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.

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

// A type, typically a collection, that satisfies sort.Interface can be
// sorted by the routines in this package.  The methods require that the
// elements of the collection be enumerated by an integer index.
type Interface interface {
	// Len is the number of elements in the collection.
	Len() int
	// Less reports whether the element with
	// index i should sort before the element with index j.
	Less(i, j int) bool
	// Swap swaps the elements with indexes i and j.
	Swap(i, j int)
}

func min(a, b int) int {
	if a < b {
		return a
	}
	return b
}

// Insertion sort
func insertionSort(data Interface, a, b int) {
	for i := a + 1; i < b; i++ {
		for j := i; j > a && data.Less(j, j-1); j-- {
			data.Swap(j, j-1)
		}
	}
}

// siftDown implements the heap property on data[lo, hi).
// first is an offset into the array where the root of the heap lies.
func siftDown(data Interface, lo, hi, first int) {
	root := lo
	for {
		child := 2*root + 1
		if child >= hi {
			break
		}
		if child+1 < hi && data.Less(first+child, first+child+1) {
			child++
		}
		if !data.Less(first+root, first+child) {
			return
		}
		data.Swap(first+root, first+child)
		root = child
	}
}

func heapSort(data Interface, a, b int) {
	first := a
	lo := 0
	hi := b - a

	// Build heap with greatest element at top.
	for i := (hi - 1) / 2; i >= 0; i-- {
		siftDown(data, i, hi, first)
	}

	// Pop elements, largest first, into end of data.
	for i := hi - 1; i >= 0; i-- {
		data.Swap(first, first+i)
		siftDown(data, lo, i, first)
	}
}

// Quicksort, loosely following Bentley and McIlroy,
// ``Engineering a Sort Function,'' SP&E November 1993.

// medianOfThree moves the median of the three values data[m0], data[m1], data[m2] into data[m1].
func medianOfThree(data Interface, m1, m0, m2 int) {
	// sort 3 elements
	if data.Less(m1, m0) {
		data.Swap(m1, m0)
	}
	// data[m0] <= data[m1]
	if data.Less(m2, m1) {
		data.Swap(m2, m1)
		// data[m0] <= data[m2] && data[m1] < data[m2]
		if data.Less(m1, m0) {
			data.Swap(m1, m0)
		}
	}
	// now data[m0] <= data[m1] <= data[m2]
}

func swapRange(data Interface, a, b, n int) {
	for i := 0; i < n; i++ {
		data.Swap(a+i, b+i)
	}
}

func doPivot(data Interface, lo, hi int) (midlo, midhi int) {
	m := lo + (hi-lo)/2 // Written like this to avoid integer overflow.
	if hi-lo > 40 {
		// Tukey's ``Ninther,'' median of three medians of three.
		s := (hi - lo) / 8
		medianOfThree(data, lo, lo+s, lo+2*s)
		medianOfThree(data, m, m-s, m+s)
		medianOfThree(data, hi-1, hi-1-s, hi-1-2*s)
	}
	medianOfThree(data, lo, m, hi-1)

	// Invariants are:
	//	data[lo] = pivot (set up by ChoosePivot)
	//	data[lo < i < a] < pivot
	//	data[a <= i < b] <= pivot
	//	data[b <= i < c] unexamined
	//	data[c <= i < hi-1] > pivot
	//	data[hi-1] >= pivot
	pivot := lo
	a, c := lo+1, hi-1

	for ; a != c && data.Less(a, pivot); a++ {
	}
	b := a
	for {
		for ; b != c && !data.Less(pivot, b); b++ { // data[b] <= pivot
		}
		for ; b != c && data.Less(pivot, c-1); c-- { // data[c-1] > pivot
		}
		if b == c {
			break
		}
		// data[b] > pivot; data[c-1] <= pivot
		data.Swap(b, c-1)
		b++
		c--
	}
	// If hi-c<3 then there are duplicates (by property of median of nine).
	// Let be a bit more conservative, and set border to 5.
	protect := hi-c < 5
	if !protect && hi-c < (hi-lo)/4 {
		// Lets test some points for equality to pivot
		dups := 0
		if !data.Less(pivot, hi-1) { // data[hi-1] = pivot
			data.Swap(c, hi-1)
			c++
			dups++
		}
		if !data.Less(b-1, pivot) { // data[b-1] = pivot
			b--
			dups++
		}
		// m-lo = (hi-lo)/2 > 6
		// b-lo > (hi-lo)*3/4-1 > 8
		// ==> m < b ==> data[m] <= pivot
		if !data.Less(m, pivot) { // data[m] = pivot
			data.Swap(m, b-1)
			b--
			dups++
		}
		// if at least 2 points are equal to pivot, assume skewed distribution
		protect = dups > 1
	}
	if protect {
		// Protect against a lot of duplicates
		// Add invariant:
		//	data[a <= i < b] unexamined
		//	data[b <= i < c] = pivot
		for {
			for ; a != b && !data.Less(b-1, pivot); b-- { // data[b] == pivot
			}
			for ; a != b && data.Less(a, pivot); a++ { // data[a] < pivot
			}
			if a == b {
				break
			}
			// data[a] == pivot; data[b-1] < pivot
			data.Swap(a, b-1)
			a++
			b--
		}
	}
	// Swap pivot into middle
	data.Swap(pivot, b-1)
	return b - 1, c
}

func quickSort(data Interface, a, b, maxDepth int) {
	for b-a > 12 { // Use ShellSort for slices <= 12 elements
		if maxDepth == 0 {
			heapSort(data, a, b)
			return
		}
		maxDepth--
		mlo, mhi := doPivot(data, a, b)
		// Avoiding recursion on the larger subproblem guarantees
		// a stack depth of at most lg(b-a).
		if mlo-a < b-mhi {
			quickSort(data, a, mlo, maxDepth)
			a = mhi // i.e., quickSort(data, mhi, b)
		} else {
			quickSort(data, mhi, b, maxDepth)
			b = mlo // i.e., quickSort(data, a, mlo)
		}
	}
	if b-a > 1 {
		// Do ShellSort pass with gap 6
		// It could be written in this simplified form cause b-a <= 12
		for i := a + 6; i < b; i++ {
			if data.Less(i, i-6) {
				data.Swap(i, i-6)
			}
		}
		insertionSort(data, a, b)
	}
}

// Sort sorts data.
// 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) {
	// Switch to heapsort if depth of 2*ceil(lg(n+1)) is reached.
	n := data.Len()
	maxDepth := 0
	for i := n; i > 0; i >>= 1 {
		maxDepth++
	}
	maxDepth *= 2
	quickSort(data, 0, n, maxDepth)
}

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

// Sort is a convenience method.
func (p IntSlice) Sort() { Sort(p) }

// Float64Slice attaches the methods of Interface to []float64, sorting in increasing order.
type Float64Slice []float64

func (p Float64Slice) Len() int           { return len(p) }
func (p Float64Slice) Less(i, j int) bool { return p[i] < p[j] || isNaN(p[i]) && !isNaN(p[j]) }
func (p Float64Slice) Swap(i, j int)      { p[i], p[j] = p[j], p[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.
func (p Float64Slice) Sort() { Sort(p) }

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

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

// Sort is a convenience method.
func (p StringSlice) Sort() { Sort(p) }

// Convenience wrappers for common cases

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

// Float64s sorts a slice of float64s in increasing order.
func Float64s(a []float64) { Sort(Float64Slice(a)) }

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

// IntsAreSorted tests whether a slice of ints is sorted in increasing order.
func IntsAreSorted(a []int) bool { return IsSorted(IntSlice(a)) }

// Float64sAreSorted tests whether a slice of float64s is sorted in increasing order.
func Float64sAreSorted(a []float64) bool { return IsSorted(Float64Slice(a)) }

// StringsAreSorted tests whether a slice of strings is sorted in increasing order.
func StringsAreSorted(a []string) bool { return IsSorted(StringSlice(a)) }

// 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 reverences in there.
//  - Often "optimal" algorithms are optimal in the number of assignments
//    but Interface has only Swap as operation.

// Stable sorts data 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) {
	n := data.Len()
	blockSize := 20 // must be > 0
	a, b := 0, blockSize
	for b <= n {
		insertionSort(data, a, b)
		a = b
		b += blockSize
	}
	insertionSort(data, a, n)

	for blockSize < n {
		a, b = 0, 2*blockSize
		for b <= n {
			symMerge(data, a, a+blockSize, b)
			a = b
			b += 2 * blockSize
		}
		if m := a + blockSize; m < n {
			symMerge(data, a, m, n)
		}
		blockSize *= 2
	}
}

// SymMerge merges the two sorted subsequences data[a:m] and data[m:b] using
// the SymMerge algorithm from Pok-Son Kim and Arne Kutzner, "Stable Minimum
// Storage Merging by Symmetric Comparisons", in Susanne Albers and Tomasz
// Radzik, editors, Algorithms - ESA 2004, volume 3221 of Lecture Notes in
// Computer Science, pages 714-723. Springer, 2004.
//
// Let M = m-a and N = b-n. Wolog M < N.
// The recursion depth is bound by ceil(log(N+M)).
// The algorithm needs O(M*log(N/M + 1)) calls to data.Less.
// The algorithm needs O((M+N)*log(M)) calls to data.Swap.
//
// The paper gives O((M+N)*log(M)) as the number of assignments assuming a
// rotation algorithm which uses O(M+N+gcd(M+N)) assignments. The argumentation
// in the paper carries through for Swap operations, especially as the block
// swapping rotate uses only O(M+N) Swaps.
//
// symMerge assumes non-degenerate arguments: a < m && m < b.
// Having the caller check this condition eliminates many leaf recursion calls,
// which improves performance.
func symMerge(data Interface, a, m, b int) {
	// Avoid unnecessary recursions of symMerge
	// by direct insertion of data[a] into data[m:b]
	// if data[a:m] only contains one element.
	if m-a == 1 {
		// Use binary search to find the lowest index i
		// such that data[i] >= data[a] for m <= i < b.
		// Exit the search loop with i == b in case no such index exists.
		i := m
		j := b
		for i < j {
			h := i + (j-i)/2
			if data.Less(h, a) {
				i = h + 1
			} else {
				j = h
			}
		}
		// Swap values until data[a] reaches the position before i.
		for k := a; k < i-1; k++ {
			data.Swap(k, k+1)
		}
		return
	}

	// Avoid unnecessary recursions of symMerge
	// by direct insertion of data[m] into data[a:m]
	// if data[m:b] only contains one element.
	if b-m == 1 {
		// Use binary search to find the lowest index i
		// such that data[i] > data[m] for a <= i < m.
		// Exit the search loop with i == m in case no such index exists.
		i := a
		j := m
		for i < j {
			h := i + (j-i)/2
			if !data.Less(m, h) {
				i = h + 1
			} else {
				j = h
			}
		}
		// Swap values until data[m] reaches the position i.
		for k := m; k > i; k-- {
			data.Swap(k, k-1)
		}
		return
	}

	mid := a + (b-a)/2
	n := mid + m
	var start, r int
	if m > mid {
		start = n - b
		r = mid
	} else {
		start = a
		r = m
	}
	p := n - 1

	for start < r {
		c := start + (r-start)/2
		if !data.Less(p-c, c) {
			start = c + 1
		} else {
			r = c
		}
	}

	end := n - start
	if start < m && m < end {
		rotate(data, start, m, end)
	}
	if a < start && start < mid {
		symMerge(data, a, start, mid)
	}
	if mid < end && end < b {
		symMerge(data, mid, end, b)
	}
}

// Rotate two consecutives blocks u = data[a:m] and v = data[m:b] in data:
// Data of the form 'x u v y' is changed to 'x v u y'.
// Rotate performs at most b-a many calls to data.Swap.
// Rotate assumes non-degenerate arguments: a < m && m < b.
func rotate(data Interface, a, m, b int) {
	i := m - a
	j := b - m

	for i != j {
		if i > j {
			swapRange(data, m-i, m, j)
			i -= j
		} else {
			swapRange(data, m-i, m+j-i, i)
			j -= i
		}
	}
	// i == j
	swapRange(data, m-i, m, i)
}

/*
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))

*/