIter<'_, T> {
        self.bit_set.iter()
    }

    #[inline]
    pub fn len(&self) -> usize {
        self.bit_set.count()
    }
}

impl<T: Idx> From<BitSet<T>> for GrowableBitSet<T> {
    fn from(bit_set: BitSet<T>) -> Self {
        Self { bit_set }
    }
}

/// A fixed-size 2D bit matrix type with a dense representation.
///
/// `R` and `C` are index types used to identify rows and columns respectively;
/// typically newtyped `usize` wrappers, but they can also just be `usize`.
///
/// All operations that involve a row and/or column index will panic if the
/// index exceeds the relevant bound.
#[cfg_attr(feature = "nightly", derive(Decodable_Generic, Encodable_Generic))]
#[derive(Clone, Eq, PartialEq, Hash)]
pub struct BitMatrix<R: Idx, C: Idx> {
    num_rows: usize,
    num_columns: usize,
    words: SmallVec<[Word; 2]>,
    marker: PhantomData<(R, C)>,
}

impl<R: Idx, C: Idx> BitMatrix<R, C> {
    /// Creates a new `rows x columns` matrix, initially empty.
    pub fn new(num_rows: usize, num_columns: usize) -> BitMatrix<R, C> {
        // For every element, we need one bit for every other
        // element. Round up to an even number of words.
        let words_per_row = num_words(num_columns);
        BitMatrix {
            num_rows,
            num_columns,
            words: smallvec![0; num_rows * words_per_row],
            marker: PhantomData,
        }
    }

    /// Creates a new matrix, with `row` used as the value for every row.
    pub fn from_row_n(row: &BitSet<C>, num_rows: usize) -> BitMatrix<R, C> {
        let num_columns = row.domain_size();
        let words_per_row = num_words(num_columns);
        assert_eq!(words_per_row, row.words.len());
        BitMatrix {
            num_rows,
            num_columns,
            words: iter::repeat(&row.words).take(num_rows).flatten().cloned().collect(),
            marker: PhantomData,
        }
    }

    pub fn rows(&self) -> impl Iterator<Item = R> {
        (0..self.num_rows).map(R::new)
    }

    /// The range of bits for a given row.
    fn range(&self, row: R) -> (usize, usize) {
        let words_per_row = num_words(self.num_columns);
        let start = row.index() * words_per_row;
        (start, start + words_per_row)
    }

    /// Sets the cell at `(row, column)` to true. Put another way, insert
    /// `column` to the bitset for `row`.
    ///
    /// Returns `true` if this changed the matrix.
    pub fn insert(&mut self, row: R, column: C) -> bool {
        assert!(row.index() < self.num_rows && column.index() < self.num_columns);
        let (start, _) = self.range(row);
        let (word_index, mask) = word_index_and_mask(column);
        let words = &mut self.words[..];
        let word = words[start + word_index];
        let new_word = word | mask;
        words[start + word_index] = new_word;
        word != new_word
    }

    /// Do the bits from `row` contain `column`? Put another way, is
    /// the matrix cell at `(row, column)` true?  Put yet another way,
    /// if the matrix represents (transitive) reachability, can
    /// `row` reach `column`?
    pub fn contains(&self, row: R, column: C) -> bool {
        assert!(row.index() < self.num_rows && column.index() < self.num_columns);
        let (start, _) = self.range(row);
        let (word_index, mask) = word_index_and_mask(column);
        (self.words[start + word_index] & mask) != 0
    }

    /// Returns those indices that are true in rows `a` and `b`. This
    /// is an *O*(*n*) operation where *n* is the number of elements
    /// (somewhat independent from the actual size of the
    /// intersection, in particular).
    pub fn intersect_rows(&self, row1: R, row2: R) -> Vec<C> {
        assert!(row1.index() < self.num_rows && row2.index() < self.num_rows);
        let (row1_start, row1_end) = self.range(row1);
        let (row2_start, row2_end) = self.range(row2);
        let mut result = Vec::with_capacity(self.num_columns);
        for (base, (i, j)) in (row1_start..row1_end).zip(row2_start..row2_end).enumerate() {
            let mut v = self.words[i] & self.words[j];
            for bit in 0..WORD_BITS {
                if v == 0 {
                    break;
                }
                if v & 0x1 != 0 {
                    result.push(C::new(base * WORD_BITS + bit));
                }
                v >>= 1;
            }
        }
        result
    }

    /// Adds the bits from row `read` to the bits from row `write`, and
    /// returns `true` if anything changed.
    ///
    /// This is used when computing transitive reachability because if
    /// you have an edge `write -> read`, because in that case
    /// `write` can reach everything that `read` can (and
    /// potentially more).
    pub fn union_rows(&mut self, read: R, write: R) -> bool {
        assert!(read.index() < self.num_rows && write.index() < self.num_rows);
        let (read_start, read_end) = self.range(read);
        let (write_start, write_end) = self.range(write);
        let words = &mut self.words[..];
        let mut changed = 0;
        for (read_index, write_index) in iter::zip(read_start..read_end, write_start..write_end) {
            let word = words[write_index];
            let new_word = word | words[read_index];
            words[write_index] = new_word;
            // See `bitwise` for the rationale.
            changed |= word ^ new_word;
        }
        changed != 0
    }

    /// Adds the bits from `with` to the bits from row `write`, and
    /// returns `true` if anything changed.
    pub fn union_row_with(&mut self, with: &BitSet<C>, write: R) -> bool {
        assert!(write.index() < self.num_rows);
        assert_eq!(with.domain_size(), self.num_columns);
        let (write_start, write_end) = self.range(write);
        bitwise(&mut self.words[write_start..write_end], &with.words, |a, b| a | b)
    }

    /// Sets every cell in `row` to true.
    pub fn insert_all_into_row(&mut self, row: R) {
        assert!(row.index() < self.num_rows);
        let (start, end) = self.range(row);
        let words = &mut self.words[..];
        for index in start..end {
            words[index] = !0;
        }
        clear_excess_bits_in_final_word(self.num_columns, &mut self.words[..end]);
    }

    /// Gets a slice of the underlying words.
    pub fn words(&self) -> &[Word] {
        &self.words
    }

    /// Iterates through all the columns set to true in a given row of
    /// the matrix.
    pub fn iter(&self, row: R) -> BitIter<'_, C> {
        assert!(row.index() < self.num_rows);
        let (start, end) = self.range(row);
        BitIter::new(&self.words[start..end])
    }

    /// Returns the number of elements in `row`.
    pub fn count(&self, row: R) -> usize {
        let (start, end) = self.range(row);
        self.words[start..end].iter().map(|e| e.count_ones() as usize).sum()
    }
}

impl<R: Idx, C: Idx> fmt::Debug for BitMatrix<R, C> {
    fn fmt(&self, fmt: &mut fmt::Formatter<'_>) -> fmt::Result {
        /// Forces its contents to print in regular mode instead of alternate mode.
        struct OneLinePrinter<T>(T);
        impl<T: fmt::Debug> fmt::Debug for OneLinePrinter<T> {
            fn fmt(&self, fmt: &mut fmt::Formatter<'_>) -> fmt::Result {
                write!(fmt, "{:?}", self.0)
            }
        }

        write!(fmt, "BitMatrix({}x{}) ", self.num_rows, self.num_columns)?;
        let items = self.rows().flat_map(|r| self.iter(r).map(move |c| (r, c)));
        fmt.debug_set().entries(items.map(OneLinePrinter)).finish()
    }
}

/// A fixed-column-size, variable-row-size 2D bit matrix with a moderately
/// sparse representation.
///
/// Initially, every row has no explicit representation. If any bit within a
/// row is set, the entire row is instantiated as `Some(<HybridBitSet>)`.
/// Furthermore, any previously uninstantiated rows prior to it will be
/// instantiated as `None`. Those prior rows may themselves become fully
/// instantiated later on if any of their bits are set.
///
/// `R` and `C` are index types used to identify rows and columns respectively;
/// typically newtyped `usize` wrappers, but they can also just be `usize`.
#[derive(Clone, Debug)]
pub struct SparseBitMatrix<R, C>
where
    R: Idx,
    C: Idx,
{
    num_columns: usize,
    rows: IndexVec<R, Option<HybridBitSet<C>>>,
}

impl<R: Idx, C: Idx> SparseBitMatrix<R, C> {
    /// Creates a new empty sparse bit matrix with no rows or columns.
    pub fn new(num_columns: usize) -> Self {
        Self { num_columns, rows: IndexVec::new() }
    }

    fn ensure_row(&mut self, row: R) -> &mut HybridBitSet<C> {
        // Instantiate any missing rows up to and including row `row` with an empty HybridBitSet.
        // Then replace row `row` with a full HybridBitSet if necessary.
        self.rows.get_or_insert_with(row, || HybridBitSet::new_empty(self.num_columns))
    }

    /// Sets the cell at `(row, column)` to true. Put another way, insert
    /// `column` to the bitset for `row`.
    ///
    /// Returns `true` if this changed the matrix.
    pub fn insert(&mut self, row: R, column: C) -> bool {
        self.ensure_row(row).insert(column)
    }

    /// Sets the cell at `(row, column)` to false. Put another way, delete
    /// `column` from the bitset for `row`. Has no effect if `row` does not
    /// exist.
    ///
    /// Returns `true` if this changed the matrix.
    pub fn remove(&mut self, row: R, column: C) -> bool {
        match self.rows.get_mut(row) {
            Some(Some(row)) => row.remove(column),
            _ => false,
        }
    }

    /// Sets all columns at `row` to false. Has no effect if `row` does
    /// not exist.
    pub fn clear(&mut self, row: R) {
        if let Some(Some(row)) = self.rows.get_mut(row) {
            row.clear();
        }
    }

    /// Do the bits from `row` contain `column`? Put another way, is
    /// the matrix cell at `(row, column)` true?  Put yet another way,
    /// if the matrix represents (transitive) reachability, can
    /// `row` reach `column`?
    pub fn contains(&self, row: R, column: C) -> bool {
        self.row(row).is_some_and(|r| r.contains(column))
    }

    /// Adds the bits from row `read` to the bits from row `write`, and
    /// returns `true` if anything changed.
    ///
    /// This is used when computing transitive reachability because if
    /// you have an edge `write -> read`, because in that case
    /// `write` can reach everything that `read` can (and
    /// potentially more).
    pub fn union_rows(&mut self, read: R, write: R) -> bool {
        if read == write || self.row(read).is_none() {
            return false;
        }

        self.ensure_row(write);
        if let (Some(read_row), Some(write_row)) = self.rows.pick2_mut(read, write) {
            write_row.union(read_row)
        } else {
            unreachable!()
        }
    }

    /// Insert all bits in the given row.
    pub fn insert_all_into_row(&mut self, row: R) {
        self.ensure_row(row).insert_all();
    }

    pub fn rows(&self) -> impl Iterator<Item = R> {
        self.rows.indices()
    }

    /// Iterates through all the columns set to true in a given row of
    /// the matrix.
    pub fn iter(&self, row: R) -> impl Iterator<Item = C> + '_ {
        self.row(row).into_iter().flat_map(|r| r.iter())
    }

    pub fn row(&self, row: R) -> Option<&HybridBitSet<C>> {
        self.rows.get(row)?.as_ref()
    }

    /// Intersects `row` with `set`. `set` can be either `BitSet` or
    /// `HybridBitSet`. Has no effect if `row` does not exist.
    ///
    /// Returns true if the row was changed.
    pub fn intersect_row<Set>(&mut self, row: R, set: &Set) -> bool
    where
        HybridBitSet<C>: BitRelations<Set>,
    {
        match self.rows.get_mut(row) {
            Some(Some(row)) => row.intersect(set),
            _ => false,
        }
    }

    /// Subtracts `set` from `row`. `set` can be either `BitSet` or
    /// `HybridBitSet`. Has no effect if `row` does not exist.
    ///
    /// Returns true if the row was changed.
    pub fn subtract_row<Set>(&mut self, row: R, set: &Set) -> bool
    where
        HybridBitSet<C>: BitRelations<Set>,
    {
        match self.rows.get_mut(row) {
            Some(Some(row)) => row.subtract(set),
            _ => false,
        }
    }

    /// Unions `row` with `set`. `set` can be either `BitSet` or
    /// `HybridBitSet`.
    ///
    /// Returns true if the row was changed.
    pub fn union_row<Set>(&mut self, row: R, set: &Set) -> bool
    where
        HybridBitSet<C>: BitRelations<Set>,
    {
        self.ensure_row(row).union(set)
    }
}

#[inline]
fn num_words<T: Idx>(domain_size: T) -> usize {
    (domain_size.index() + WORD_BITS - 1) / WORD_BITS
}

#[inline]
fn num_chunks<T: Idx>(domain_size: T) -> usize {
    assert!(domain_size.index() > 0);
    (domain_size.index() + CHUNK_BITS - 1) / CHUNK_BITS
}

#[inline]
fn word_index_and_mask<T: Idx>(elem: T) -> (usize, Word) {
    let elem = elem.index();
    let word_index = elem / WORD_BITS;
    let mask = 1 << (elem % WORD_BITS);
    (word_index, mask)
}

#[inline]
fn chunk_index<T: Idx>(elem: T) -> usize {
    elem.index() / CHUNK_BITS
}

#[inline]
fn chunk_word_index_and_mask<T: Idx>(elem: T) -> (usize, Word) {
    let chunk_elem = elem.index() % CHUNK_BITS;
    word_index_and_mask(chunk_elem)
}

fn clear_excess_bits_in_final_word(domain_size: usize, words: &mut [Word]) {
    let num_bits_in_final_word = domain_size % WORD_BITS;
    if num_bits_in_final_word > 0 {
        let mask = (1 << num_bits_in_final_word) - 1;
        words[words.len() - 1] &= mask;
    }
}

#[inline]
fn max_bit(word: Word) -> usize {
    WORD_BITS - 1 - word.leading_zeros() as usize
}

/// Integral type used to represent the bit set.
pub trait FiniteBitSetTy:
    BitAnd<Output = Self>
    + BitAndAssign
    + BitOrAssign
    + Clone
    + Copy
    + Shl
    + Not<Output = Self>
    + PartialEq
    + Sized
{
    /// Size of the domain representable by this type, e.g. 64 for `u64`.
    const DOMAIN_SIZE: u32;

    /// Value which represents the `FiniteBitSet` having every bit set.
    const FILLED: Self;
    /// Value which represents the `FiniteBitSet` having no bits set.
    const EMPTY: Self;

    /// Value for one as the integral type.
    const ONE: Self;
    /// Value for zero as the integral type.
    const ZERO: Self;

    /// Perform a checked left shift on the integral type.
    fn checked_shl(self, rhs: u32) -> Option<Self>;
    /// Perform a checked right shift on the integral type.
    fn checked_shr(self, rhs: u32) -> Option<Self>;
}

impl FiniteBitSetTy for u32 {
    const DOMAIN_SIZE: u32 = 32;

    const FILLED: Self = Self::MAX;
    const EMPTY: Self = Self::MIN;

    const ONE: Self = 1u32;
    const ZERO: Self = 0u32;

    fn checked_shl(self, rhs: u32) -> Option<Self> {
        self.checked_shl(rhs)
    }

    fn checked_shr(self, rhs: u32) -> Option<Self> {
        self.checked_shr(rhs)
    }
}

impl std::fmt::Debug for FiniteBitSet<u32> {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        write!(f, "{:032b}", self.0)
    }
}

/// A fixed-sized bitset type represented by an integer type. Indices outwith than the range
/// representable by `T` are considered set.
#[cfg_attr(feature = "nightly", derive(Decodable_Generic, Encodable_Generic))]
#[derive(Copy, Clone, Eq, PartialEq)]
pub struct FiniteBitSet<T: FiniteBitSetTy>(pub T);

impl<T: FiniteBitSetTy> FiniteBitSet<T> {
    /// Creates a new, empty bitset.
    pub fn new_empty() -> Self {
        Self(T::EMPTY)
    }

    /// Sets the `index`th bit.
    pub fn set(&mut self, index: u32) {
        self.0 |= T::ONE.checked_shl(index).unwrap_or(T::ZERO);
    }

    /// Unsets the `index`th bit.
    pub fn clear(&mut self, index: u32) {
        self.0 &= !T::ONE.checked_shl(index).unwrap_or(T::ZERO);
    }

    /// Sets the `i`th to `j`th bits.
    pub fn set_range(&mut self, range: Range<u32>) {
        let bits = T::FILLED
            .checked_shl(range.end - range.start)
            .unwrap_or(T::ZERO)
            .not()
            .checked_shl(range.start)
            .unwrap_or(T::ZERO);
        self.0 |= bits;
    }

    /// Is the set empty?
    pub fn is_empty(&self) -> bool {
        self.0 == T::EMPTY
    }

    /// Returns the domain size of the bitset.
    pub fn within_domain(&self, index: u32) -> bool {
        index < T::DOMAIN_SIZE
    }

    /// Returns if the `index`th bit is set.
    pub fn contains(&self, index: u32) -> Option<bool> {
        self.within_domain(index)
            .then(|| ((self.0.checked_shr(index).unwrap_or(T::ONE)) & T::ONE) == T::ONE)
    }
}

impl<T: FiniteBitSetTy> Default for FiniteBitSet<T> {
    fn default() -> Self {
        Self::new_empty()
    }
}