src/sort/sort.go

// Copyright 2009 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.

//go:generate go run gen_sort_variants.go

// Package sort provides primitives for sorting slices and user-defined collections.
package sort

import "math/bits"

// An implementation of Interface can be sorted by the routines in this package.
// The methods refer to elements of the underlying collection by integer index.
type Interface interface {
	// Len is the number of elements in the collection.
	Len() int

	// Less reports whether the element with index i
	// must sort before the element with index j.
	//
	// If both Less(i, j) and Less(j, i) are false,
	// then the elements at index i and j are considered equal.
	// Sort may place equal elements in any order in the final result,
	// while Stable preserves the original input order of equal elements.
	//
	// Less must describe a transitive ordering:
	//  - if both Less(i, j) and Less(j, k) are true, then Less(i, k) must be true as well.
	//  - if both Less(i, j) and Less(j, k) are false, then Less(i, k) must be false as well.
	//
	// Note that floating-point comparison (the < operator on float32 or float64 values)
	// is not a transitive ordering when not-a-number (NaN) values are involved.
	// See Float64Slice.Less for a correct implementation for floating-point values.
	Less(i, j int) bool

	// Swap swaps the elements with indexes i and j.
	Swap(i, j int)
}

// Sort sorts data in ascending order as determined by the Less method.
// It makes one call to data.Len to determine n and O(n*log(n)) calls to
// data.Less and data.Swap. The sort is not guaranteed to be stable.
func Sort(data Interface) {
	n := data.Len()
	if n <= 1 {
		return
	}
	limit := bits.Len(uint(n))
	pdqsort(data, 0, n, limit)
}

type sortedHint int // hint for pdqsort when choosing the pivot

const (
	unknownHint sortedHint = iota
	increasingHint
	decreasingHint
)

// xorshift paper: https://www.jstatsoft.org/article/view/v008i14/xorshift.pdf
type xorshift uint64

func (r *xorshift) Next() uint64 {
	*r ^= *r << 13
	*r ^= *r >> 17
	*r ^= *r << 5
	return uint64(*r)
}

func nextPowerOfTwo(length int) uint {
	shift := uint(bits.Len(uint(length)))
	return uint(1 << shift)
}

// lessSwap is a pair of Less and Swap function for use with the
// auto-generated func-optimized variant of sort.go in
// zfuncversion.go.
type lessSwap struct {
	Less func(i, j int) bool
	Swap func(i, j int)
}

type reverse struct {
	// This embedded Interface permits Reverse to use the methods of
	// another Interface implementation.
	Interface
}

// Less returns the opposite of the embedded implementation's Less method.
func (r reverse) Less(i, j int) bool {
	return r.Interface.Less(j, i)
}

// Reverse returns the reverse order for data.
func Reverse(data Interface) Interface {
	return &reverse{data}
}

// IsSorted reports whether data is sorted.
func IsSorted(data Interface) bool {
	n := data.Len()
	for i := n - 1; i > 0; i-- {
		if data.Less(i, i-1) {
			return false
		}
	}
	return true
}

// Convenience types for common cases

// IntSlice attaches the methods of Interface to []int, sorting in increasing order.
type IntSlice []int

func (x IntSlice) Len() int           { return len(x) }
func (x IntSlice) Less(i, j int) bool { return x[i] < x[j] }
func (x IntSlice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }

// Sort is a convenience method: x.Sort() calls Sort(x).
func (x IntSlice) Sort() { Sort(x) }

// Float64Slice implements Interface for a []float64, sorting in increasing order,
// with not-a-number (NaN) values ordered before other values.
type Float64Slice []float64

func (x Float64Slice) Len() int { return len(x) }

// Less reports whether x[i] should be ordered before x[j], as required by the sort Interface.
// Note that floating-point comparison by itself is not a transitive relation: it does not
// report a consistent ordering for not-a-number (NaN) values.
// This implementation of Less places NaN values before any others, by using:
//
//	x[i] < x[j] || (math.IsNaN(x[i]) && !math.IsNaN(x[j]))
func (x Float64Slice) Less(i, j int) bool { return x[i] < x[j] || (isNaN(x[i]) && !isNaN(x[j])) }
func (x Float64Slice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }

// isNaN is a copy of math.IsNaN to avoid a dependency on the math package.
func isNaN(f float64) bool {
	return f != f
}

// Sort is a convenience method: x.Sort() calls Sort(x).
func (x Float64Slice) Sort() { Sort(x) }

// StringSlice attaches the methods of Interface to []string, sorting in increasing order.
type StringSlice []string

func (x StringSlice) Len() int           { return len(x) }
func (x StringSlice) Less(i, j int) bool { return x[i] < x[j] }
func (x StringSlice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }

// Sort is a convenience method: x.Sort() calls Sort(x).
func (x StringSlice) Sort() { Sort(x) }

// Convenience wrappers for common cases

// Ints sorts a slice of ints in increasing order.
func Ints(x []int) { Sort(IntSlice(x)) }

// Float64s sorts a slice of float64s in increasing order.
// Not-a-number (NaN) values are ordered before other values.
func Float64s(x []float64) { Sort(Float64Slice(x)) }

// Strings sorts a slice of strings in increasing order.
func Strings(x []string) { Sort(StringSlice(x)) }

// IntsAreSorted reports whether the slice x is sorted in increasing order.
func IntsAreSorted(x []int) bool { return IsSorted(IntSlice(x)) }

// Float64sAreSorted reports whether the slice x is sorted in increasing order,
// with not-a-number (NaN) values before any other values.
func Float64sAreSorted(x []float64) bool { return IsSorted(Float64Slice(x)) }

// StringsAreSorted reports whether the slice x is sorted in increasing order.
func StringsAreSorted(x []string) bool { return IsSorted(StringSlice(x)) }

// Notes on stable sorting:
// The used algorithms are simple and provable correct on all input and use
// only logarithmic additional stack space. They perform well if compared
// experimentally to other stable in-place sorting algorithms.
//
// Remarks on other algorithms evaluated:
//  - GCC's 4.6.3 stable_sort with merge_without_buffer from libstdc++:
//    Not faster.
//  - GCC's __rotate for block rotations: Not faster.
//  - "Practical in-place mergesort" from  Jyrki Katajainen, Tomi A. Pasanen
//    and Jukka Teuhola; Nordic Journal of Computing 3,1 (1996), 27-40:
//    The given algorithms are in-place, number of Swap and Assignments
//    grow as n log n but the algorithm is not stable.
//  - "Fast Stable In-Place Sorting with O(n) Data Moves" J.I. Munro and
//    V. Raman in Algorithmica (1996) 16, 115-160:
//    This algorithm either needs additional 2n bits or works only if there
//    are enough different elements available to encode some permutations
//    which have to be undone later (so not stable on any input).
//  - All the optimal in-place sorting/merging algorithms I found are either
//    unstable or rely on enough different elements in each step to encode the
//    performed block rearrangements. See also "In-Place Merging Algorithms",
//    Denham Coates-Evely, Department of Computer Science, Kings College,
//    January 2004 and the references in there.
//  - Often "optimal" algorithms are optimal in the number of assignments
//    but Interface has only Swap as operation.

// Stable sorts data in ascending order as determined by the Less method,
// while keeping the original order of equal elements.
//
// It makes one call to data.Len to determine n, O(n*log(n)) calls to
// data.Less and O(n*log(n)*log(n)) calls to data.Swap.
func Stable(data Interface) {
	stable(data, data.Len())
}

/*
Complexity of Stable Sorting


Complexity of block swapping rotation

Each Swap puts one new element into its correct, final position.
Elements which reach their final position are no longer moved.
Thus block swapping rotation needs |u|+|v| calls to Swaps.
This is best possible as each element might need a move.

Pay attention when comparing to other optimal algorithms which
typically count the number of assignments instead of swaps:
E.g. the optimal algorithm of Dudzinski and Dydek for in-place
rotations uses O(u + v + gcd(u,v)) assignments which is
better than our O(3 * (u+v)) as gcd(u,v) <= u.


Stable sorting by SymMerge and BlockSwap rotations

SymMerg complexity for same size input M = N:
Calls to Less:  O(M*log(N/M+1)) = O(N*log(2)) = O(N)
Calls to Swap:  O((M+N)*log(M)) = O(2*N*log(N)) = O(N*log(N))

(The following argument does not fuzz over a missing -1 or
other stuff which does not impact the final result).

Let n = data.Len(). Assume n = 2^k.

Plain merge sort performs log(n) = k iterations.
On iteration i the algorithm merges 2^(k-i) blocks, each of size 2^i.

Thus iteration i of merge sort performs:
Calls to Less  O(2^(k-i) * 2^i) = O(2^k) = O(2^log(n)) = O(n)
Calls to Swap  O(2^(k-i) * 2^i * log(2^i)) = O(2^k * i) = O(n*i)

In total k = log(n) iterations are performed; so in total:
Calls to Less O(log(n) * n)
Calls to Swap O(n + 2*n + 3*n + ... + (k-1)*n + k*n)
   = O((k/2) * k * n) = O(n * k^2) = O(n * log^2(n))


Above results should generalize to arbitrary n = 2^k + p
and should not be influenced by the initial insertion sort phase:
Insertion sort is O(n^2) on Swap and Less, thus O(bs^2) per block of
size bs at n/bs blocks:  O(bs*n) Swaps and Less during insertion sort.
Merge sort iterations start at i = log(bs). With t = log(bs) constant:
Calls to Less O((log(n)-t) * n + bs*n) = O(log(n)*n + (bs-t)*n)
   = O(n * log(n))
Calls to Swap O(n * log^2(n) - (t^2+t)/2*n) = O(n * log^2(n))

*/