Sorting and Related Functions

Julia has an extensive, flexible API for sorting and interacting with already-sorted arrays of values. By default, Julia picks reasonable algorithms and sorts in standard ascending order:

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

You can easily sort in reverse order as well:

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

To sort an array in-place, use the “bang” version of the sort function:

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

Instead of directly sorting an array, you can compute a permutation of the array’s indices that puts the array into sorted order:

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

Arrays can easily be sorted according to an arbitrary transformation of their values:

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

Or in reverse order by a transformation:

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

If needed, the sorting algorithm can be chosen:

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

All the sorting and order related functions rely on a “less than” relation defining a total order on the values to be manipulated. The isless function is invoked by default, but the relation can be specified via the lt keyword.

Sorting Functions

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. 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 (note that for every x and y, only one of lt(x,y) and lt(y,x) can return true); 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

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")

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 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.sort — Function

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=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.

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.sortperm — Function

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!, partialsortperm, invperm, indexin.

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

source

Base.Sort.InsertionSort — Constant

InsertionSort

Indicate that a sorting function should 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 list.

Characteristics:

• stable: preserves the ordering of elements which compare equal (e.g. “a” and “A” in a sort of letters which 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.MergeSort — Constant

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 which compare equal (e.g. “a” and “A” in a sort of letters which ignores case).
• not in-place in memory.
• divide-and-conquer sort strategy.

source

Base.Sort.QuickSort — Constant

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 which compare equal (e.g. “a” and “A” in a sort of letters which ignores case).
• in-place in memory.
• divide-and-conquer: sort strategy similar to MergeSort.
• good performance for large collections.

source

Base.Sort.PartialQuickSort — Type

PartialQuickSort{T <: Union{Integer,OrdinalRange}}

Indicate that a sorting function should use the partial quick sort algorithm. Partial quick sort returns the smallest k elements sorted from smallest to largest, finding them and sorting them using QuickSort.

Characteristics:

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

source

Base.Sort.sortperm! — Function

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

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

source

Base.sortslices — Function

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

Order-Related Functions

Base.issorted — Function

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

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

source

Base.Sort.searchsorted — Function

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.

See also: insorted, searchsortedfirst, sort, 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

source

Base.Sort.searchsortedfirst — Function

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.

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

source

Base.Sort.searchsortedlast — Function

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

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

source

Base.Sort.insorted — Function

insorted(a, x; by=<transform>, lt=<comparison>, rev=false) -> Bool

Determine whether an item is in the given sorted collection, in the sense that it is \== to one of the values of the collection according to the order specified by the by, lt and rev keywords, assuming that a is already sorted in that order, see sort for the keywords.

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 1.6

insorted was added in Julia 1.6.

source

Base.Sort.partialsort! — Function

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

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.partialsort — Function

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

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

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=<transform>, lt=<comparison>, rev=false, initialized=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.

If the index vector ix is initialized with the indices of v (or a permutation thereof), initialized should be set to true.

If initialized is false (the default), then ix is initialized to contain the indices of v.

If initialized is true, but ix does not contain (a permutation of) the indices of v, the behavior of partialsortperm! is undefined.

(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 also be an OffsetArray with the same indices, and must contain as values (a permutation of) these 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.

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, initialized=true)
2-element view(::Vector{Int64}, 2:3) with eltype Int64:
 4
 3

source

Sorting Algorithms

There are currently four sorting algorithms available in base Julia:

• InsertionSort
• QuickSort
• PartialQuickSort(k)
• MergeSort

InsertionSort is an O(n^2) stable sorting algorithm. It is efficient for very small n, and is used internally by QuickSort.

QuickSort is an O(n log n) sorting algorithm which is in-place, very fast, but not stable – i.e. elements which are considered equal will not remain in the same order in which they originally appeared in the array to be sorted. QuickSort is the default algorithm for numeric values, including integers and floats.

PartialQuickSort(k) is similar to QuickSort, but the output array is only sorted up to index k if k is an integer, or in the range of k if k is an OrdinalRange. For example:

x = rand(1:500, 100)
k = 50
k2 = 50:100
s = sort(x; alg=QuickSort)
ps = sort(x; alg=PartialQuickSort(k))
qs = sort(x; alg=PartialQuickSort(k2))
map(issorted, (s, ps, qs))             # => (true, false, false)
map(x->issorted(x[1:k]), (s, ps, qs))  # => (true, true, false)
map(x->issorted(x[k2]), (s, ps, qs))   # => (true, false, true)
s[1:k] == ps[1:k]                      # => true
s[k2] == qs[k2]                        # => true

MergeSort is an O(n log n) stable sorting algorithm but is not in-place – it requires a temporary array of half the size of the input array – and is typically not quite as fast as QuickSort. It is the default algorithm for non-numeric data.

The default sorting algorithms are chosen on the basis that they are fast and stable, or appear to be so. For numeric types indeed, QuickSort is selected as it is faster and indistinguishable in this case from a stable sort (unless the array records its mutations in some way). The stability property comes at a non-negligible cost, so if you don’t need it, you may want to explicitly specify your preferred algorithm, e.g. sort!(v, alg=QuickSort).

The mechanism by which Julia picks default sorting algorithms is implemented via the Base.Sort.defalg function. It allows a particular algorithm to be registered as the default in all sorting functions for specific arrays. For example, here are the two default methods from sort.jl:

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

As for numeric arrays, choosing a non-stable default algorithm for array types for which the notion of a stable sort is meaningless (i.e. when two values comparing equal can not be distinguished) may make sense.

Alternate orderings

By default, sort and related functions use isless to compare two elements in order to determine which should come first. The Base.Order.Ordering abstract type provides a mechanism for defining alternate orderings on the same set of elements. Instances of Ordering define a total order on a set of elements, so that for any elements a, b, c the following hold:

• Exactly one of the following is true: a is less than b, b is less than a, or a and b are equal (according to isequal).
• The relation is transitive - if a is less than b and b is less than c then a is less than c.

The Base.Order.lt function works as a generalization of isless to test whether a is less than b according to a given order.

Base.Order.Ordering — Type

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.

source

Base.Order.lt — Function

lt(o::Ordering, a, b)

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

source

Base.Order.ord — Function

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 which obeys similar rules. 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.

source

Base.Order.Forward — Constant

Base.Order.Forward

Default ordering according to isless.

source

Base.Order.ReverseOrdering — Type

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)

source

Base.Order.Reverse — Constant

Base.Order.Reverse

Reverse ordering according to isless.

source

Base.Order.By — Type

By(by, order::Ordering=Forward)

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

source

Base.Order.Lt — Type

Lt(lt)

Ordering which calls lt(a, b) to compare elements. lt should obey the same rules as implementations of isless.

source

Base.Order.Perm — Type

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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