# Collections and Data Structures¶

## Iteration¶

Sequential iteration is implemented by the methods start, done, and next. The general for loop:

for i = I   # or  "for i in I"
# body
end


is translated to:

state = start(I)
while !done(I, state)
(i, state) = next(I, state)
# body
end


The state object may be anything, and should be chosen appropriately for each iterable type.

start(iter) → state

Get initial iteration state for an iterable object

done(iter, state) → Bool

Test whether we are done iterating

next(iter, state) → item, state

For a given iterable object and iteration state, return the current item and the next iteration state

zip(iters...)

For a set of iterable objects, returns an iterable of tuples, where the ith tuple contains the ith component of each input iterable.

Note that zip is its own inverse: [zip(zip(a...)...)...] == [a...].

enumerate(iter)

Return an iterator that yields (i, x) where i is an index starting at 1, and x is the ith value from the given iterator. It’s useful when you need not only the values x over which you are iterating, but also the index i of the iterations.

julia> a = ["a", "b", "c"];

julia> for (index, value) in enumerate(a)
println("$index$value")
end
1 a
2 b
3 c


Fully implemented by: Range, UnitRange, NDRange, Tuple, Real, AbstractArray, IntSet, ObjectIdDict, Dict, WeakKeyDict, EachLine, String, Set, Task.

## General Collections¶

isempty(collection) → Bool

Determine whether a collection is empty (has no elements).

julia> isempty([])
true

julia> isempty([1 2 3])
false

empty!(collection) → collection

Remove all elements from a collection.

length(collection) → Integer

For ordered, indexable collections, the maximum index i for which getindex(collection, i) is valid. For unordered collections, the number of elements.

endof(collection) → Integer

Returns the last index of the collection.

julia> endof([1,2,4])
3


Fully implemented by: Range, UnitRange, Tuple, Number, AbstractArray, IntSet, Dict, WeakKeyDict, String, Set.

## Iterable Collections¶

in(item, collection) → Bool
(item, collection) → Bool
(collection, item) → Bool
(item, collection) → Bool
(collection, item) → Bool

Determine whether an item is in the given collection, in the sense that it is == to one of the values generated by iterating over the collection. Some collections need a slightly different definition; for example Sets check whether the item is isequal to one of the elements. Dicts look for (key,value) pairs, and the key is compared using isequal. To test for the presence of a key in a dictionary, use haskey or k in keys(dict).

eltype(collection)

Determine the type of the elements generated by iterating collection. For associative collections, this will be a (key,value) tuple type.

indexin(a, b)

Returns a vector containing the highest index in b for each value in a that is a member of b . The output vector contains 0 wherever a is not a member of b.

findin(a, b)

Returns the indices of elements in collection a that appear in collection b

unique(itr[, dim])

Returns an array containing only the unique elements of the iterable itr, in the order that the first of each set of equivalent elements originally appears. If dim is specified, returns unique regions of the array itr along dim.

reduce(op, v0, itr)

Reduce the given collection ìtr with the given binary operator op. v0 must be a neutral element for op that will be returned for empty collections. It is unspecified whether v0 is used for non-empty collections.

Reductions for certain commonly-used operators have special implementations which should be used instead: maximum(itr), minimum(itr), sum(itr), prod(itr), any(itr), all(itr).

The associativity of the reduction is implementation-dependent. This means that you can’t use non-associative operations like - because it is undefined whether reduce(-,[1,2,3]) should be evaluated as (1-2)-3 or 1-(2-3). Use foldl or foldr instead for guaranteed left or right associativity.

Some operations accumulate error, and parallelism will also be easier if the reduction can be executed in groups. Future versions of Julia might change the algorithm. Note that the elements are not reordered if you use an ordered collection.

reduce(op, itr)

Like reduce(op, v0, itr). This cannot be used with empty collections, except for some special cases (e.g. when op is one of +, *, max, min, &, |) when Julia can determine the neutral element of op.

foldl(op, v0, itr)

Like reduce, but with guaranteed left associativity. v0 will be used exactly once.

foldl(op, itr)

Like foldl(op, v0, itr), but using the first element of itr as v0. In general, this cannot be used with empty collections (see reduce(op, itr)).

foldr(op, v0, itr)

Like reduce, but with guaranteed right associativity. v0 will be used exactly once.

foldr(op, itr)

Like foldr(op, v0, itr), but using the last element of itr as v0. In general, this cannot be used with empty collections (see reduce(op, itr)).

maximum(itr)

Returns the largest element in a collection.

maximum(A, dims)

Compute the maximum value of an array over the given dimensions.

maximum!(r, A)

Compute the maximum value of A over the singleton dimensions of r, and write results to r.

minimum(itr)

Returns the smallest element in a collection.

minimum(A, dims)

Compute the minimum value of an array over the given dimensions.

minimum!(r, A)

Compute the minimum value of A over the singleton dimensions of r, and write results to r.

extrema(itr)

Compute both the minimum and maximum element in a single pass, and return them as a 2-tuple.

indmax(itr) → Integer

Returns the index of the maximum element in a collection.

indmin(itr) → Integer

Returns the index of the minimum element in a collection.

findmax(itr) -> (x, index)

Returns the maximum element and its index.

findmax(A, dims) -> (maxval, index)

For an array input, returns the value and index of the maximum over the given dimensions.

findmin(itr) -> (x, index)

Returns the minimum element and its index.

findmin(A, dims) -> (minval, index)

For an array input, returns the value and index of the minimum over the given dimensions.

maxabs(itr)

Compute the maximum absolute value of a collection of values.

maxabs(A, dims)

Compute the maximum absolute values over given dimensions.

maxabs!(r, A)

Compute the maximum absolute values over the singleton dimensions of r, and write values to r.

minabs(itr)

Compute the minimum absolute value of a collection of values.

minabs(A, dims)

Compute the minimum absolute values over given dimensions.

minabs!(r, A)

Compute the minimum absolute values over the singleton dimensions of r, and write values to r.

sum(itr)

Returns the sum of all elements in a collection.

sum(A, dims)

Sum elements of an array over the given dimensions.

sum!(r, A)

Sum elements of A over the singleton dimensions of r, and write results to r.

sum(f, itr)

Sum the results of calling function f on each element of itr.

sumabs(itr)

Sum absolute values of all elements in a collection. This is equivalent to sum(abs(itr)) but faster.

sumabs(A, dims)

Sum absolute values of elements of an array over the given dimensions.

sumabs!(r, A)

Sum absolute values of elements of A over the singleton dimensions of r, and write results to r.

sumabs2(itr)

Sum squared absolute values of all elements in a collection. This is equivalent to sum(abs2(itr)) but faster.

sumabs2(A, dims)

Sum squared absolute values of elements of an array over the given dimensions.

sumabs2!(r, A)

Sum squared absolute values of elements of A over the singleton dimensions of r, and write results to r.

prod(itr)

Returns the product of all elements of a collection.

prod(A, dims)

Multiply elements of an array over the given dimensions.

prod!(r, A)

Multiply elements of A over the singleton dimensions of r, and write results to r.

any(itr) → Bool

Test whether any elements of a boolean collection are true.

any(A, dims)

Test whether any values along the given dimensions of an array are true.

any!(r, A)

Test whether any values in A along the singleton dimensions of r are true, and write results to r.

all(itr) → Bool

Test whether all elements of a boolean collection are true.

all(A, dims)

Test whether all values along the given dimensions of an array are true.

all!(r, A)

Test whether all values in A along the singleton dimensions of r are true, and write results to r.

count(p, itr) → Integer

Count the number of elements in itr for which predicate p returns true.

any(p, itr) → Bool

Determine whether predicate p returns true for any elements of itr.

all(p, itr) → Bool

Determine whether predicate p returns true for all elements of itr.

julia> all(i->(4<=i<=6), [4,5,6])
true

map(f, c...) → collection

Transform collection c by applying f to each element. For multiple collection arguments, apply f elementwise.

julia> map((x) -> x * 2, [1, 2, 3])
3-element Array{Int64,1}:
2
4
6

julia> map(+, [1, 2, 3], [10, 20, 30])
3-element Array{Int64,1}:
11
22
33

map!(function, collection)

In-place version of map().

map!(function, destination, collection...)

Like map(), but stores the result in destination rather than a new collection. destination must be at least as large as the first collection.

mapreduce(f, op, v0, itr)

Apply function f to each element in itr, and then reduce the result using the binary function op. v0 must be a neutral element for op that will be returned for empty collections. It is unspecified whether v0 is used for non-empty collections.

mapreduce is functionally equivalent to calling reduce(op, v0, map(f, itr)), but will in general execute faster since no intermediate collection needs to be created. See documentation for reduce and map.

julia> mapreduce(x->x^2, +, [1:3]) # == 1 + 4 + 9
14


The associativity of the reduction is implementation-dependent. Use mapfoldl() or mapfoldr() instead for guaranteed left or right associativity.

mapreduce(f, op, itr)

Like mapreduce(f, op, v0, itr). In general, this cannot be used with empty collections (see reduce(op, itr)).

mapfoldl(f, op, v0, itr)

Like mapreduce, but with guaranteed left associativity. v0 will be used exactly once.

mapfoldl(f, op, itr)

Like mapfoldl(f, op, v0, itr), but using the first element of itr as v0. In general, this cannot be used with empty collections (see reduce(op, itr)).

mapfoldr(f, op, v0, itr)

Like mapreduce, but with guaranteed right associativity. v0 will be used exactly once.

mapfoldr(f, op, itr)

Like mapfoldr(f, op, v0, itr), but using the first element of itr as v0. In general, this cannot be used with empty collections (see reduce(op, itr)).

first(coll)

Get the first element of an iterable collection.

last(coll)

Get the last element of an ordered collection, if it can be computed in O(1) time. This is accomplished by calling endof to get the last index.

step(r)

Get the step size of a Range object.

collect(collection)

Return an array of all items in a collection. For associative collections, returns (key, value) tuples.

collect(element_type, collection)

Return an array of type Array{element_type,1} of all items in a collection.

issubset(a, b)
(A, S) → Bool
(A, S) → Bool
(A, S) → Bool

Determine whether every element of a is also in b, using the in function.

filter(function, collection)

Return a copy of collection, removing elements for which function is false. For associative collections, the function is passed two arguments (key and value).

filter!(function, collection)

Update collection, removing elements for which function is false. For associative collections, the function is passed two arguments (key and value).

## Indexable Collections¶

getindex(collection, key...)

Retrieve the value(s) stored at the given key or index within a collection. The syntax a[i,j,...] is converted by the compiler to getindex(a, i, j, ...).

setindex!(collection, value, key...)

Store the given value at the given key or index within a collection. The syntax a[i,j,...] = x is converted by the compiler to setindex!(a, x, i, j, ...).

Fully implemented by: Array, DArray, BitArray, AbstractArray, SubArray, ObjectIdDict, Dict, WeakKeyDict, String.

Partially implemented by: Range, UnitRange, Tuple.

## Associative Collections¶

Dict is the standard associative collection. Its implementation uses the hash(x) as the hashing function for the key, and isequal(x,y) to determine equality. Define these two functions for custom types to override how they are stored in a hash table.

ObjectIdDict is a special hash table where the keys are always object identities. WeakKeyDict is a hash table implementation where the keys are weak references to objects, and thus may be garbage collected even when referenced in a hash table.

Dicts can be created using a literal syntax: {"A"=>1, "B"=>2}. Use of curly brackets will create a Dict of type Dict{Any,Any}. Use of square brackets will attempt to infer type information from the keys and values (i.e. ["A"=>1, "B"=>2] creates a Dict{ASCIIString, Int64}). To explicitly specify types use the syntax: (KeyType=>ValueType)[...]. For example, (ASCIIString=>Int32)["A"=>1, "B"=>2].

As with arrays, Dicts may be created with comprehensions. For example, {i => f(i) for i = 1:10}.

Given a dictionary D, the syntax D[x] returns the value of key x (if it exists) or throws an error, and D[x] = y stores the key-value pair x => y in D (replacing any existing value for the key x). Multiple arguments to D[...] are converted to tuples; for example, the syntax D[x,y] is equivalent to D[(x,y)], i.e. it refers to the value keyed by the tuple (x,y).

Dict()

Dict{K,V}() constructs a hash

table with keys of type K and values of type V. The literal syntax is {"A"=>1, "B"=>2} for a Dict{Any,Any}, or ["A"=>1, "B"=>2] for a Dict of inferred type.

Determine whether a collection has a mapping for a given key.

get(collection, key, default)

Return the value stored for the given key, or the given default value if no mapping for the key is present.

get(f::Function, collection, key)

Return the value stored for the given key, or if no mapping for the key is present, return f(). Use get! to also store the default value in the dictionary.

This is intended to be called using do block syntax:

get(dict, key) do
# default value calculated here
time()
end

get!(collection, key, default)

Return the value stored for the given key, or if no mapping for the key is present, store key => default, and return default.

get!(f::Function, collection, key)

Return the value stored for the given key, or if no mapping for the key is present, store key => f(), and return f().

This is intended to be called using do block syntax:

get!(dict, key) do
# default value calculated here
time()
end

getkey(collection, key, default)

Return the key matching argument key if one exists in collection, otherwise return default.

delete!(collection, key)

Delete the mapping for the given key in a collection, and return the colection.

pop!(collection, key[, default])

Delete and return the mapping for key if it exists in collection, otherwise return default, or throw an error if default is not specified.

keys(collection)

Return an iterator over all keys in a collection. collect(keys(d)) returns an array of keys.

values(collection)

Return an iterator over all values in a collection. collect(values(d)) returns an array of values.

merge(collection, others...)

Construct a merged collection from the given collections.

merge!(collection, others...)

Update collection with pairs from the other collections

sizehint(s, n)

Suggest that collection s reserve capacity for at least n elements. This can improve performance.

Fully implemented by: ObjectIdDict, Dict, WeakKeyDict.

Partially implemented by: IntSet, Set, EnvHash, Array, BitArray.

## Set-Like Collections¶

Set([itr])

Construct a Set of the values generated by the given iterable object, or an empty set. Should be used instead of IntSet for sparse integer sets, or for sets of arbitrary objects.

IntSet([itr])

Construct a sorted set of the integers generated by the given iterable object, or an empty set. Implemented as a bit string, and therefore designed for dense integer sets. Only non-negative integers can be stored. If the set will be sparse (for example holding a single very large integer), use Set instead.

union(s1, s2...)
(s1, s2)

Construct the union of two or more sets. Maintains order with arrays.

union!(s, iterable)

Union each element of iterable into set s in-place.

intersect(s1, s2...)
(s1, s2)

Construct the intersection of two or more sets. Maintains order and multiplicity of the first argument for arrays and ranges.

setdiff(s1, s2)

Construct the set of elements in s1 but not s2. Maintains order with arrays. Note that both arguments must be collections, and both will be iterated over. In particular, setdiff(set,element) where element is a potential member of set, will not work in general.

setdiff!(s, iterable)

Remove each element of iterable from set s in-place.

symdiff(s1, s2...)

Construct the symmetric difference of elements in the passed in sets or arrays. Maintains order with arrays.

symdiff!(s, n)

IntSet s is destructively modified to toggle the inclusion of integer n.

symdiff!(s, itr)

For each element in itr, destructively toggle its inclusion in set s.

symdiff!(s1, s2)

Construct the symmetric difference of IntSets s1 and s2, storing the result in s1.

complement(s)

Returns the set-complement of IntSet s.

complement!(s)

Mutates IntSet s into its set-complement.

intersect!(s1, s2)

Intersects IntSets s1 and s2 and overwrites the set s1 with the result. If needed, s1 will be expanded to the size of s2.

issubset(A, S) → Bool
(A, S) → Bool

True if A is a subset of or equal to S.

Fully implemented by: IntSet, Set.

Partially implemented by: Array.

## Dequeues¶

push!(collection, items...) → collection

Insert items at the end of a collection.

pop!(collection) → item

Remove the last item in a collection and return it.

unshift!(collection, items...) → collection

Insert items at the beginning of a collection.

shift!(collection) → item

Remove the first item in a collection.

insert!(collection, index, item)

Insert an item at the given index.

deleteat!(collection, index)

Remove the item at the given index, and return the modified collection. Subsequent items are shifted to fill the resulting gap.

deleteat!(collection, itr)

Remove the items at the indices given by itr, and return the modified collection. Subsequent items are shifted to fill the resulting gap. itr must be sorted and unique.

splice!(collection, index[, replacement]) → item

Remove the item at the given index, and return the removed item. Subsequent items are shifted down to fill the resulting gap. If specified, replacement values from an ordered collection will be spliced in place of the removed item.

To insert replacement before an index n without removing any items, use splice!(collection, n:n-1, replacement).

splice!(collection, range[, replacement]) → items

Remove items in the specified index range, and return a collection containing the removed items. Subsequent items are shifted down to fill the resulting gap. If specified, replacement values from an ordered collection will be spliced in place of the removed items.

To insert replacement before an index n without removing any items, use splice!(collection, n:n-1, replacement).

resize!(collection, n) → collection

Resize collection to contain n elements.

append!(collection, items) → collection.

Add the elements of items to the end of a collection.

julia> append!([1],[2,3])
3-element Array{Int64,1}:
1
2
3

prepend!(collection, items) → collection

Insert the elements of items to the beginning of a collection.

julia> prepend!([3],[1,2])
3-element Array{Int64,1}:
1
2
3


Fully implemented by: Vector (aka 1-d Array), BitVector (aka 1-d BitArray).

## PriorityQueue¶

The PriorityQueue type is available from the Collections module. It provides a basic priority queue implementation allowing for arbitrary key and priority types. Multiple identical keys are not permitted, but the priority of existing keys can be changed efficiently.

PriorityQueue{K,V}([ord])

Construct a new PriorityQueue, with keys of type K and values/priorites of type V. If an order is not given, the priority queue is min-ordered using the default comparison for V.

enqueue!(pq, k, v)

Insert the a key k into a priority queue pq with priority v.

dequeue!(pq)

Remove and return the lowest priority key from a priority queue.

peek(pq)

Return the lowest priority key from a priority queue without removing that key from the queue.

PriorityQueue also behaves similarly to a Dict so that keys can be inserted and priorities accessed or changed using indexing notation:

# Julia code
pq = Collections.PriorityQueue()

# Insert keys with associated priorities
pq["a"] = 10
pq["b"] = 5
pq["c"] = 15

# Change the priority of an existing key
pq["a"] = 0


## Heap Functions¶

Along with the PriorityQueue type, the Collections module provides lower level functions for performing binary heap operations on arrays. Each function takes an optional ordering argument. If not given, default ordering is used, so that elements popped from the heap are given in ascending order.

heapify(v[, ord])

Return a new vector in binary heap order, optionally using the given ordering.

heapify!(v[, ord])

In-place heapify.

isheap(v[, ord])

Return true iff an array is heap-ordered according to the given order.

heappush!(v, x[, ord])

Given a binary heap-ordered array, push a new element x, preserving the heap property. For efficiency, this function does not check that the array is indeed heap-ordered.

heappop!(v[, ord])

Given a binary heap-ordered array, remove and return the lowest ordered element. For efficiency, this function does not check that the array is indeed heap-ordered.