# 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  //
43  // Note: in many situations, the newer slices.SortFunc function is more
44  // ergonomic and runs faster.
45  func Sort(data Interface) {
46  	n := data.Len()
47  	if n <= 1 {
48  		return
49  	}
50  	limit := bits.Len(uint(n))
51  	pdqsort(data, 0, n, limit)
52  }
53
54  type sortedHint int // hint for pdqsort when choosing the pivot
55
56  const (
57  	unknownHint sortedHint = iota
58  	increasingHint
59  	decreasingHint
60  )
61
62  // xorshift paper: https://www.jstatsoft.org/article/view/v008i14/xorshift.pdf
63  type xorshift uint64
64
65  func (r *xorshift) Next() uint64 {
66  	*r ^= *r << 13
67  	*r ^= *r >> 17
68  	*r ^= *r << 5
69  	return uint64(*r)
70  }
71
72  func nextPowerOfTwo(length int) uint {
73  	shift := uint(bits.Len(uint(length)))
74  	return uint(1 << shift)
75  }
76
77  // lessSwap is a pair of Less and Swap function for use with the
78  // auto-generated func-optimized variant of sort.go in
79  // zfuncversion.go.
80  type lessSwap struct {
81  	Less func(i, j int) bool
82  	Swap func(i, j int)
83  }
84
85  type reverse struct {
86  	// This embedded Interface permits Reverse to use the methods of
87  	// another Interface implementation.
88  	Interface
89  }
90
91  // Less returns the opposite of the embedded implementation's Less method.
92  func (r reverse) Less(i, j int) bool {
93  	return r.Interface.Less(j, i)
94  }
95
96  // Reverse returns the reverse order for data.
97  func Reverse(data Interface) Interface {
98  	return &reverse{data}
99  }
100
101  // IsSorted reports whether data is sorted.
102  //
103  // Note: in many situations, the newer slices.IsSortedFunc function is more
104  // ergonomic and runs faster.
105  func IsSorted(data Interface) bool {
106  	n := data.Len()
107  	for i := n - 1; i > 0; i-- {
108  		if data.Less(i, i-1) {
109  			return false
110  		}
111  	}
112  	return true
113  }
114
115  // Convenience types for common cases
116
117  // IntSlice attaches the methods of Interface to []int, sorting in increasing order.
118  type IntSlice []int
119
120  func (x IntSlice) Len() int           { return len(x) }
121  func (x IntSlice) Less(i, j int) bool { return x[i] < x[j] }
122  func (x IntSlice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }
123
124  // Sort is a convenience method: x.Sort() calls Sort(x).
125  func (x IntSlice) Sort() { Sort(x) }
126
127  // Float64Slice implements Interface for a []float64, sorting in increasing order,
128  // with not-a-number (NaN) values ordered before other values.
129  type Float64Slice []float64
130
131  func (x Float64Slice) Len() int { return len(x) }
132
133  // Less reports whether x[i] should be ordered before x[j], as required by the sort Interface.
134  // Note that floating-point comparison by itself is not a transitive relation: it does not
135  // report a consistent ordering for not-a-number (NaN) values.
136  // This implementation of Less places NaN values before any others, by using:
137  //
138  //	x[i] < x[j] || (math.IsNaN(x[i]) && !math.IsNaN(x[j]))
139  func (x Float64Slice) Less(i, j int) bool { return x[i] < x[j] || (isNaN(x[i]) && !isNaN(x[j])) }
140  func (x Float64Slice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }
141
142  // isNaN is a copy of math.IsNaN to avoid a dependency on the math package.
143  func isNaN(f float64) bool {
144  	return f != f
145  }
146
147  // Sort is a convenience method: x.Sort() calls Sort(x).
148  func (x Float64Slice) Sort() { Sort(x) }
149
150  // StringSlice attaches the methods of Interface to []string, sorting in increasing order.
151  type StringSlice []string
152
153  func (x StringSlice) Len() int           { return len(x) }
154  func (x StringSlice) Less(i, j int) bool { return x[i] < x[j] }
155  func (x StringSlice) Swap(i, j int)      { x[i], x[j] = x[j], x[i] }
156
157  // Sort is a convenience method: x.Sort() calls Sort(x).
158  func (x StringSlice) Sort() { Sort(x) }
159
160  // Convenience wrappers for common cases
161
162  // Ints sorts a slice of ints in increasing order.
163  //
164  // Note: consider using the newer slices.Sort function, which runs faster.
165  func Ints(x []int) { Sort(IntSlice(x)) }
166
167  // Float64s sorts a slice of float64s in increasing order.
168  // Not-a-number (NaN) values are ordered before other values.
169  //
170  // Note: consider using the newer slices.Sort function, which runs faster.
171  func Float64s(x []float64) { Sort(Float64Slice(x)) }
172
173  // Strings sorts a slice of strings in increasing order.
174  //
175  // Note: consider using the newer slices.Sort function, which runs faster.
176  func Strings(x []string) { Sort(StringSlice(x)) }
177
178  // IntsAreSorted reports whether the slice x is sorted in increasing order.
179  //
180  // Note: consider using the newer slices.IsSorted function, which runs faster.
181  func IntsAreSorted(x []int) bool { return IsSorted(IntSlice(x)) }
182
183  // Float64sAreSorted reports whether the slice x is sorted in increasing order,
184  // with not-a-number (NaN) values before any other values.
185  //
186  // Note: consider using the newer slices.IsSorted function, which runs faster.
187  func Float64sAreSorted(x []float64) bool { return IsSorted(Float64Slice(x)) }
188
189  // StringsAreSorted reports whether the slice x is sorted in increasing order.
190  //
191  // Note: consider using the newer slices.IsSorted function, which runs faster.
192  func StringsAreSorted(x []string) bool { return IsSorted(StringSlice(x)) }
193
194  // Notes on stable sorting:
195  // The used algorithms are simple and provable correct on all input and use
196  // only logarithmic additional stack space. They perform well if compared
197  // experimentally to other stable in-place sorting algorithms.
198  //
199  // Remarks on other algorithms evaluated:
200  //  - GCC's 4.6.3 stable_sort with merge_without_buffer from libstdc++:
201  //    Not faster.
202  //  - GCC's __rotate for block rotations: Not faster.
203  //  - "Practical in-place mergesort" from  Jyrki Katajainen, Tomi A. Pasanen
204  //    and Jukka Teuhola; Nordic Journal of Computing 3,1 (1996), 27-40:
205  //    The given algorithms are in-place, number of Swap and Assignments
206  //    grow as n log n but the algorithm is not stable.
207  //  - "Fast Stable In-Place Sorting with O(n) Data Moves" J.I. Munro and
208  //    V. Raman in Algorithmica (1996) 16, 115-160:
209  //    This algorithm either needs additional 2n bits or works only if there
210  //    are enough different elements available to encode some permutations
211  //    which have to be undone later (so not stable on any input).
212  //  - All the optimal in-place sorting/merging algorithms I found are either
213  //    unstable or rely on enough different elements in each step to encode the
215  //    Denham Coates-Evely, Department of Computer Science, Kings College,
216  //    January 2004 and the references in there.
217  //  - Often "optimal" algorithms are optimal in the number of assignments
218  //    but Interface has only Swap as operation.
219
220  // Stable sorts data in ascending order as determined by the Less method,
221  // while keeping the original order of equal elements.
222  //
223  // It makes one call to data.Len to determine n, O(n*log(n)) calls to
224  // data.Less and O(n*log(n)*log(n)) calls to data.Swap.
225  //
226  // Note: in many situations, the newer slices.SortStableFunc function is more
227  // ergonomic and runs faster.
228  func Stable(data Interface) {
229  	stable(data, data.Len())
230  }
231
232  /*
233  Complexity of Stable Sorting
234
235
236  Complexity of block swapping rotation
237
238  Each Swap puts one new element into its correct, final position.
239  Elements which reach their final position are no longer moved.
240  Thus block swapping rotation needs |u|+|v| calls to Swaps.
241  This is best possible as each element might need a move.
242
243  Pay attention when comparing to other optimal algorithms which
244  typically count the number of assignments instead of swaps:
245  E.g. the optimal algorithm of Dudzinski and Dydek for in-place
246  rotations uses O(u + v + gcd(u,v)) assignments which is
247  better than our O(3 * (u+v)) as gcd(u,v) <= u.
248
249
250  Stable sorting by SymMerge and BlockSwap rotations
251
252  SymMerg complexity for same size input M = N:
253  Calls to Less:  O(M*log(N/M+1)) = O(N*log(2)) = O(N)
254  Calls to Swap:  O((M+N)*log(M)) = O(2*N*log(N)) = O(N*log(N))
255
256  (The following argument does not fuzz over a missing -1 or
257  other stuff which does not impact the final result).
258
259  Let n = data.Len(). Assume n = 2^k.
260
261  Plain merge sort performs log(n) = k iterations.
262  On iteration i the algorithm merges 2^(k-i) blocks, each of size 2^i.
263
264  Thus iteration i of merge sort performs:
265  Calls to Less  O(2^(k-i) * 2^i) = O(2^k) = O(2^log(n)) = O(n)
266  Calls to Swap  O(2^(k-i) * 2^i * log(2^i)) = O(2^k * i) = O(n*i)
267
268  In total k = log(n) iterations are performed; so in total:
269  Calls to Less O(log(n) * n)
270  Calls to Swap O(n + 2*n + 3*n + ... + (k-1)*n + k*n)
271     = O((k/2) * k * n) = O(n * k^2) = O(n * log^2(n))
272
273
274  Above results should generalize to arbitrary n = 2^k + p
275  and should not be influenced by the initial insertion sort phase:
276  Insertion sort is O(n^2) on Swap and Less, thus O(bs^2) per block of
277  size bs at n/bs blocks:  O(bs*n) Swaps and Less during insertion sort.
278  Merge sort iterations start at i = log(bs). With t = log(bs) constant:
279  Calls to Less O((log(n)-t) * n + bs*n) = O(log(n)*n + (bs-t)*n)
280     = O(n * log(n))
281  Calls to Swap O(n * log^2(n) - (t^2+t)/2*n) = O(n * log^2(n))
282
283  */
284
```

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