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 genzfunc.go

// 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)
}

// 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.
	// 堆排序的第一步：(hi - 1) / 2 到0 的元素进行 向下堆化 heapify
	for i := (hi - 1) / 2; i >= 0; i-- {
		siftDown(data, i, hi, first)
	}

	// Pop elements, largest first, into end of data.
	// 堆排序的第二步：排序，将最大的值放到最后，剩余的进行堆化
	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 := int(uint(lo+hi) >> 1) // Written like this to avoid integer overflow.
	// 元素大于40时，进行三次三值取中
	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)
	}
	// 这时，要注意lo/m/hi 并不是字面意思
	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
	
	// lo设置为pivot，也就是中值被设置为pivot
	pivot := lo
	// 接下来要排序的是(lo,hi-1]
	a, c := lo+1, hi-1

	// 对应上面 data[lo < i < a] < pivot
	for ; a < c && data.Less(a, pivot); a++ {
	}
	b := a
	for {
		// 对应上面 data[a <= i < b] <= pivot
		for ; b < c && !data.Less(pivot, b); b++ { // data[b] <= pivot
		}
		// 对应上面 data[c <= i < hi-1] > pivot
		for ; b < c && data.Less(pivot, c-1); c-- { // data[c-1] > pivot
		}
		// 初始条件，满足 data[hi-1] >= pivot
		// 所以，还剩 data[b <= i < c] unexamined
		// 如果b>=c，那么这一段可以直接跳过
		if b >= c {
			break
		}
		// data[b] > pivot; data[c-1] <= pivot
		// 由于：data[b] > pivot,data[c-1] <= pivot
		// 所以可以通过交换这两个元素，让 data[b] <= pivot，data[c-1] > pivot
		// 再次进行上面的排序
		// 不断进行下去的话，data[b <= i < c] unexamined这一段会被排序完成
		// 注意，这里就是不稳定的原因!
		data.Swap(b, c-1)
		b++
		c--
	}
	// 总结一下：
	// 快速排序思想是以pivot为中心值，将后面的元素划分为2段：
	// 左边为<=pivot的，右边为>=pivot
	
	// If hi-c<3 then there are duplicates (by property of median of nine).
	// Let's be a bit more conservative, and set border to 5.
	
	// 由于可能存在大量等于pivot的情况，所以整个排序后的分布可能是比较"偏"的
	
	// 因为c为划分pivot的大小的临界值，所以在9值划分时，正常来说，应该是两边各4个
	// 由于左边是<=，多了个相等的情况，所以5，3分布，也是没有问题
	// 如果hi-c<3，c的值明显偏向于hi，说明有多个和pivot重复值
	// 为了更保守一点，所以设置为5(反正只是多校验一次而已)
	protect := hi-c < 5
	// 即便大于等于5，也可能是因为元素总值很多，所以对比hi-c是否小于总数量的1/4
	if !protect && hi-c < (hi-lo)/4 {
		// Lets test some points for equality to pivot
		dups := 0
		// 以hi-1/b-1/m为样例，查看重复情况
		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
		// 如果dups > 1，表示超过了不少
		protect = dups > 1
	}
	// 思路基本和上面的一致，这里划分的是<pivot和=pivot的两组
	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
	// 将pivot放到中间
	data.Swap(pivot, b-1)
	return b - 1, c
}

func quickSort(data Interface, a, b, maxDepth int) {
	// 元素多于12时，循环处理
	for b-a > 12 { // Use ShellSort for slices <= 12 elements
		// 对应前面的maxDepth分析，当其为0时，进行堆排序
		if maxDepth == 0 {
			// TODO：怎么进行堆排序?
			heapSort(data, a, b)
			return
		}
		maxDepth--
		// 从字面来看，取出了两个值：lo和hi
		// TODO: doPivot 是怎么取值的？
		// pivot：快速排序的分区点
		mlo, mhi := doPivot(data, a, b)
		// Avoiding recursion on the larger subproblem guarantees
		// a stack depth of at most lg(b-a).
		// 数据规模较小的进行递归
		// 数据规模较大的继续在循环中进行
		// 为什么这里不进行两个递归？
		// 其实用两个递归也是可以实现的，但这里是偏工程化的思想，我个人能想到的有两点：
		// 1. 减少一个递归，就能减少一次压栈、出栈的操作
		// 2. 那为什么选择小规模的数据进行递归呢？我觉得小规模的函数，更容易快速完成，释放空间
		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)
		}
	}
	// b-a = 1时代表只有一个元素，无需排序
	// 元素个数小于等于12时，进行一次gap为6的Shell排序（间隔为6的元素之间进行一次）,保证数据整体的一致性
	// 再进行一次插入排序
	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)
			}
		}
		// TODO: 插入排序
		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.

// 前置知识点：插入排序、堆排序、快速排序、Shell排序
// 稳定性 sort.Stable
// pivot 快速排序的分区点
// heapify 堆化
func Sort(data Interface) {
	n := data.Len()
	// TODO: 快速排序，真的跟它的名字一模一样吗？
	quickSort(data, 0, n, maxDepth(n))
}

// maxDepth returns a threshold at which quicksort should switch
// to heapsort. It returns 2*ceil(lg(n+1)).
// 设置快速排序到堆排序的阀值
// 为什么要设置这个阀值?
// 以快排为例，需要选择一个pivot-中心点，划分所有元素到到pivot的左右两边，再进行排序
// 如果pivot选择得好，那就是需要复杂度O(logN)次即可。但在极端情况下，每次都选择到最边缘的元素，那就需要O(N)次。
func maxDepth(n int) int {
	var depth int
	for i := n; i > 0; i >>= 1 {
		depth++
	}
	return depth * 2
}

// 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 (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
// (not-a-number values are treated as less than other values).
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
// (not-a-number values are treated as less than other values).
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
// (not-a-number values are treated as less than other values).
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 references 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) {
	stable(data, data.Len())
}

func stable(data Interface, n int) {
	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 := int(uint(i+j) >> 1)
			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 := int(uint(i+j) >> 1)
			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 := int(uint(a+b) >> 1)
	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 := int(uint(start+r) >> 1)
		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 consecutive 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))

*/