排序及相关函数

Julia 拥有广泛而灵活的应用程序接口,可用于对已排序的数组值进行排序和交互。 默认情况下,Julia 会选择合理的算法并按升序排序:

julia> sort([2,3,1])
3-element Vector{Int64}:
 1
 2
 3

或者指定使用逆序:

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

sort 构造一个已排序的副本,保持输入不变。 使用排序函数的"感叹号"版本来修改现有数组:

julia> a = [2,3,1];

julia> sort!(a);

julia> a
3-element Vector{Int64}:
 1
 2
 3

除了直接对数组进行排序,你还可以计算一个数组索引的置换,使数组按排序顺序排列:

julia> v = randn(5)
5-element Array{Float64,1}:
  0.297288
  0.382396
 -0.597634
 -0.0104452
 -0.839027

julia> p = sortperm(v)
5-element Array{Int64,1}:
 5
 3
 4
 1
 2

julia> v[p]
5-element Array{Float64,1}:
 -0.839027
 -0.597634
 -0.0104452
  0.297288
  0.382396

数组可以根据其值的任意转换进行排序:

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

或者指定使用逆序:

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

如有必要,可以选择排序算法:

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

所有排序和顺序相关的函数都依赖于一个"小于"关系,该关系在要操作的值上定义了一个 严格弱序。 默认调用 isless 函数,但可以通过 lt 关键字指定关系,这是一个接受两个数组元素的函数, 当且仅当第一个参数"小于"第二个参数时返回 true。 更多信息请参见 sort!替代排序

排序函数

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

Sort the vector v in place. A stable algorithm is used by default: the ordering of elements that compare equal is preserved. A specific algorithm can be selected via the alg keyword (see Sorting Algorithms for available algorithms).

Elements are first transformed with the function by and then compared according to either the function lt or the ordering order. Finally, the resulting order is reversed if rev=true (this preserves forward stability: elements that compare equal are not reversed). The current implemention applies the by transformation before each comparison rather than once per element.

Passing an lt other than isless along with an order other than Base.Order.Forward or Base.Order.Reverse is not permitted, otherwise all options are independent and can be used together in all possible combinations. Note that order can also include a "by" transformation, in which case it is applied after that defined with the by keyword. For more information on order values see the documentation on Alternate Orderings.

Relations between two elements are defined as follows (with "less" and "greater" exchanged when rev=true):

  • x is less than y if lt(by(x), by(y)) (or Base.Order.lt(order, by(x), by(y))) yields true.
  • x is greater than y if y is less than x.
  • x and y are equivalent if neither is less than the other ("incomparable" is sometimes used as a synonym for "equivalent").

The result of sort! is sorted in the sense that every element is greater than or equivalent to the previous one.

The lt function must define a strict weak order, that is, it must be

  • irreflexive: lt(x, x) always yields false,
  • asymmetric: if lt(x, y) yields true then lt(y, x) yields false,
  • transitive: lt(x, y) && lt(y, z) implies lt(x, z),
  • transitive in equivalence: !lt(x, y) && !lt(y, x) and !lt(y, z) && !lt(z, y) together imply !lt(x, z) && !lt(z, x). In words: if x and y are equivalent and y and z are equivalent then x and z must be equivalent.

For example < is a valid lt function for Int values but is not: it violates irreflexivity. For Float64 values even < is invalid as it violates the fourth condition: 1.0 and NaN are equivalent and so are NaN and 2.0 but 1.0 and 2.0 are not equivalent.

See also sort, sortperm, sortslices, partialsort!, partialsortperm, issorted, searchsorted, insorted, Base.Order.ord.

Examples

julia> v = [3, 1, 2]; sort!(v); v
3-element Vector{Int64}:
 1
 2
 3

julia> v = [3, 1, 2]; sort!(v, rev = true); v
3-element Vector{Int64}:
 3
 2
 1

julia> v = [(1, "c"), (3, "a"), (2, "b")]; sort!(v, by = x -> x[1]); v
3-element Vector{Tuple{Int64, String}}:
 (1, "c")
 (2, "b")
 (3, "a")

julia> v = [(1, "c"), (3, "a"), (2, "b")]; sort!(v, by = x -> x[2]); v
3-element Vector{Tuple{Int64, String}}:
 (3, "a")
 (2, "b")
 (1, "c")

julia> sort(0:3, by=x->x-2, order=Base.Order.By(abs)) # same as sort(0:3, by=abs(x->x-2))
4-element Vector{Int64}:
 2
 1
 3
 0

julia> sort([2, NaN, 1, NaN, 3]) # correct sort with default lt=isless
5-element Vector{Float64}:
   1.0
   2.0
   3.0
 NaN
 NaN

julia> sort([2, NaN, 1, NaN, 3], lt=<) # wrong sort due to invalid lt. This behavior is undefined.
5-element Vector{Float64}:
   2.0
 NaN
   1.0
 NaN
   3.0
source
sort!(A; dims::Integer, alg::Algorithm=defalg(A), lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward)

Sort the multidimensional array A along dimension dims. See the one-dimensional version of sort! for a description of possible keyword arguments.

To sort slices of an array, refer to sortslices.

Julia 1.1

This function requires at least Julia 1.1.

Examples

julia> A = [4 3; 1 2]
2×2 Matrix{Int64}:
 4  3
 1  2

julia> sort!(A, dims = 1); A
2×2 Matrix{Int64}:
 1  2
 4  3

julia> sort!(A, dims = 2); A
2×2 Matrix{Int64}:
 1  2
 3  4
source
Base.sortFunction
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

julia> v = [3, 1, 2];

julia> sort(v)
3-element Vector{Int64}:
 1
 2
 3

julia> v
3-element Vector{Int64}:
 3
 1
 2
source
sort(A; dims::Integer, alg::Algorithm=defalg(A), 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.

To sort slices of an array, refer to sortslices.

Examples

julia> A = [4 3; 1 2]
2×2 Matrix{Int64}:
 4  3
 1  2

julia> sort(A, dims = 1)
2×2 Matrix{Int64}:
 1  2
 4  3

julia> sort(A, dims = 2)
2×2 Matrix{Int64}:
 3  4
 1  2
source
Base.sortpermFunction
sortperm(A; alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward, [dims::Integer])

Return a permutation vector or array I that puts A[I] in sorted order along the given dimension. If A has more than one dimension, then the dims keyword argument must be specified. The order is specified using the same keywords as sort!. The permutation is guaranteed to be stable even if the sorting algorithm is unstable: the indices of equal elements will appear in ascending order.

See also sortperm!, partialsortperm, invperm, indexin. To sort slices of an array, refer to sortslices.

Julia 1.9

The method accepting dims requires at least Julia 1.9.

Examples

julia> v = [3, 1, 2];

julia> p = sortperm(v)
3-element Vector{Int64}:
 2
 3
 1

julia> v[p]
3-element Vector{Int64}:
 1
 2
 3

julia> A = [8 7; 5 6]
2×2 Matrix{Int64}:
 8  7
 5  6

julia> sortperm(A, dims = 1)
2×2 Matrix{Int64}:
 2  4
 1  3

julia> sortperm(A, dims = 2)
2×2 Matrix{Int64}:
 3  1
 2  4
source
Base.Sort.InsertionSortConstant
InsertionSort

Use the insertion sort algorithm.

Insertion sort traverses the collection one element at a time, inserting each element into its correct, sorted position in the output vector.

Characteristics:

  • stable: preserves the ordering of elements that compare equal

(e.g. "a" and "A" in a sort of letters that ignores case).

  • in-place in memory.
  • quadratic performance in the number of elements to be sorted:

it is well-suited to small collections but should not be used for large ones.

source
Base.Sort.MergeSortConstant
MergeSort

Indicate that a sorting function should use the merge sort algorithm. Merge sort divides the collection into subcollections and repeatedly merges them, sorting each subcollection at each step, until the entire collection has been recombined in sorted form.

Characteristics:

  • stable: preserves the ordering of elements that compare equal (e.g. "a" and "A" in a sort of letters that ignores case).
  • not in-place in memory.
  • divide-and-conquer sort strategy.
  • good performance for large collections but typically not quite as fast as QuickSort.
source
Base.Sort.QuickSortConstant
QuickSort

Indicate that a sorting function should use the quick sort algorithm, which is not stable.

Characteristics:

  • not stable: does not preserve the ordering of elements that compare equal (e.g. "a" and "A" in a sort of letters that ignores case).
  • in-place in memory.
  • divide-and-conquer: sort strategy similar to MergeSort.
  • good performance for large collections.
source
Base.Sort.PartialQuickSortType
PartialQuickSort{T <: Union{Integer,OrdinalRange}}

Indicate that a sorting function should use the partial quick sort algorithm. PartialQuickSort(k) is like QuickSort, but is only required to find and sort the elements that would end up in v[k] were v fully sorted.

Characteristics:

  • not stable: does not preserve the ordering of elements that compare equal (e.g. "a" and "A" in a sort of letters that ignores case).
  • in-place in memory.
  • divide-and-conquer: sort strategy similar to MergeSort.

Note that PartialQuickSort(k) does not necessarily sort the whole array. For example,

julia> x = rand(100);

julia> k = 50:100;

julia> s1 = sort(x; alg=QuickSort);

julia> s2 = sort(x; alg=PartialQuickSort(k));

julia> map(issorted, (s1, s2))
(true, false)

julia> map(x->issorted(x[k]), (s1, s2))
(true, true)

julia> s1[k] == s2[k]
true
source
Base.Sort.sortperm!Function
sortperm!(ix, A; alg::Algorithm=DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Ordering=Forward, [dims::Integer])

Like sortperm, but accepts a preallocated index vector or array ix with the same axes as A. ix is initialized to contain the values LinearIndices(A).

Warning

Behavior can be unexpected when any mutated argument shares memory with any other argument.

Julia 1.9

The method accepting dims requires at least Julia 1.9.

Examples

julia> v = [3, 1, 2]; p = zeros(Int, 3);

julia> sortperm!(p, v); p
3-element Vector{Int64}:
 2
 3
 1

julia> v[p]
3-element Vector{Int64}:
 1
 2
 3

julia> A = [8 7; 5 6]; p = zeros(Int,2, 2);

julia> sortperm!(p, A; dims=1); p
2×2 Matrix{Int64}:
 2  4
 1  3

julia> sortperm!(p, A; dims=2); p
2×2 Matrix{Int64}:
 3  1
 2  4
source
Base.sortslicesFunction
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

julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1) # Sort rows
3×3 Matrix{Int64}:
 -1   6  4
  7   3  5
  9  -2  8

julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1, lt=(x,y)->isless(x[2],y[2]))
3×3 Matrix{Int64}:
  9  -2  8
  7   3  5
 -1   6  4

julia> sortslices([7 3 5; -1 6 4; 9 -2 8], dims=1, rev=true)
3×3 Matrix{Int64}:
  9  -2  8
  7   3  5
 -1   6  4

julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2) # Sort columns
3×3 Matrix{Int64}:
  3   5  7
 -1  -4  6
 -2   8  9

julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2, alg=InsertionSort, lt=(x,y)->isless(x[2],y[2]))
3×3 Matrix{Int64}:
  5   3  7
 -4  -1  6
  8  -2  9

julia> sortslices([7 3 5; 6 -1 -4; 9 -2 8], dims=2, rev=true)
3×3 Matrix{Int64}:
 7   5   3
 6  -4  -1
 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 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

julia> A = permutedims(reshape([4 3; 2 1; 'A' 'B'; 'C' 'D'], (2, 2, 2)), (1, 3, 2))
2×2×2 Array{Any, 3}:
[:, :, 1] =
 4  3
 2  1

[:, :, 2] =
 'A'  'B'
 'C'  'D'

julia> sortslices(A, dims=(1,2))
2×2×2 Array{Any, 3}:
[:, :, 1] =
 1  3
 2  4

[:, :, 2] =
 'D'  'B'
 'C'  'A'

julia> sortslices(A, dims=(2,1))
2×2×2 Array{Any, 3}:
[:, :, 1] =
 1  2
 3  4

[:, :, 2] =
 'D'  'C'
 'B'  'A'

julia> sortslices(reshape([5; 4; 3; 2; 1], (1,1,5)), dims=3, by=x->x[1,1])
1×1×5 Array{Int64, 3}:
[:, :, 1] =
 1

[:, :, 2] =
 2

[:, :, 3] =
 3

[:, :, 4] =
 4

[:, :, 5] =
 5
source

排列顺序相关的函数

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

Test whether a collection is in sorted order. The keywords modify what order is considered sorted, as described in the sort! documentation.

Examples

julia> issorted([1, 2, 3])
true

julia> issorted([(1, "b"), (2, "a")], by = x -> x[1])
true

julia> issorted([(1, "b"), (2, "a")], by = x -> x[2])
false

julia> issorted([(1, "b"), (2, "a")], by = x -> x[2], rev=true)
true

julia> issorted([1, 2, -2, 3], by=abs)
true
source
Base.Sort.searchsortedFunction
searchsorted(v, x; by=identity, lt=isless, rev=false)

Return the range of indices in v where values are equivalent to x, or an empty range located at the insertion point if v does not contain values equivalent to x. The vector v must be sorted according to the order defined by the keywords. Refer to sort! for the meaning of the keywords and the definition of equivalence. Note that the by function is applied to the searched value x as well as the values in v.

The range is generally found using binary search, but there are optimized implementations for some inputs.

See also: searchsortedfirst, sort!, insorted, findall.

Examples

julia> searchsorted([1, 2, 4, 5, 5, 7], 4) # single match
3:3

julia> searchsorted([1, 2, 4, 5, 5, 7], 5) # multiple matches
4:5

julia> searchsorted([1, 2, 4, 5, 5, 7], 3) # no match, insert in the middle
3:2

julia> searchsorted([1, 2, 4, 5, 5, 7], 9) # no match, insert at end
7:6

julia> searchsorted([1, 2, 4, 5, 5, 7], 0) # no match, insert at start
1:0

julia> searchsorted([1=>"one", 2=>"two", 2=>"two", 4=>"four"], 2=>"two", by=first) # compare the keys of the pairs
2:3
source
Base.Sort.searchsortedfirstFunction
searchsortedfirst(v, x; by=identity, lt=isless, rev=false)

Return the index of the first value in v greater than or equivalent to x. If x is greater than all values in v, return lastindex(v) + 1.

The vector v must be sorted according to the order defined by the keywords. insert!ing x at the returned index will maintain the sorted order. Refer to sort! for the meaning of the keywords and the definition of "greater than" and equivalence. Note that the by function is applied to the searched value x as well as the values in v.

The index is generally found using binary search, but there are optimized implementations for some inputs.

See also: searchsortedlast, searchsorted, findfirst.

Examples

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 4) # single match
3

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 5) # multiple matches
4

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 3) # no match, insert in the middle
3

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 9) # no match, insert at end
7

julia> searchsortedfirst([1, 2, 4, 5, 5, 7], 0) # no match, insert at start
1

julia> searchsortedfirst([1=>"one", 2=>"two", 4=>"four"], 3=>"three", by=first) # compare the keys of the pairs
3
source
Base.Sort.searchsortedlastFunction
searchsortedlast(v, x; by=identity, lt=isless, rev=false)

Return the index of the last value in v less than or equivalent to x. If x is less than all values in v the function returns firstindex(v) - 1.

The vector v must be sorted according to the order defined by the keywords. Refer to sort! for the meaning of the keywords and the definition of "less than" and equivalence. Note that the by function is applied to the searched value x as well as the values in v.

The index is generally found using binary search, but there are optimized implementations for some inputs

Examples

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 4) # single match
3

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 5) # multiple matches
5

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 3) # no match, insert in the middle
2

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 9) # no match, insert at end
6

julia> searchsortedlast([1, 2, 4, 5, 5, 7], 0) # no match, insert at start
0

julia> searchsortedlast([1=>"one", 2=>"two", 4=>"four"], 3=>"three", by=first) # compare the keys of the pairs
2
source
Base.Sort.insortedFunction
insorted(x, v; by=identity, lt=isless, rev=false) -> Bool

Determine whether a vector v contains any value equivalent to x. The vector v must be sorted according to the order defined by the keywords. Refer to sort! for the meaning of the keywords and the definition of equivalence. Note that the by function is applied to the searched value x as well as the values in v.

The check is generally done using binary search, but there are optimized implementations for some inputs.

See also in.

Examples

julia> insorted(4, [1, 2, 4, 5, 5, 7]) # single match
true

julia> insorted(5, [1, 2, 4, 5, 5, 7]) # multiple matches
true

julia> insorted(3, [1, 2, 4, 5, 5, 7]) # no match
false

julia> insorted(9, [1, 2, 4, 5, 5, 7]) # no match
false

julia> insorted(0, [1, 2, 4, 5, 5, 7]) # no match
false

julia> insorted(2=>"TWO", [1=>"one", 2=>"two", 4=>"four"], by=first) # compare the keys of the pairs
true
Julia 1.6

insorted was added in Julia 1.6.

source
Base.Sort.partialsort!Function
partialsort!(v, k; by=identity, lt=isless, rev=false)

Partially sort the vector v in place 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. 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! may not fully sort the input array.

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

Examples

julia> a = [1, 2, 4, 3, 4]
5-element Vector{Int64}:
 1
 2
 4
 3
 4

julia> partialsort!(a, 4)
4

julia> a
5-element Vector{Int64}:
 1
 2
 3
 4
 4

julia> a = [1, 2, 4, 3, 4]
5-element Vector{Int64}:
 1
 2
 4
 3
 4

julia> partialsort!(a, 4, rev=true)
2

julia> a
5-element Vector{Int64}:
 4
 4
 3
 2
 1
source
Base.Sort.partialsortFunction
partialsort(v, k, by=identity, lt=isless, rev=false)

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

source
Base.Sort.partialsortpermFunction
partialsortperm(v, k; by=ientity, lt=isless, 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: the indices of equal elements will appear in ascending order.

This function is equivalent to, but more efficient than, calling sortperm(...)[k].

Examples

julia> v = [3, 1, 2, 1];

julia> v[partialsortperm(v, 1)]
1

julia> p = partialsortperm(v, 1:3)
3-element view(::Vector{Int64}, 1:3) with eltype Int64:
 2
 4
 3

julia> v[p]
3-element Vector{Int64}:
 1
 1
 2
source
Base.Sort.partialsortperm!Function
partialsortperm!(ix, v, k; by=identity, lt=isless, rev=false)

Like partialsortperm, but accepts a preallocated index vector ix the same size as v, which is used to store (a permutation of) the indices of v.

ix is initialized to contain the indices of v.

(Typically, the indices of v will be 1:length(v), although if v has an alternative array type with non-one-based indices, such as an OffsetArray, ix must share those same indices)

Upon return, ix is guaranteed to have the indices k in their sorted positions, such that

partialsortperm!(ix, v, k);
v[ix[k]] == partialsort(v, k)

The return value is the kth element of ix if k is an integer, or view into ix if k is a range.

Warning

Behavior can be unexpected when any mutated argument shares memory with any other argument.

Examples

julia> v = [3, 1, 2, 1];

julia> ix = Vector{Int}(undef, 4);

julia> partialsortperm!(ix, v, 1)
2

julia> ix = [1:4;];

julia> partialsortperm!(ix, v, 2:3)
2-element view(::Vector{Int64}, 2:3) with eltype Int64:
 4
 3
source

排序算法

基础 Julia 中目前有四种公开可用的排序算法:

默认情况下,sort 系列函数使用稳定的排序算法,这些算法在大多数输入上都很快。 具体的算法选择是实现细节,以便于未来的性能改进。

目前,基于输入类型、大小和组成,使用了 RadixSortScratchQuickSortInsertionSortCountingSort 的混合算法。 实现细节可能会发生变化,但目前可以在 ??Base.DEFAULT_STABLE 的扩展帮助和其中列出的内部排序算法的文档字符串中找到。

你可以使用 alg 关键字显式指定你偏好的算法(例如 sort!(v, alg=PartialQuickSort(10:20))), 或者通过向 Base.Sort.defalg 函数添加专门的方法来重新配置自定义类型的默认排序算法。 例如,InlineStrings.jl 定义了以下方法:

Base.Sort.defalg(::AbstractArray{<:Union{SmallInlineStrings, Missing}}) = InlineStringSort
Julia 1.9

默认排序算法(由 Base.Sort.defalg 返回)自 Julia 1.9 起保证是稳定的。 之前的版本在排序数值数组时有不稳定的边缘情况。

替代排序

默认情况下,sortsearchsorted 和相关函数使用 isless 来比较两个元素,以确定哪个应该排在前面。 Base.Order.Ordering 抽象类型提供了一种机制,用于在相同的元素集上定义替代排序: 当调用像 sort! 这样的排序函数时,可以通过关键字参数 order 提供一个 Ordering 实例。

Ordering 的实例通过 Base.Order.lt 函数定义排序,该函数作为 isless 的泛化。 这个函数在自定义 Ordering 上的行为必须满足 严格弱序 的所有条件。 有关有效和无效 lt 函数的详细信息和示例,请参见 sort!

Base.Order.OrderingType
Base.Order.Ordering

Abstract type which represents a total order on some set of elements.

Use Base.Order.lt to compare two elements according to the ordering.

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Base.Order.ltFunction
lt(o::Ordering, a, b)

Test whether a is less than b according to the ordering o.

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Base.Order.ordFunction
ord(lt, by, rev::Union{Bool, Nothing}, order::Ordering=Forward)

Construct an Ordering object from the same arguments used by sort!. Elements are first transformed by the function by (which may be identity) and are then compared according to either the function lt or an existing ordering order. lt should be isless or a function that obeys the same rules as the lt parameter of sort!. Finally, the resulting order is reversed if rev=true.

Passing an lt other than isless along with an order other than Base.Order.Forward or Base.Order.Reverse is not permitted, otherwise all options are independent and can be used together in all possible combinations.

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Base.Order.ReverseOrderingType
ReverseOrdering(fwd::Ordering=Forward)

A wrapper which reverses an ordering.

For a given Ordering o, the following holds for all a, b:

lt(ReverseOrdering(o), a, b) == lt(o, b, a)
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Base.Order.ByType
By(by, order::Ordering=Forward)

Ordering which applies order to elements after they have been transformed by the function by.

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Base.Order.LtType
Lt(lt)

Ordering that calls lt(a, b) to compare elements. lt must obey the same rules as the lt parameter of sort!.

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Base.Order.PermType
Perm(order::Ordering, data::AbstractVector)

Ordering on the indices of data where i is less than j if data[i] is less than data[j] according to order. In the case that data[i] and data[j] are equal, i and j are compared by numeric value.

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