# Source file src/sort/sort.go

```     1  // Copyright 2009 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
4
5  //go:generate go run gen_sort_variants.go
6
7  // Package sort provides primitives for sorting slices and user-defined collections.
8  package sort
9
10  import "math/bits"
11
12  // An implementation of Interface can be sorted by the routines in this package.
13  // The methods refer to elements of the underlying collection by integer index.
14  type Interface interface {
15  	// Len is the number of elements in the collection.
16  	Len() int
17
18  	// Less reports whether the element with index i
19  	// must sort before the element with index j.
20  	//
21  	// If both Less(i, j) and Less(j, i) are false,
22  	// then the elements at index i and j are considered equal.
23  	// Sort may place equal elements in any order in the final result,
24  	// while Stable preserves the original input order of equal elements.
25  	//
26  	// Less must describe a transitive ordering:
27  	//  - if both Less(i, j) and Less(j, k) are true, then Less(i, k) must be true as well.
28  	//  - if both Less(i, j) and Less(j, k) are false, then Less(i, k) must be false as well.
29  	//
30  	// Note that floating-point comparison (the < operator on float32 or float64 values)
31  	// is not a transitive ordering when not-a-number (NaN) values are involved.
32  	// See Float64Slice.Less for a correct implementation for floating-point values.
33  	Less(i, j int) bool
34
35  	// Swap swaps the elements with indexes i and j.
36  	Swap(i, j int)
37  }
38
39  // Sort sorts data in ascending order as determined by the Less method.
40  // It makes one call to data.Len to determine n and O(n*log(n)) calls to
41  // data.Less and data.Swap. The sort is not guaranteed to be stable.
42  func Sort(data Interface) {
43  	n := data.Len()
44  	if n <= 1 {
45  		return
46  	}
47  	limit := bits.Len(uint(n))
48  	pdqsort(data, 0, n, limit)
49  }
50
51  type sortedHint int // hint for pdqsort when choosing the pivot
52
53  const (
54  	unknownHint sortedHint = iota
55  	increasingHint
56  	decreasingHint
57  )
58
59  // xorshift paper: https://www.jstatsoft.org/article/view/v008i14/xorshift.pdf
60  type xorshift uint64
61
62  func (r *xorshift) Next() uint64 {
63  	*r ^= *r << 13
64  	*r ^= *r >> 17
65  	*r ^= *r << 5
66  	return uint64(*r)
67  }
68
69  func nextPowerOfTwo(length int) uint {
70  	shift := uint(bits.Len(uint(length)))
71  	return uint(1 << shift)
72  }
73
74  // lessSwap is a pair of Less and Swap function for use with the
75  // auto-generated func-optimized variant of sort.go in
76  // zfuncversion.go.
77  type lessSwap struct {
78  	Less func(i, j int) bool
79  	Swap func(i, j int)
80  }
81
82  type reverse struct {
83  	// This embedded Interface permits Reverse to use the methods of
84  	// another Interface implementation.
85  	Interface
86  }
87
88  // Less returns the opposite of the embedded implementation's Less method.
89  func (r reverse) Less(i, j int) bool {
90  	return r.Interface.Less(j, i)
91  }
92
93  // Reverse returns the reverse order for data.
94  func Reverse(data Interface) Interface {
95  	return &reverse{data}
96  }
97
98  // IsSorted reports whether data is sorted.
99  func IsSorted(data Interface) bool {
100  	n := data.Len()
101  	for i := n - 1; i > 0; i-- {
102  		if data.Less(i, i-1) {
103  			return false
104  		}
105  	}
106  	return true
107  }
108
109  // Convenience types for common cases
110
111  // IntSlice attaches the methods of Interface to []int, sorting in increasing order.
112  type IntSlice []int
113
114  func (x IntSlice) Len() int           { return len(x) }
115  func (x IntSlice) Less(i, j int) bool { return x[i] < x[j] }
116  func (x IntSlice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }
117
118  // Sort is a convenience method: x.Sort() calls Sort(x).
119  func (x IntSlice) Sort() { Sort(x) }
120
121  // Float64Slice implements Interface for a []float64, sorting in increasing order,
122  // with not-a-number (NaN) values ordered before other values.
123  type Float64Slice []float64
124
125  func (x Float64Slice) Len() int { return len(x) }
126
127  // Less reports whether x[i] should be ordered before x[j], as required by the sort Interface.
128  // Note that floating-point comparison by itself is not a transitive relation: it does not
129  // report a consistent ordering for not-a-number (NaN) values.
130  // This implementation of Less places NaN values before any others, by using:
131  //
132  //	x[i] < x[j] || (math.IsNaN(x[i]) && !math.IsNaN(x[j]))
133  func (x Float64Slice) Less(i, j int) bool { return x[i] < x[j] || (isNaN(x[i]) && !isNaN(x[j])) }
134  func (x Float64Slice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }
135
136  // isNaN is a copy of math.IsNaN to avoid a dependency on the math package.
137  func isNaN(f float64) bool {
138  	return f != f
139  }
140
141  // Sort is a convenience method: x.Sort() calls Sort(x).
142  func (x Float64Slice) Sort() { Sort(x) }
143
144  // StringSlice attaches the methods of Interface to []string, sorting in increasing order.
145  type StringSlice []string
146
147  func (x StringSlice) Len() int           { return len(x) }
148  func (x StringSlice) Less(i, j int) bool { return x[i] < x[j] }
149  func (x StringSlice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }
150
151  // Sort is a convenience method: x.Sort() calls Sort(x).
152  func (x StringSlice) Sort() { Sort(x) }
153
154  // Convenience wrappers for common cases
155
156  // Ints sorts a slice of ints in increasing order.
157  func Ints(x []int) { Sort(IntSlice(x)) }
158
159  // Float64s sorts a slice of float64s in increasing order.
160  // Not-a-number (NaN) values are ordered before other values.
161  func Float64s(x []float64) { Sort(Float64Slice(x)) }
162
163  // Strings sorts a slice of strings in increasing order.
164  func Strings(x []string) { Sort(StringSlice(x)) }
165
166  // IntsAreSorted reports whether the slice x is sorted in increasing order.
167  func IntsAreSorted(x []int) bool { return IsSorted(IntSlice(x)) }
168
169  // Float64sAreSorted reports whether the slice x is sorted in increasing order,
170  // with not-a-number (NaN) values before any other values.
171  func Float64sAreSorted(x []float64) bool { return IsSorted(Float64Slice(x)) }
172
173  // StringsAreSorted reports whether the slice x is sorted in increasing order.
174  func StringsAreSorted(x []string) bool { return IsSorted(StringSlice(x)) }
175
176  // Notes on stable sorting:
177  // The used algorithms are simple and provable correct on all input and use
178  // only logarithmic additional stack space. They perform well if compared
179  // experimentally to other stable in-place sorting algorithms.
180  //
181  // Remarks on other algorithms evaluated:
182  //  - GCC's 4.6.3 stable_sort with merge_without_buffer from libstdc++:
183  //    Not faster.
184  //  - GCC's __rotate for block rotations: Not faster.
185  //  - "Practical in-place mergesort" from  Jyrki Katajainen, Tomi A. Pasanen
186  //    and Jukka Teuhola; Nordic Journal of Computing 3,1 (1996), 27-40:
187  //    The given algorithms are in-place, number of Swap and Assignments
188  //    grow as n log n but the algorithm is not stable.
189  //  - "Fast Stable In-Place Sorting with O(n) Data Moves" J.I. Munro and
190  //    V. Raman in Algorithmica (1996) 16, 115-160:
191  //    This algorithm either needs additional 2n bits or works only if there
192  //    are enough different elements available to encode some permutations
193  //    which have to be undone later (so not stable on any input).
194  //  - All the optimal in-place sorting/merging algorithms I found are either
195  //    unstable or rely on enough different elements in each step to encode the
197  //    Denham Coates-Evely, Department of Computer Science, Kings College,
198  //    January 2004 and the references in there.
199  //  - Often "optimal" algorithms are optimal in the number of assignments
200  //    but Interface has only Swap as operation.
201
202  // Stable sorts data in ascending order as determined by the Less method,
203  // while keeping the original order of equal elements.
204  //
205  // It makes one call to data.Len to determine n, O(n*log(n)) calls to
206  // data.Less and O(n*log(n)*log(n)) calls to data.Swap.
207  func Stable(data Interface) {
208  	stable(data, data.Len())
209  }
210
211  /*
212  Complexity of Stable Sorting
213
214
215  Complexity of block swapping rotation
216
217  Each Swap puts one new element into its correct, final position.
218  Elements which reach their final position are no longer moved.
219  Thus block swapping rotation needs |u|+|v| calls to Swaps.
220  This is best possible as each element might need a move.
221
222  Pay attention when comparing to other optimal algorithms which
223  typically count the number of assignments instead of swaps:
224  E.g. the optimal algorithm of Dudzinski and Dydek for in-place
225  rotations uses O(u + v + gcd(u,v)) assignments which is
226  better than our O(3 * (u+v)) as gcd(u,v) <= u.
227
228
229  Stable sorting by SymMerge and BlockSwap rotations
230
231  SymMerg complexity for same size input M = N:
232  Calls to Less:  O(M*log(N/M+1)) = O(N*log(2)) = O(N)
233  Calls to Swap:  O((M+N)*log(M)) = O(2*N*log(N)) = O(N*log(N))
234
235  (The following argument does not fuzz over a missing -1 or
236  other stuff which does not impact the final result).
237
238  Let n = data.Len(). Assume n = 2^k.
239
240  Plain merge sort performs log(n) = k iterations.
241  On iteration i the algorithm merges 2^(k-i) blocks, each of size 2^i.
242
243  Thus iteration i of merge sort performs:
244  Calls to Less  O(2^(k-i) * 2^i) = O(2^k) = O(2^log(n)) = O(n)
245  Calls to Swap  O(2^(k-i) * 2^i * log(2^i)) = O(2^k * i) = O(n*i)
246
247  In total k = log(n) iterations are performed; so in total:
248  Calls to Less O(log(n) * n)
249  Calls to Swap O(n + 2*n + 3*n + ... + (k-1)*n + k*n)
250     = O((k/2) * k * n) = O(n * k^2) = O(n * log^2(n))
251
252
253  Above results should generalize to arbitrary n = 2^k + p
254  and should not be influenced by the initial insertion sort phase:
255  Insertion sort is O(n^2) on Swap and Less, thus O(bs^2) per block of
256  size bs at n/bs blocks:  O(bs*n) Swaps and Less during insertion sort.
257  Merge sort iterations start at i = log(bs). With t = log(bs) constant:
258  Calls to Less O((log(n)-t) * n + bs*n) = O(log(n)*n + (bs-t)*n)
259     = O(n * log(n))
260  Calls to Swap O(n * log^2(n) - (t^2+t)/2*n) = O(n * log^2(n))
261
262  */
263
```

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