go/container/intsets/sparse.go

// Copyright 2014 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

// Package intsets provides Sparse, a compact and fast representation
// for sparse sets of int values.
//
// The time complexity of the operations Len, Insert, Remove and Has
// is in O(n) but in practice those methods are faster and more
// space-efficient than equivalent operations on sets based on the Go
// map type.  The IsEmpty, Min, Max, Clear and TakeMin operations
// require constant time.
//
package intsets

// TODO(adonovan):
// - Add SymmetricDifference(x, y *Sparse), i.e. x ∆ y.
// - Add InsertAll(...int), RemoveAll(...int)
// - Add 'bool changed' results for {Intersection,Difference}With too.
//
// TODO(adonovan): implement Dense, a dense bit vector with a similar API.
// The space usage would be proportional to Max(), not Len(), and the
// implementation would be based upon big.Int.
//
// TODO(adonovan): experiment with making the root block indirect (nil
// iff IsEmpty).  This would reduce the memory usage when empty and
// might simplify the aliasing invariants.

import (
	"bytes"
	"fmt"
)

// A Sparse is a set of int values.
// Sparse operations (even queries) are not concurrency-safe.
//
// The zero value for Sparse is a valid empty set.
//
// Sparse sets must be copied using the Copy method, not by assigning
// a Sparse value.
//
type Sparse struct {
	// An uninitialized Sparse represents an empty set.
	// An empty set may also be represented by
	//  root.next == root.prev == &root.
	// In a non-empty set, root.next points to the first block and
	// root.prev to the last.
	// root.offset and root.bits are unused.
	root block
}

type word uintptr

const (
	_m            = ^word(0)
	bitsPerWord   = 8 << (_m>>8&1 + _m>>16&1 + _m>>32&1)
	bitsPerBlock  = 256 // optimal value for go/pointer solver performance
	wordsPerBlock = bitsPerBlock / bitsPerWord
)

// Limit values of implementation-specific int type.
const (
	MaxInt = int(^uint(0) >> 1)
	MinInt = -MaxInt - 1
)

// -- block ------------------------------------------------------------

// A set is represented as a circular doubly-linked list of blocks,
// each containing an offset and a bit array of fixed size
// bitsPerBlock; the blocks are ordered by increasing offset.
//
// The set contains an element x iff the block whose offset is x - (x
// mod bitsPerBlock) has the bit (x mod bitsPerBlock) set, where mod
// is the Euclidean remainder.
//
// A block may only be empty transiently.
//
type block struct {
	offset     int                 // offset mod bitsPerBlock == 0
	bits       [wordsPerBlock]word // contains at least one set bit
	next, prev *block              // doubly-linked list of blocks
}

// wordMask returns the word index (in block.bits)
// and single-bit mask for the block's ith bit.
func wordMask(i uint) (w uint, mask word) {
	w = i / bitsPerWord
	mask = 1 << (i % bitsPerWord)
	return
}

// insert sets the block b's ith bit and
// returns true if it was not already set.
//
func (b *block) insert(i uint) bool {
	w, mask := wordMask(i)
	if b.bits[w]&mask == 0 {
		b.bits[w] |= mask
		return true
	}
	return false
}

// remove clears the block's ith bit and
// returns true if the bit was previously set.
// NB: may leave the block empty.
//
func (b *block) remove(i uint) bool {
	w, mask := wordMask(i)
	if b.bits[w]&mask != 0 {
		b.bits[w] &^= mask
		return true
	}
	return false
}

// has reports whether the block's ith bit is set.
func (b *block) has(i uint) bool {
	w, mask := wordMask(i)
	return b.bits[w]&mask != 0
}

// empty reports whether b.len()==0, but more efficiently.
func (b *block) empty() bool {
	for _, w := range b.bits {
		if w != 0 {
			return false
		}
	}
	return true
}

// len returns the number of set bits in block b.
func (b *block) len() int {
	var l int
	for _, w := range b.bits {
		l += popcount(w)
	}
	return l
}

// max returns the maximum element of the block.
// The block must not be empty.
//
func (b *block) max() int {
	bi := b.offset + bitsPerBlock
	// Decrement bi by number of high zeros in last.bits.
	for i := len(b.bits) - 1; i >= 0; i-- {
		if w := b.bits[i]; w != 0 {
			return bi - nlz(w) - 1
		}
		bi -= bitsPerWord
	}
	panic("BUG: empty block")
}

// min returns the minimum element of the block,
// and also removes it if take is set.
// The block must not be initially empty.
// NB: may leave the block empty.
//
func (b *block) min(take bool) int {
	for i, w := range b.bits {
		if w != 0 {
			tz := ntz(w)
			if take {
				b.bits[i] = w &^ (1 << uint(tz))
			}
			return b.offset + int(i*bitsPerWord) + tz
		}
	}
	panic("BUG: empty block")
}

// forEach calls f for each element of block b.
// f must not mutate b's enclosing Sparse.
func (b *block) forEach(f func(int)) {
	for i, w := range b.bits {
		offset := b.offset + i*bitsPerWord
		// TODO(adonovan): opt: uses subword
		// masks to avoid testing every bit.
		for bi := 0; w != 0 && bi < bitsPerWord; bi++ {
			if w&1 != 0 {
				f(offset)
			}
			offset++
			w >>= 1
		}
	}
}

// offsetAndBitIndex returns the offset of the block that would
// contain x and the bit index of x within that block.
//
func offsetAndBitIndex(x int) (int, uint) {
	mod := x % bitsPerBlock
	if mod < 0 {
		// Euclidean (non-negative) remainder
		mod += bitsPerBlock
	}
	return x - mod, uint(mod)
}

// -- Sparse --------------------------------------------------------------

// start returns the root's next block, which is the root block
// (if s.IsEmpty()) or the first true block otherwise.
// start has the side effect of ensuring that s is properly
// initialized.
//
func (s *Sparse) start() *block {
	if s.root.next == nil {
		s.root.next = &s.root
		s.root.prev = &s.root
	} else if s.root.next.prev != &s.root {
		// Copying a Sparse x leads to pernicious corruption: the
		// new Sparse y shares the old linked list, but iteration
		// on y will never encounter &y.root so it goes into a
		// loop.  Fail fast before this occurs.
		panic("A Sparse has been copied without (*Sparse).Copy()")
	}

	return s.root.next
}

// IsEmpty reports whether the set s is empty.
func (s *Sparse) IsEmpty() bool {
	return s.start() == &s.root
}

// Len returns the number of elements in the set s.
func (s *Sparse) Len() int {
	var l int
	for b := s.start(); b != &s.root; b = b.next {
		l += b.len()
	}
	return l
}

// Max returns the maximum element of the set s, or MinInt if s is empty.
func (s *Sparse) Max() int {
	if s.IsEmpty() {
		return MinInt
	}
	return s.root.prev.max()
}

// Min returns the minimum element of the set s, or MaxInt if s is empty.
func (s *Sparse) Min() int {
	if s.IsEmpty() {
		return MaxInt
	}
	return s.root.next.min(false)
}

// block returns the block that would contain offset,
// or nil if s contains no such block.
//
func (s *Sparse) block(offset int) *block {
	b := s.start()
	for b != &s.root && b.offset <= offset {
		if b.offset == offset {
			return b
		}
		b = b.next
	}
	return nil
}

// Insert adds x to the set s, and reports whether the set grew.
func (s *Sparse) Insert(x int) bool {
	offset, i := offsetAndBitIndex(x)
	b := s.start()
	for b != &s.root && b.offset <= offset {
		if b.offset == offset {
			return b.insert(i)
		}
		b = b.next
	}

	// Insert new block before b.
	new := &block{offset: offset}
	new.next = b
	new.prev = b.prev
	new.prev.next = new
	new.next.prev = new
	return new.insert(i)
}

func (s *Sparse) removeBlock(b *block) {
	b.prev.next = b.next
	b.next.prev = b.prev
}

// Remove removes x from the set s, and reports whether the set shrank.
func (s *Sparse) Remove(x int) bool {
	offset, i := offsetAndBitIndex(x)
	if b := s.block(offset); b != nil {
		if !b.remove(i) {
			return false
		}
		if b.empty() {
			s.removeBlock(b)
		}
		return true
	}
	return false
}

// Clear removes all elements from the set s.
func (s *Sparse) Clear() {
	s.root.next = &s.root
	s.root.prev = &s.root
}

// If set s is non-empty, TakeMin sets *p to the minimum element of
// the set s, removes that element from the set and returns true.
// Otherwise, it returns false and *p is undefined.
//
// This method may be used for iteration over a worklist like so:
//
// 	var x int
// 	for worklist.TakeMin(&x) { use(x) }
//
func (s *Sparse) TakeMin(p *int) bool {
	head := s.start()
	if head == &s.root {
		return false
	}
	*p = head.min(true)
	if head.empty() {
		s.removeBlock(head)
	}
	return true
}

// Has reports whether x is an element of the set s.
func (s *Sparse) Has(x int) bool {
	offset, i := offsetAndBitIndex(x)
	if b := s.block(offset); b != nil {
		return b.has(i)
	}
	return false
}

// forEach applies function f to each element of the set s in order.
//
// f must not mutate s.  Consequently, forEach is not safe to expose
// to clients.  In any case, using "range s.AppendTo()" allows more
// natural control flow with continue/break/return.
//
func (s *Sparse) forEach(f func(int)) {
	for b := s.start(); b != &s.root; b = b.next {
		b.forEach(f)
	}
}

// Copy sets s to the value of x.
func (s *Sparse) Copy(x *Sparse) {
	if s == x {
		return
	}

	xb := x.start()
	sb := s.start()
	for xb != &x.root {
		if sb == &s.root {
			sb = s.insertBlockBefore(sb)
		}
		sb.offset = xb.offset
		sb.bits = xb.bits
		xb = xb.next
		sb = sb.next
	}
	s.discardTail(sb)
}

// insertBlockBefore returns a new block, inserting it before next.
func (s *Sparse) insertBlockBefore(next *block) *block {
	b := new(block)
	b.next = next
	b.prev = next.prev
	b.prev.next = b
	next.prev = b
	return b
}

// discardTail removes block b and all its successors from s.
func (s *Sparse) discardTail(b *block) {
	if b != &s.root {
		b.prev.next = &s.root
		s.root.prev = b.prev
	}
}

// IntersectionWith sets s to the intersection s ∩ x.
func (s *Sparse) IntersectionWith(x *Sparse) {
	if s == x {
		return
	}

	xb := x.start()
	sb := s.start()
	for xb != &x.root && sb != &s.root {
		switch {
		case xb.offset < sb.offset:
			xb = xb.next

		case xb.offset > sb.offset:
			sb = sb.next
			s.removeBlock(sb.prev)

		default:
			var sum word
			for i := range sb.bits {
				r := xb.bits[i] & sb.bits[i]
				sb.bits[i] = r
				sum |= r
			}
			if sum != 0 {
				sb = sb.next
			} else {
				// sb will be overwritten or removed
			}

			xb = xb.next
		}
	}

	s.discardTail(sb)
}

// Intersection sets s to the intersection x ∩ y.
func (s *Sparse) Intersection(x, y *Sparse) {
	switch {
	case s == x:
		s.IntersectionWith(y)
		return
	case s == y:
		s.IntersectionWith(x)
		return
	case x == y:
		s.Copy(x)
		return
	}

	xb := x.start()
	yb := y.start()
	sb := s.start()
	for xb != &x.root && yb != &y.root {
		switch {
		case xb.offset < yb.offset:
			xb = xb.next
			continue
		case xb.offset > yb.offset:
			yb = yb.next
			continue
		}

		if sb == &s.root {
			sb = s.insertBlockBefore(sb)
		}
		sb.offset = xb.offset

		var sum word
		for i := range sb.bits {
			r := xb.bits[i] & yb.bits[i]
			sb.bits[i] = r
			sum |= r
		}
		if sum != 0 {
			sb = sb.next
		} else {
			// sb will be overwritten or removed
		}

		xb = xb.next
		yb = yb.next
	}

	s.discardTail(sb)
}

// UnionWith sets s to the union s ∪ x, and reports whether s grew.
func (s *Sparse) UnionWith(x *Sparse) bool {
	if s == x {
		return false
	}

	var changed bool
	xb := x.start()
	sb := s.start()
	for xb != &x.root {
		if sb != &s.root && sb.offset == xb.offset {
			for i := range xb.bits {
				if sb.bits[i] != xb.bits[i] {
					sb.bits[i] |= xb.bits[i]
					changed = true
				}
			}
			xb = xb.next
		} else if sb == &s.root || sb.offset > xb.offset {
			sb = s.insertBlockBefore(sb)
			sb.offset = xb.offset
			sb.bits = xb.bits
			changed = true

			xb = xb.next
		}
		sb = sb.next
	}
	return changed
}

// Union sets s to the union x ∪ y.
func (s *Sparse) Union(x, y *Sparse) {
	switch {
	case x == y:
		s.Copy(x)
		return
	case s == x:
		s.UnionWith(y)
		return
	case s == y:
		s.UnionWith(x)
		return
	}

	xb := x.start()
	yb := y.start()
	sb := s.start()
	for xb != &x.root || yb != &y.root {
		if sb == &s.root {
			sb = s.insertBlockBefore(sb)
		}
		switch {
		case yb == &y.root || (xb != &x.root && xb.offset < yb.offset):
			sb.offset = xb.offset
			sb.bits = xb.bits
			xb = xb.next

		case xb == &x.root || (yb != &y.root && yb.offset < xb.offset):
			sb.offset = yb.offset
			sb.bits = yb.bits
			yb = yb.next

		default:
			sb.offset = xb.offset
			for i := range xb.bits {
				sb.bits[i] = xb.bits[i] | yb.bits[i]
			}
			xb = xb.next
			yb = yb.next
		}
		sb = sb.next
	}

	s.discardTail(sb)
}

// DifferenceWith sets s to the difference s ∖ x.
func (s *Sparse) DifferenceWith(x *Sparse) {
	if s == x {
		s.Clear()
		return
	}

	xb := x.start()
	sb := s.start()
	for xb != &x.root && sb != &s.root {
		switch {
		case xb.offset > sb.offset:
			sb = sb.next

		case xb.offset < sb.offset:
			xb = xb.next

		default:
			var sum word
			for i := range sb.bits {
				r := sb.bits[i] & ^xb.bits[i]
				sb.bits[i] = r
				sum |= r
			}
			sb = sb.next
			xb = xb.next

			if sum == 0 {
				s.removeBlock(sb.prev)
			}
		}
	}
}

// Difference sets s to the difference x ∖ y.
func (s *Sparse) Difference(x, y *Sparse) {
	switch {
	case x == y:
		s.Clear()
		return
	case s == x:
		s.DifferenceWith(y)
		return
	case s == y:
		var y2 Sparse
		y2.Copy(y)
		s.Difference(x, &y2)
		return
	}

	xb := x.start()
	yb := y.start()
	sb := s.start()
	for xb != &x.root && yb != &y.root {
		if xb.offset > yb.offset {
			// y has block, x has none
			yb = yb.next
			continue
		}

		if sb == &s.root {
			sb = s.insertBlockBefore(sb)
		}
		sb.offset = xb.offset

		switch {
		case xb.offset < yb.offset:
			// x has block, y has none
			sb.bits = xb.bits

			sb = sb.next

		default:
			// x and y have corresponding blocks
			var sum word
			for i := range sb.bits {
				r := xb.bits[i] & ^yb.bits[i]
				sb.bits[i] = r
				sum |= r
			}
			if sum != 0 {
				sb = sb.next
			} else {
				// sb will be overrwritten or removed
			}

			yb = yb.next
		}
		xb = xb.next
	}

	for xb != &x.root {
		if sb == &s.root {
			sb = s.insertBlockBefore(sb)
		}
		sb.offset = xb.offset
		sb.bits = xb.bits
		sb = sb.next

		xb = xb.next
	}

	s.discardTail(sb)
}

// Equals reports whether the sets s and t have the same elements.
func (s *Sparse) Equals(t *Sparse) bool {
	if s == t {
		return true
	}
	sb := s.start()
	tb := t.start()
	for {
		switch {
		case sb == &s.root && tb == &t.root:
			return true
		case sb == &s.root || tb == &t.root:
			return false
		case sb.offset != tb.offset:
			return false
		case sb.bits != tb.bits:
			return false
		}

		sb = sb.next
		tb = tb.next
	}
}

// String returns a human-readable description of the set s.
func (s *Sparse) String() string {
	var buf bytes.Buffer
	buf.WriteByte('{')
	s.forEach(func(x int) {
		if buf.Len() > 1 {
			buf.WriteByte(' ')
		}
		fmt.Fprintf(&buf, "%d", x)
	})
	buf.WriteByte('}')
	return buf.String()
}

// BitString returns the set as a string of 1s and 0s denoting the sum
// of the i'th powers of 2, for each i in s.  A radix point, always
// preceded by a digit, appears if the sum is non-integral.
//
// Examples:
//              {}.BitString() =      "0"
//	     {4,5}.BitString() = "110000"
//            {-3}.BitString() =      "0.001"
//      {-3,0,4,5}.BitString() = "110001.001"
//
func (s *Sparse) BitString() string {
	if s.IsEmpty() {
		return "0"
	}

	min, max := s.Min(), s.Max()
	var nbytes int
	if max > 0 {
		nbytes = max
	}
	nbytes++ // zero bit
	radix := nbytes
	if min < 0 {
		nbytes += len(".") - min
	}

	b := make([]byte, nbytes)
	for i := range b {
		b[i] = '0'
	}
	if radix < nbytes {
		b[radix] = '.'
	}
	s.forEach(func(x int) {
		if x >= 0 {
			x += len(".")
		}
		b[radix-x] = '1'
	})
	return string(b)
}

// GoString returns a string showing the internal representation of
// the set s.
//
func (s *Sparse) GoString() string {
	var buf bytes.Buffer
	for b := s.start(); b != &s.root; b = b.next {
		fmt.Fprintf(&buf, "block %p {offset=%d next=%p prev=%p",
			b, b.offset, b.next, b.prev)
		for _, w := range b.bits {
			fmt.Fprintf(&buf, " 0%016x", w)
		}
		fmt.Fprintf(&buf, "}\n")
	}
	return buf.String()
}

// AppendTo returns the result of appending the elements of s to slice
// in order.
func (s *Sparse) AppendTo(slice []int) []int {
	s.forEach(func(x int) {
		slice = append(slice, x)
	})
	return slice
}

// -- Testing/debugging ------------------------------------------------

// check returns an error if the representation invariants of s are violated.
func (s *Sparse) check() error {
	if !s.root.empty() {
		return fmt.Errorf("non-empty root block")
	}
	if s.root.offset != 0 {
		return fmt.Errorf("root block has non-zero offset %d", s.root.offset)
	}
	for b := s.start(); b != &s.root; b = b.next {
		if b.offset%bitsPerBlock != 0 {
			return fmt.Errorf("bad offset modulo: %d", b.offset)
		}
		if b.empty() {
			return fmt.Errorf("empty block")
		}
		if b.prev.next != b {
			return fmt.Errorf("bad prev.next link")
		}
		if b.next.prev != b {
			return fmt.Errorf("bad next.prev link")
		}
		if b.prev != &s.root {
			if b.offset <= b.prev.offset {
				return fmt.Errorf("bad offset order: b.offset=%d, prev.offset=%d",
					b.offset, b.prev.offset)
			}
		}
	}
	return nil
}