Sorting and Related Functions

sort!(A; dims::Integer, alg::Base.Sort.Algorithm=Base.Sort.defalg(A), lt=isless, by=identity, rev::Bool=false, order::Base.Order.Ordering=Base.Order.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.

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

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

sort(A; dims::Integer, alg::Base.Sort.Algorithm=Base.Sort.defalg(A), lt=isless, by=identity, rev::Bool=false, order::Base.Order.Ordering=Base.Order.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

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

sortperm(A; alg::Base.Sort.Algorithm=Base.Sort.DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Base.Order.Ordering=Base.Order.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.

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

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.

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.

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.

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

sortperm!(ix, A; alg::Base.Sort.Algorithm=Base.Sort.DEFAULT_UNSTABLE, lt=isless, by=identity, rev::Bool=false, order::Base.Order.Ordering=Base.Order.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).

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

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 = [4 3; 2 1 ;;; 'A' 'B'; 'C' 'D']
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

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

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

searchsortedfirst(v, x; by=identity, lt=isless, rev=false)

Return the index of the first value in v that is not ordered before x. If all values in v are ordered before x, 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 and use of the keywords. 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

searchsortedlast(v, x; by=identity, lt=isless, rev=false)

Return the index of the last value in v that is not ordered after x. If all values in v are ordered after x, return firstindex(v) - 1.

The vector v must be sorted according to the order defined by the keywords. insert!ing x immediately after the returned index will maintain the sorted order. Refer to sort! for the meaning and use of the keywords. 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

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

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

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.

partialsortperm(v, k; by=identity, 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

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.

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

Base.Order.Ordering

Abstract type which represents a strict weak order on some set of elements. See sort! for more.

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

lt(o::Ordering, a, b) -> Bool

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

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.

Base.Order.Forward

Default ordering according to isless.

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)

Base.Order.Reverse

Reverse ordering according to isless.

By(by, order::Ordering=Forward)

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

Lt(lt)

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

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.