排序及相关函数

Julia 拥有为数众多的灵活的 API，用于对已经排序的值数组进行排序和交互。默认情况下，Julia 会选择合理的算法并按标准升序进行排序：

  1. julia> sort([2,3,1])
  2. 3-element Array{Int64,1}:
  3.  1
  4.  2
  5.  3

你同样可以轻松实现逆序排序：

  1. julia> sort([2,3,1], rev=true)
  2. 3-element Array{Int64,1}:
  3.  3
  4.  2
  5.  1

对数组进行 in-place 排序时，要使用 ! 版的排序函数：

  1. julia> a = [2,3,1];
  3. julia> sort!(a);
  5. julia> a
  6. 3-element Array{Int64,1}:
  7.  1
  8.  2
  9.  3

你可以计算用于排列的索引，而不是直接对数组进行排序：

  1. julia> v = randn(5)
  2. 5-element Array{Float64,1}:
  3.   0.297288
  4.   0.382396
  5.  -0.597634
  6.  -0.0104452
  7.  -0.839027
  9. julia> p = sortperm(v)
  10. 5-element Array{Int64,1}:
  11.  5
  12.  3
  13.  4
  14.  1
  15.  2
  17. julia> v[p]
  18. 5-element Array{Float64,1}:
  19.  -0.839027
  20.  -0.597634
  21.  -0.0104452
  22.   0.297288
  23.   0.382396

数组可以根据对其值任意的转换结果来进行排序；

  1. julia> sort(v, by=abs)
  2. 5-element Array{Float64,1}:
  3.  -0.0104452
  4.   0.297288
  5.   0.382396
  6.  -0.597634
  7.  -0.839027

或者通过转换来进行逆序排序

  1. julia> sort(v, by=abs, rev=true)
  2. 5-element Array{Float64,1}:
  3.  -0.839027
  4.  -0.597634
  5.   0.382396
  6.   0.297288
  7.  -0.0104452

如有必要，可以选择排序算法：

  1. julia> sort(v, alg=InsertionSort)
  2. 5-element Array{Float64,1}:
  3.  -0.839027
  4.  -0.597634
  5.  -0.0104452
  6.   0.297288
  7.   0.382396

所有与排序和顺序相关的函数依赖于“小于”关系，该关系定义了要操纵的值的总顺序。默认情况下会调用 isless 函数，但可以通过 lt 关键字指定关系。

排序函数

Base.sort! — Function.

  1. sort!(v; alg::Algorithm=defalg(v), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort the vector v in place. QuickSort is used by default for numeric arrays while MergeSort is used for other arrays. You can specify an algorithm to use via the alg keyword (see Sorting Algorithms for available algorithms). The by keyword lets you provide a function that will be applied to each element before comparison; the lt keyword allows providing a custom "less than" function; use rev=true to reverse the sorting order. These options are independent and can be used together in all possible combinations: if both by and lt are specified, the lt function is applied to the result of the by function; rev=true reverses whatever ordering specified via the by and lt keywords.

Examples

  1. julia> v = [3, 1, 2]; sort!(v); v
  2. 3-element Array{Int64,1}:
  3.  1
  4.  2
  5.  3
  7. julia> v = [3, 1, 2]; sort!(v, rev = true); v
  8. 3-element Array{Int64,1}:
  9.  3
  10.  2
  11.  1
  13. julia> v = [(1, "c"), (3, "a"), (2, "b")]; sort!(v, by = x -> x[1]); v
  14. 3-element Array{Tuple{Int64,String},1}:
  15.  (1, "c")
  16.  (2, "b")
  17.  (3, "a")
  19. julia> v = [(1, "c"), (3, "a"), (2, "b")]; sort!(v, by = x -> x[2]); v
  20. 3-element Array{Tuple{Int64,String},1}:
  21.  (3, "a")
  22.  (2, "b")
  23.  (1, "c")

source

Base.sort — Function.

  1. sort(v; alg::Algorithm=defalg(v), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Variant of sort! that returns a sorted copy of v leaving v itself unmodified.

Examples

  1. julia> v = [3, 1, 2];
  3. julia> sort(v)
  4. 3-element Array{Int64,1}:
  5.  1
  6.  2
  7.  3
  9. julia> v
  10. 3-element Array{Int64,1}:
  11.  3
  12.  1
  13.  2

source

  1. sort(A; dims::Integer, alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort a multidimensional array A along the given dimension. See sort! for a description of possible keyword arguments.

Examples

  1. julia> A = [4 3; 1 2]
  2. 2×2 Array{Int64,2}:
  3.  4  3
  4.  1  2
  6. julia> sort(A, dims = 1)
  7. 2×2 Array{Int64,2}:
  8.  1  2
  9.  4  3
  11. julia> sort(A, dims = 2)
  12. 2×2 Array{Int64,2}:
  13.  3  4
  14.  1  2

source

Base.sortperm — Function.

  1. sortperm(v; alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Return a permutation vector I that puts v[I] in sorted order. The order is specified using the same keywords as sort!. The permutation is guaranteed to be stable even if the sorting algorithm is unstable, meaning that indices of equal elements appear in ascending order.

See also sortperm!.

Examples

  1. julia> v = [3, 1, 2];
  3. julia> p = sortperm(v)
  4. 3-element Array{Int64,1}:
  5.  2
  6.  3
  7.  1
  9. julia> v[p]
  10. 3-element Array{Int64,1}:
  11.  1
  12.  2
  13.  3

source

Base.Sort.sortperm! — Function.

  1. sortperm!(ix, v; alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward, initialized::Bool=false)

Like sortperm, but accepts a preallocated index vector ix. If initialized is false (the default), ix is initialized to contain the values 1:length(v).

Examples

  1. julia> v = [3, 1, 2]; p = zeros(Int, 3);
  3. julia> sortperm!(p, v); p
  4. 3-element Array{Int64,1}:
  5.  2
  6.  3
  7.  1
  9. julia> v[p]
  10. 3-element Array{Int64,1}:
  11.  1
  12.  2
  13.  3

source

Base.sortslices — Function.

  1. sortslices(A; dims, alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort slices of an array A. The required keyword argument dims must be either an integer or a tuple of integers. It specifies the dimension(s) over which the slices are sorted.

E.g., if A is a matrix, dims=1 will sort rows, dims=2 will sort columns. Note that the default comparison function on one dimensional slices sorts lexicographically.

For the remaining keyword arguments, see the documentation of sort!.

Examples

  1. julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1) # Sort rows
  2. 3×3 Array{Int64,2}:
  3.  -1   6  4
  4.   7   3  5
  5.   9  -2  8
  7. julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1, lt=(x,y)->isless(x[2],y[2]))
  8. 3×3 Array{Int64,2}:
  9.   9  -2  8
  10.   7   3  5
  11.  -1   6  4
  13. julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1, rev=true)
  14. 3×3 Array{Int64,2}:
  15.   9  -2  8
  16.   7   3  5
  17.  -1   6  4
  19. julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2) # Sort columns
  20. 3×3 Array{Int64,2}:
  21.   3   5  7
  22.  -1  -4  6
  23.  -2   8  9
  25. julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2, alg=InsertionSort, lt=(x,y)->isless(x[2],y[2]))
  26. 3×3 Array{Int64,2}:
  27.   5   3  7
  28.  -4  -1  6
  29.   8  -2  9
  31. julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2, rev=true)
  32. 3×3 Array{Int64,2}:
  33.  7   5   3
  34.  6  -4  -1
  35.  9   8  -2

Higher dimensions

sortslices extends naturally to higher dimensions. E.g., if A is a a 2x2x2 array, sortslices(A, dims=3) will sort slices within the 3rd dimension, passing the 2x2 slices A[:, :, 1] and A[:, :, 2] to the comparison function. Note that while there is no default order on higher-dimensional slices, you may use the by or lt keyword argument to specify such an order.

If dims is a tuple, the order of the dimensions in dims is relevant and specifies the linear order of the slices. E.g., if A is three dimensional and dims is (1, 2), the orderings of the first two dimensions are re-arranged such such that the slices (of the remaining third dimension) are sorted. If dims is (2, 1) instead, the same slices will be taken, but the result order will be row-major instead.

Higher dimensional examples

  1. julia> A = permutedims(reshape([4 3; 2 1; 'A' 'B'; 'C' 'D'], (2, 2, 2)), (1, 3, 2))
  2. 2×2×2 Array{Any,3}:
  3. [:, :, 1] =
  4.  4  3
  5.  2  1
  7. [:, :, 2] =
  8.  'A'  'B'
  9.  'C'  'D'
  11. julia> sortslices(A, dims=(1,2))
  12. 2×2×2 Array{Any,3}:
  13. [:, :, 1] =
  14.  1  3
  15.  2  4
  17. [:, :, 2] =
  18.  'D'  'B'
  19.  'C'  'A'
  21. julia> sortslices(A, dims=(2,1))
  22. 2×2×2 Array{Any,3}:
  23. [:, :, 1] =
  24.  1  2
  25.  3  4
  27. [:, :, 2] =
  28.  'D'  'C'
  29.  'B'  'A'
  31. julia> sortslices(reshape([5; 4; 3; 2; 1], (1,1,5)), dims=3, by=x->x[1,1])
  32. 1×1×5 Array{Int64,3}:
  33. [:, :, 1] =
  34.  1
  36. [:, :, 2] =
  37.  2
  39. [:, :, 3] =
  40.  3
  42. [:, :, 4] =
  43.  4
  45. [:, :, 5] =
  46.  5

source

排列顺序相关的函数

Base.issorted — Function.

  1. issorted(v, lt=isless, by=identity, rev:Bool=false, order::Ordering=Forward)

Test whether a vector is in sorted order. The lt, by and rev keywords modify what order is considered to be sorted just as they do for sort.

Examples

  1. julia> issorted([1, 2, 3])
  2. true
  4. julia> issorted([(1, "b"), (2, "a")], by = x -> x[1])
  5. true
  7. julia> issorted([(1, "b"), (2, "a")], by = x -> x[2])
  8. false
  10. julia> issorted([(1, "b"), (2, "a")], by = x -> x[2], rev=true)
  11. true

source

Base.Sort.searchsorted — Function.

  1. searchsorted(a, x; by=<transform>, lt=<comparison>, rev=false)

Return the range of indices of a which compare as equal to x (using binary search) according to the order specified by the by, lt and rev keywords, assuming that a is already sorted in that order. Return an empty range located at the insertion point if a does not contain values equal to x.

Examples

  1. julia> a = [4, 3, 2, 1]
  2. 4-element Array{Int64,1}:
  3.  4
  4.  3
  5.  2
  6.  1
  8. julia> searchsorted(a, 4)
  9. 5:4
  11. julia> searchsorted(a, 4, rev=true)
  12. 1:1

source

Base.Sort.searchsortedfirst — Function.

  1. searchsortedfirst(a, x; by=<transform>, lt=<comparison>, rev=false)

Return the index of the first value in a greater than or equal to x, according to the specified order. Return length(a) + 1 if x is greater than all values in a. a is assumed to be sorted.

Examples

  1. julia> searchsortedfirst([1, 2, 4, 5, 14], 4)
  2. 3
  4. julia> searchsortedfirst([1, 2, 4, 5, 14], 4, rev=true)
  5. 1
  7. julia> searchsortedfirst([1, 2, 4, 5, 14], 15)
  8. 6

source

Base.Sort.searchsortedlast — Function.

  1. searchsortedlast(a, x; by=<transform>, lt=<comparison>, rev=false)

Return the index of the last value in a less than or equal to x, according to the specified order. Return 0 if x is less than all values in a. a is assumed to be sorted.

Examples

  1. julia> searchsortedlast([1, 2, 4, 5, 14], 4)
  2. 3
  4. julia> searchsortedlast([1, 2, 4, 5, 14], 4, rev=true)
  5. 5
  7. julia> searchsortedlast([1, 2, 4, 5, 14], -1)
  8. 0

source

Base.Sort.partialsort! — Function.

  1. partialsort!(v, k; by=<transform>, lt=<comparison>, rev=false)

Partially sort the vector v in place, according to the order specified by by, lt and rev so that the value at index k (or range of adjacent values if k is a range) occurs at the position where it would appear if the array were fully sorted via a non-stable algorithm. If k is a single index, that value is returned; if k is a range, an array of values at those indices is returned. Note that partialsort! does not fully sort the input array.

Examples

  1. julia> a = [1, 2, 4, 3, 4]
  2. 5-element Array{Int64,1}:
  3.  1
  4.  2
  5.  4
  6.  3
  7.  4
  9. julia> partialsort!(a, 4)
  10. 4
  12. julia> a
  13. 5-element Array{Int64,1}:
  14.  1
  15.  2
  16.  3
  17.  4
  18.  4
  20. julia> a = [1, 2, 4, 3, 4]
  21. 5-element Array{Int64,1}:
  22.  1
  23.  2
  24.  4
  25.  3
  26.  4
  28. julia> partialsort!(a, 4, rev=true)
  29. 2
  31. julia> a
  32. 5-element Array{Int64,1}:
  33.  4
  34.  4
  35.  3
  36.  2
  37.  1

source

Base.Sort.partialsort — Function.

  1. partialsort(v, k, by=<transform>, lt=<comparison>, rev=false)

Variant of partialsort! which copies v before partially sorting it, thereby returning the same thing as partialsort! but leaving v unmodified.

source

Base.Sort.partialsortperm — Function.

  1. partialsortperm(v, k; by=<transform>, lt=<comparison>, rev=false)

Return a partial permutation I of the vector v, so that v[I] returns values of a fully sorted version of v at index k. If k is a range, a vector of indices is returned; if k is an integer, a single index is returned. The order is specified using the same keywords as sort!. The permutation is stable, meaning that indices of equal elements appear in ascending order.

Note that this function is equivalent to, but more efficient than, calling sortperm(…)[k].

Examples

  1. julia> v = [3, 1, 2, 1];
  3. julia> v[partialsortperm(v, 1)]
  4. 1
  6. julia> p = partialsortperm(v, 1:3)
  7. 3-element view(::Array{Int64,1}, 1:3) with eltype Int64:
  8.  2
  9.  4
  10.  3
  12. julia> v[p]
  13. 3-element Array{Int64,1}:
  14.  1
  15.  1
  16.  2

source

Base.Sort.partialsortperm! — Function.

  1. partialsortperm!(ix, v, k; by=<transform>, lt=<comparison>, rev=false, initialized=false)

Like partialsortperm, but accepts a preallocated index vector ix. If initialized is false (the default), ix is initialized to contain the values 1:length(ix).

source

排序算法

目前，Julia Base 中有四种可用的排序算法：

  • InsertionSort
  • QuickSort
  • PartialQuickSort(k)
  • MergeSortInsertionSort 是一个在 QuickSort 中使用的时间复杂度为 O(n^2) 的稳定的排序算法，它通常在 n 比较小的时候才具有较高的效率。

QuickSort 是一个内置并且非常快，但是不稳定的时间复杂度为 O(n log n）的排序算法，例如即使数组两个元素相等的，它们排序之后的顺序也可能和在原数组中顺序不一致。QuickSort 是内置的包括整数和浮点数在内的数字值的默认排序算法。

PartialQuickSort(k) 类似于 QuickSort，但是如果 k 是一个整数，输出数组只排序到索引 k，如果 k 是 OrdinalRange，则输出数组排在 k 范围内。 例如：

  1. x = rand(1:500, 100)
  2. k = 50
  3. k2 = 50:100
  4. s = sort(x; alg=QuickSort)
  5. ps = sort(x; alg=PartialQuickSort(k))
  6. qs = sort(x; alg=PartialQuickSort(k2))
  7. map(issorted, (s, ps, qs))             # => (true, false, false)
  8. map(x->issorted(x[1:k]), (s, ps, qs))  # => (true, true, false)
  9. map(x->issorted(x[k2]), (s, ps, qs))   # => (true, false, true)
  10. s[1:k] == ps[1:k]                      # => true
  11. s[k2] == qs[k2]                        # => true

MergeSort 是一个时间复杂度为 O(n log n) 的稳定但是非 in-place 的算法，它需要一个大小为输入数组一般的临时数组——同时也不像 QuickSort 一样快。MergeSort 是非数值型数据的默认排序算法。

默认排序算法的选择是基于它们的快速稳定，或者 appear 之类的。对于数值类型，实际上选择了 QuickSort，因为在这种情况下，它更快，与稳定排序没有区别(除非数组以某种方式记录了突变)

Julia选择默认排序算法的机制是通过 Base.Sort.defalg 来实现的，其允许将特定算法注册为特定数组的所有排序函数中的默认值。例如，这有两个默认算法 sort.jl:

  1. defalg(v::AbstractArray) = MergeSort
  2. defalg(v::AbstractArray{<:Number}) = QuickSort

对于数值型数组，选择非稳定的默认排序算法的原则是稳定的排序算法没有必要的（例如：但两个值相比较时相等且不可区分时）。