803 lines
17 KiB
Go
803 lines
17 KiB
Go
// Copyright 2014 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Package intsets provides Sparse, a compact and fast representation
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// for sparse sets of int values.
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//
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// The time complexity of the operations Len, Insert, Remove and Has
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// is in O(n) but in practice those methods are faster and more
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// space-efficient than equivalent operations on sets based on the Go
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// map type. The IsEmpty, Min, Max, Clear and TakeMin operations
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// require constant time.
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//
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package intsets
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// TODO(adonovan):
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// - Add SymmetricDifference(x, y *Sparse), i.e. x ∆ y.
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// - Add InsertAll(...int), RemoveAll(...int)
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// - Add 'bool changed' results for {Intersection,Difference}With too.
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//
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// TODO(adonovan): implement Dense, a dense bit vector with a similar API.
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// The space usage would be proportional to Max(), not Len(), and the
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// implementation would be based upon big.Int.
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//
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// TODO(adonovan): experiment with making the root block indirect (nil
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// iff IsEmpty). This would reduce the memory usage when empty and
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// might simplify the aliasing invariants.
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import (
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"bytes"
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"fmt"
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)
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// A Sparse is a set of int values.
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// Sparse operations (even queries) are not concurrency-safe.
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//
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// The zero value for Sparse is a valid empty set.
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//
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// Sparse sets must be copied using the Copy method, not by assigning
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// a Sparse value.
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//
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type Sparse struct {
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// An uninitialized Sparse represents an empty set.
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// An empty set may also be represented by
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// root.next == root.prev == &root.
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// In a non-empty set, root.next points to the first block and
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// root.prev to the last.
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// root.offset and root.bits are unused.
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root block
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}
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type word uintptr
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const (
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_m = ^word(0)
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bitsPerWord = 8 << (_m>>8&1 + _m>>16&1 + _m>>32&1)
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bitsPerBlock = 256 // optimal value for go/pointer solver performance
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wordsPerBlock = bitsPerBlock / bitsPerWord
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)
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// Limit values of implementation-specific int type.
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const (
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MaxInt = int(^uint(0) >> 1)
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MinInt = -MaxInt - 1
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)
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// -- block ------------------------------------------------------------
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// A set is represented as a circular doubly-linked list of blocks,
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// each containing an offset and a bit array of fixed size
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// bitsPerBlock; the blocks are ordered by increasing offset.
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//
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// The set contains an element x iff the block whose offset is x - (x
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// mod bitsPerBlock) has the bit (x mod bitsPerBlock) set, where mod
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// is the Euclidean remainder.
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//
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// A block may only be empty transiently.
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//
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type block struct {
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offset int // offset mod bitsPerBlock == 0
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bits [wordsPerBlock]word // contains at least one set bit
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next, prev *block // doubly-linked list of blocks
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}
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// wordMask returns the word index (in block.bits)
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// and single-bit mask for the block's ith bit.
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func wordMask(i uint) (w uint, mask word) {
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w = i / bitsPerWord
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mask = 1 << (i % bitsPerWord)
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return
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}
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// insert sets the block b's ith bit and
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// returns true if it was not already set.
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//
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func (b *block) insert(i uint) bool {
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w, mask := wordMask(i)
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if b.bits[w]&mask == 0 {
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b.bits[w] |= mask
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return true
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}
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return false
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}
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// remove clears the block's ith bit and
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// returns true if the bit was previously set.
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// NB: may leave the block empty.
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//
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func (b *block) remove(i uint) bool {
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w, mask := wordMask(i)
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if b.bits[w]&mask != 0 {
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b.bits[w] &^= mask
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return true
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}
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return false
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}
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// has reports whether the block's ith bit is set.
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func (b *block) has(i uint) bool {
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w, mask := wordMask(i)
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return b.bits[w]&mask != 0
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}
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// empty reports whether b.len()==0, but more efficiently.
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func (b *block) empty() bool {
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for _, w := range b.bits {
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if w != 0 {
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return false
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}
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}
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return true
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}
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// len returns the number of set bits in block b.
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func (b *block) len() int {
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var l int
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for _, w := range b.bits {
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l += popcount(w)
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}
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return l
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}
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// max returns the maximum element of the block.
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// The block must not be empty.
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//
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func (b *block) max() int {
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bi := b.offset + bitsPerBlock
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// Decrement bi by number of high zeros in last.bits.
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for i := len(b.bits) - 1; i >= 0; i-- {
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if w := b.bits[i]; w != 0 {
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return bi - nlz(w) - 1
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}
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bi -= bitsPerWord
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}
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panic("BUG: empty block")
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}
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// min returns the minimum element of the block,
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// and also removes it if take is set.
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// The block must not be initially empty.
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// NB: may leave the block empty.
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//
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func (b *block) min(take bool) int {
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for i, w := range b.bits {
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if w != 0 {
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tz := ntz(w)
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if take {
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b.bits[i] = w &^ (1 << uint(tz))
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}
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return b.offset + int(i*bitsPerWord) + tz
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}
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}
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panic("BUG: empty block")
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}
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// forEach calls f for each element of block b.
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// f must not mutate b's enclosing Sparse.
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func (b *block) forEach(f func(int)) {
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for i, w := range b.bits {
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offset := b.offset + i*bitsPerWord
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for bi := 0; w != 0 && bi < bitsPerWord; bi++ {
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if w&1 != 0 {
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f(offset)
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}
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offset++
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w >>= 1
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}
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}
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}
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// offsetAndBitIndex returns the offset of the block that would
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// contain x and the bit index of x within that block.
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//
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func offsetAndBitIndex(x int) (int, uint) {
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mod := x % bitsPerBlock
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if mod < 0 {
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// Euclidean (non-negative) remainder
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mod += bitsPerBlock
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}
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return x - mod, uint(mod)
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}
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// -- Sparse --------------------------------------------------------------
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// start returns the root's next block, which is the root block
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// (if s.IsEmpty()) or the first true block otherwise.
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// start has the side effect of ensuring that s is properly
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// initialized.
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//
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func (s *Sparse) start() *block {
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root := &s.root
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if root.next == nil {
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root.next = root
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root.prev = root
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} else if root.next.prev != root {
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// Copying a Sparse x leads to pernicious corruption: the
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// new Sparse y shares the old linked list, but iteration
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// on y will never encounter &y.root so it goes into a
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// loop. Fail fast before this occurs.
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panic("A Sparse has been copied without (*Sparse).Copy()")
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}
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return root.next
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}
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// IsEmpty reports whether the set s is empty.
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func (s *Sparse) IsEmpty() bool {
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return s.start() == &s.root
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}
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// Len returns the number of elements in the set s.
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func (s *Sparse) Len() int {
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var l int
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for b := s.start(); b != &s.root; b = b.next {
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l += b.len()
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}
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return l
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}
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// Max returns the maximum element of the set s, or MinInt if s is empty.
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func (s *Sparse) Max() int {
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if s.IsEmpty() {
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return MinInt
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}
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return s.root.prev.max()
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}
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// Min returns the minimum element of the set s, or MaxInt if s is empty.
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func (s *Sparse) Min() int {
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if s.IsEmpty() {
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return MaxInt
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}
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return s.root.next.min(false)
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}
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// block returns the block that would contain offset,
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// or nil if s contains no such block.
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//
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func (s *Sparse) block(offset int) *block {
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b := s.start()
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for b != &s.root && b.offset <= offset {
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if b.offset == offset {
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return b
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}
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b = b.next
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}
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return nil
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}
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// Insert adds x to the set s, and reports whether the set grew.
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func (s *Sparse) Insert(x int) bool {
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offset, i := offsetAndBitIndex(x)
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b := s.start()
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for b != &s.root && b.offset <= offset {
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if b.offset == offset {
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return b.insert(i)
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}
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b = b.next
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}
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// Insert new block before b.
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new := &block{offset: offset}
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new.next = b
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new.prev = b.prev
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new.prev.next = new
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new.next.prev = new
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return new.insert(i)
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}
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func (s *Sparse) removeBlock(b *block) {
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b.prev.next = b.next
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b.next.prev = b.prev
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}
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// Remove removes x from the set s, and reports whether the set shrank.
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func (s *Sparse) Remove(x int) bool {
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offset, i := offsetAndBitIndex(x)
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if b := s.block(offset); b != nil {
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if !b.remove(i) {
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return false
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}
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if b.empty() {
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s.removeBlock(b)
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}
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return true
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}
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return false
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}
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// Clear removes all elements from the set s.
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func (s *Sparse) Clear() {
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s.root.next = &s.root
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s.root.prev = &s.root
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}
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// If set s is non-empty, TakeMin sets *p to the minimum element of
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// the set s, removes that element from the set and returns true.
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// Otherwise, it returns false and *p is undefined.
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//
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// This method may be used for iteration over a worklist like so:
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//
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// var x int
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// for worklist.TakeMin(&x) { use(x) }
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//
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func (s *Sparse) TakeMin(p *int) bool {
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head := s.start()
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if head == &s.root {
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return false
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}
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*p = head.min(true)
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if head.empty() {
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s.removeBlock(head)
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}
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return true
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}
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// Has reports whether x is an element of the set s.
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func (s *Sparse) Has(x int) bool {
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offset, i := offsetAndBitIndex(x)
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if b := s.block(offset); b != nil {
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return b.has(i)
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}
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return false
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}
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// forEach applies function f to each element of the set s in order.
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//
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// f must not mutate s. Consequently, forEach is not safe to expose
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// to clients. In any case, using "range s.AppendTo()" allows more
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// natural control flow with continue/break/return.
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//
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func (s *Sparse) forEach(f func(int)) {
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for b := s.start(); b != &s.root; b = b.next {
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b.forEach(f)
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}
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}
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// Copy sets s to the value of x.
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func (s *Sparse) Copy(x *Sparse) {
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if s == x {
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return
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}
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xb := x.start()
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sb := s.start()
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for xb != &x.root {
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if sb == &s.root {
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sb = s.insertBlockBefore(sb)
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}
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sb.offset = xb.offset
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sb.bits = xb.bits
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xb = xb.next
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sb = sb.next
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}
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s.discardTail(sb)
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}
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// insertBlockBefore returns a new block, inserting it before next.
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func (s *Sparse) insertBlockBefore(next *block) *block {
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b := new(block)
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b.next = next
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b.prev = next.prev
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b.prev.next = b
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next.prev = b
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return b
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}
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// discardTail removes block b and all its successors from s.
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func (s *Sparse) discardTail(b *block) {
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if b != &s.root {
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b.prev.next = &s.root
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s.root.prev = b.prev
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}
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}
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// IntersectionWith sets s to the intersection s ∩ x.
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func (s *Sparse) IntersectionWith(x *Sparse) {
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if s == x {
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return
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}
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xb := x.start()
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sb := s.start()
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for xb != &x.root && sb != &s.root {
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switch {
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case xb.offset < sb.offset:
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xb = xb.next
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case xb.offset > sb.offset:
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sb = sb.next
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s.removeBlock(sb.prev)
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default:
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var sum word
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for i := range sb.bits {
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r := xb.bits[i] & sb.bits[i]
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sb.bits[i] = r
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sum |= r
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}
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if sum != 0 {
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sb = sb.next
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} else {
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// sb will be overwritten or removed
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}
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xb = xb.next
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}
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}
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s.discardTail(sb)
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}
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// Intersection sets s to the intersection x ∩ y.
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func (s *Sparse) Intersection(x, y *Sparse) {
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switch {
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case s == x:
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s.IntersectionWith(y)
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return
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case s == y:
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s.IntersectionWith(x)
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return
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case x == y:
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s.Copy(x)
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return
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}
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xb := x.start()
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yb := y.start()
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sb := s.start()
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for xb != &x.root && yb != &y.root {
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switch {
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case xb.offset < yb.offset:
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xb = xb.next
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continue
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case xb.offset > yb.offset:
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yb = yb.next
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continue
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}
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if sb == &s.root {
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sb = s.insertBlockBefore(sb)
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}
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sb.offset = xb.offset
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var sum word
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for i := range sb.bits {
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r := xb.bits[i] & yb.bits[i]
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sb.bits[i] = r
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sum |= r
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}
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if sum != 0 {
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sb = sb.next
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} else {
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// sb will be overwritten or removed
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}
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xb = xb.next
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yb = yb.next
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}
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s.discardTail(sb)
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}
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// UnionWith sets s to the union s ∪ x, and reports whether s grew.
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func (s *Sparse) UnionWith(x *Sparse) bool {
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if s == x {
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return false
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}
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var changed bool
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xb := x.start()
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sb := s.start()
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for xb != &x.root {
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if sb != &s.root && sb.offset == xb.offset {
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for i := range xb.bits {
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if sb.bits[i] != xb.bits[i] {
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sb.bits[i] |= xb.bits[i]
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changed = true
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}
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}
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xb = xb.next
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} else if sb == &s.root || sb.offset > xb.offset {
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sb = s.insertBlockBefore(sb)
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sb.offset = xb.offset
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sb.bits = xb.bits
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changed = true
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xb = xb.next
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}
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sb = sb.next
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}
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return changed
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}
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// Union sets s to the union x ∪ y.
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func (s *Sparse) Union(x, y *Sparse) {
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switch {
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case x == y:
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s.Copy(x)
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return
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case s == x:
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s.UnionWith(y)
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return
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case s == y:
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s.UnionWith(x)
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return
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}
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xb := x.start()
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yb := y.start()
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sb := s.start()
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for xb != &x.root || yb != &y.root {
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if sb == &s.root {
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sb = s.insertBlockBefore(sb)
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}
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switch {
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case yb == &y.root || (xb != &x.root && xb.offset < yb.offset):
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sb.offset = xb.offset
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sb.bits = xb.bits
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xb = xb.next
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case xb == &x.root || (yb != &y.root && yb.offset < xb.offset):
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sb.offset = yb.offset
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sb.bits = yb.bits
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yb = yb.next
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default:
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sb.offset = xb.offset
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for i := range xb.bits {
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sb.bits[i] = xb.bits[i] | yb.bits[i]
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}
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xb = xb.next
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yb = yb.next
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}
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sb = sb.next
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}
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s.discardTail(sb)
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}
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// DifferenceWith sets s to the difference s ∖ x.
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func (s *Sparse) DifferenceWith(x *Sparse) {
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if s == x {
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s.Clear()
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return
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}
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xb := x.start()
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sb := s.start()
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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
|
||
}
|