DataStructures.jl

This package implements a variety of data structures, including

• Deque (based on block-list)
• Stack
• Queue
• Accumulators and Counters
• Disjoint Sets
• Binary Heap
• Mutable Binary Heap
• Ordered Dicts and Sets
• Dictionaries with Defaults
• Trie
• Linked List

Deque

The Deque type implements a double-ended queue using a list of blocks. This data structure supports constant-time insertion/removal of elements at both ends of a sequence.

Usage:

a = Deque{Int}()
isempty(a)          # test whether the dequeue is empty
length(a)           # get the number of elements
push!(a, 10)        # add an element to the back
pop!(a)             # remove an element from the back
unshift!(a, 20)     # add an element to the front
shift!(a)           # remove an element from the front
front(a)            # get the element at the front
back(a)             # get the element at the back

Note: Julia's Vector type also provides this interface, and thus can be used as a deque. However, the Deque type in this package is implemented as a list of contiguous blocks (default size = 2K). As a deque grows, new blocks may be created and linked to existing blocks. This way avoids the copying when growing a vector.

Benchmark shows that the performance of Deque is comparable to Vector on push!, and is noticeably faster on unshift! (by about 30% to 40%).

Stack and Queue

The Stack and Queue types are a light-weight wrapper of a deque type, which respectively provide interfaces for FILO and FIFO access.

Usage of Stack:

s = Stack(Int)
push!(s, x)
x = top(s)
x = pop!(s)

Usage of Queue:

q = Queue(Int)
enqueue!(q, x)
x = front(q)
x = back(q)
x = dequeue!(q)

Accumulators and Counters

A accumulator, as defined below, is a data structure that maintains an accumulated number for each key. This is a counter when the accumulated values reflect the counts:

type Accumulator{K, V<:Number}
    map::Dict{K, V}
end

There are different ways to construct an accumulator/counter:

a = accumulator(K, V)    # construct an accumulator with key-type K and
                         # accumulated value type V

a = accumulator(dict)    # construct an accumulator from a dictionary

a = counter(K)           # construct a counter, i.e. an accumulator with
                         # key type K and value type Int

a = counter(dict)        # construct a counter from a dictionary

a = counter(seq)         # construct a counter by counting keys in a sequence

Usage of an accumulator/counter:

# let a and a2 be accumulators/counters

a[x]             # get the current value/count for x.
                 # if x was not added to a, it returns zero(V)

push!(a, x)       # add the value/count for x by 1
push!(a, x, v)    # add the value/count for x by v
push!(a, a2)      # add all counts from a2 to a1

pop!(a, x)       # remove a key x from a, and returns its current value

merge(a, a2)     # return a new accumulator/counter that combines the
                 # values/counts in both a and a2

Disjoint Sets

Some algorithms, such as finding connected components in undirected graph and Kruskal's method of finding minimum spanning tree, require a data structure that can efficiently represent a collection of disjoint subsets. A widely used data structure for this purpose is the Disjoint set forest.

Usage:

a = IntDisjointSets(10)      # creates a forest comprised of 10 singletons
union!(a, 3, 5)             # merges the sets that contain 3 and 5 into one
in_same_set(a, x, y)        # determines whether x and y are in the same set
elem = push!(a)             # adds a single element in a new set; returns the new element
                            # (this operation is often called MakeSet)

One may also use other element types:

a = DisjointSets{String}(["a", "b", "c", "d"])
union!(a, "a", "b")
in_same_set(a, "c", "d")
push!(a, "f")

Note that the internal implementation of IntDisjointSets is based on vectors, and is very efficient. DisjointSets{T} is a wrapper of IntDisjointSets, which uses a dictionary to map input elements to an internal index.

Heaps

Heaps are data structures that efficiently maintain the minimum (or maximum) for a set of data that may dynamically change.

All heaps in this package are derived from AbstractHeap, and provide the following interface:

# Let h be a heap, i be a handle, and v be a value.

length(h)         # returns the number of elements

isempty(h)        # returns whether the heap is empty

push!(h, v)       # add a value to the heap

top(h)            # return the top value of a heap

pop!(h)           # removes the top value, and returns it

Mutable heaps (values can be changed after being pushed to a heap) are derived from AbstractMutableHeap <: AbstractHeap, and additionally provides the following interface:

i = push!(h, v)       # adds a value to the heap and and returns a handle to v

update!(h, i, v)      # updates the value of an element (referred to by the handle i)

Currently, both min/max versions of binary heap (type BinaryHeap) and mutable binary heap (type MutableBinaryHeap) have been implemented.

Examples of constructing a heap:

h = binary_minheap(Int)
h = binary_maxheap(Int)            # create an empty min/max binary heap of integers

h = binary_minheap([1,4,3,2])
h = binary_maxheap([1,4,3,2])      # create a min/max heap from a vector

h = mutable_binary_minheap(Int)
h = mutable_binary_maxheap(Int)    # create an empty mutable min/max heap

h = mutable_binary_minheap([1,4,3,2])
h = mutable_binary_maxheap([1,4,3,2])    # create a mutable min/max heap from a vector

Functions using heaps

Heaps can be used to extract the largest or smallest elements of an array without sorting the entire array first:

nlargest(3, [0,21,-12,68,-25,14]) # => [68,21,14]
nsmallest(3, [0,21,-12,68,-25,14]) # => [-25,-12,0]

nlargest(n, a) is equivalent to sort(a, lt = >)[1:min(n, end)], and nsmallest(n, a) is equivalent to sort(a, lt = <)[1:min(n, end)].

OrderedDicts and OrderedSets

OrderedDicts are simply dictionaries whose entries have a particular order. For OrderedDicts (and OrderedSets), order refers to insertion order, which allows deterministic iteration over the dictionary or set:

```julia
d = OrderedDict(Char,Int)
for c in 'a':'e'
    d[c] = c-'a'+1
end
collect(d) # => [('a',1),('b',2),('c',3),('d',4),('e',5)]

s = OrderedSet(π,e,γ,catalan,φ)
collect(s) # => [π = 3.1415926535897...,
           #     e = 2.7182818284590...,
           #     γ = 0.5772156649015...,
                 #     catalan = 0.9159655941772...,
                 #     φ = 1.6180339887498...]

All standard Associative and Dict functions are available for OrderedDicts, and all Set operations are available for OrderedSets.

Note that to create an OrderedSet of a particular type, you must specify the type in curly-braces:

# create an OrderedSet of Strings
strs = OrderedSet{String}()

DefaultDict and DefaultOrderedDict

A DefaultDict allows specification of a default value to return when a requested key is not in a dictionary.

While the implementation is slightly different, a DefaultDict can be thought to provide a normal Dict with a default value. A DefaultOrderedDict does the same for an OrderedDict.

Constructors:

DefaultDict(default, kv)                        # create a DefaultDict with a default value or function,
                                                # optionally wrapping an existing dictionary
                                                                                       # or array of key-value pairs

DefaultDict(KeyType, ValueType, default)        # create a DefaultDict with Dict type (KeyType,ValueType)

DefaultOrderedDict(default, kv)                 # create a DefaultOrderedDict with a default value or function,
                                                # optionally wrapping an existing dictionary
                                                                              # or array of key-value pairs

DefaultOrderedDict(KeyType, ValueType, default) # create a DefaultOrderedDict with Dict type (KeyType,ValueType)

Examples using DefaultDict:

dd = DefaultDict(1)               # create an (Any=>Any) DefaultDict with a default value of 1
dd = DefaultDict(String, Int, 0)  # create a (String=>Int) DefaultDict with a default value of 0

d = ['a'=>1, 'b'=>2]
dd = DefaultDict(0, d)            # provide a default value to an existing dictionary
dd['c'] == 0                      # true
#d['c'] == 0                      # false

dd = DefaultOrderedDict(time)     # call time() to provide the default value for an OrderedDict
dd = DefaultDict(Dict)            # Create a dictionary of dictionaries
                                  # Dict() is called to provide the default value
dd = DefaultDict(()->myfunc())    # call function myfunc to provide the default value

# create a Dictionary of type String=>DefaultDict{String, Int}, where the default of the
# inner set of DefaultDicts is zero
dd = DefaultDict(String, DefaultDict, ()->DefaultDict(String,Int,0))

```

Note that in the last example, we need to use a function to create each new DefaultDict. If we forget, we will end up using the same DefaultDict for all default values:

julia> dd = DefaultDict(String, DefaultDict, DefaultDict(String,Int,0));

julia> dd["a"]
DefaultDict{String,Int64,Int64,Dict{K,V}}()

julia> dd["b"]["a"] = 1
1

julia> dd["a"]
["a"=>1]

Trie

An implementation of the Trie data structure. This is an associative structure, with String keys:

t=Trie{Int}()
t["Rob"]=42
t["Roger"]=24
haskey(t,"Rob") #true
get(t,"Rob",nothing) #42
keys(t) # "Rob", "Roger"

Constructors:

Trie(keys, values)                  # construct a Trie with the given keys and values
Trie(keys)                          # construct a Trie{Nothing} with the given keys and with values = nothing
Trie(kvs::AbstractVector{(K, V)})   # construct a Trie from the given vector of (key, value) pairs
Trie(kvs::Associative{K, V})        # construct a Trie from the given associative structure

This package also provides an iterator path(t::Trie, str) for looping over all the nodes encountered in searching for the given string str. This obviates much of the boilerplate code needed in writing many trie algorithms. For example, to test whether a trie contains any prefix of a given string, use:

seen_prefix(t::Trie, str) = any(v -> v.is_key, path(t, str))

Linked List

A list of sequentially linked nodes. This allows efficient insertion of nodes to the front of the list:

julia> l1 = nil()
nil()

julia> l2 = cons(1, l1)
list(1)

julia> l3 = list(2, 3)
list(2, 3)

julia> l4 = cat(l1, l2, l3)
list(1, 2, 3)

julia> l5 = map((x) -> x*2, l4)
list(2, 4, 6)

julia> for i in l5; print(i); end
246

Overview of Sorted Containers

Currently one sorted container is provided: SortedDict. SortedDict is similar to the built-in Julia type Dict with the additional feature that the keys are stored in sorted order and can be efficiently iterated in this order. SortedDict is a subtype of Associative. SortedDict is a parameterized type with three parameters, the key type K, the value type V, and the ordering type O.

SortedDict internally uses a 2-3 tree, which is a kind of balanced tree and is described in many elementary data structure textbook.

This container requires two functions to compare keys: a less-than and equals function. With the default ordering argument, the comparison functions are isless(a,b) and isequal(a,b) where a and b are keys. User-specified ordering functions are discussed below.

Tokens for Sorted Containers

The SortedDict type is accompanied by an auxiliary type called the token and is defined as type SDToken. A token is an item that stores the address of a single data item in the SortedDict and can be dereferenced in time O(1). This notion of token is similar to the concept of iterators used by C++ standard containers. Tokens can be explicitly advanced or regressed through the data in the sorted order; they are implicitly advanced or regressed via iteration loops defined below. A token may take two special values: the before-start value and the past-end value. These values act as lower and upper bounds on the actual data. The before-start token can be advanced, while the past-end token can be regressed. A dereferencing operation on either leads to an error.

A token has two parts: the first part refers to the container as a whole, and the second part refers to the particular item. The second part is called a semitoken. In some applications, one might need an auxiliary data structure that contains thousands of tokens addressing the same container. In this case, it may be more efficient to store semitokens rather than tokens and reconstruct the full tokens as needed. In the current implementation, semitokens are internally stored as integers. However, for the purpose of future compatibility, the user should not extract this internal representation; these integers do not have any direct interpretation in terms of the container.

Constructors for Sorted Containers

SortedDict(d)

Argument d is an ordinary Julia dict (or any associative type) used to initialize the container, e.g.:

c = SortedDict(Dict("New York" => 1788, "Illinois" => 1818))

In this example the key-type is deduced to be ASCIIString, while the value-type is Int.

SortedDict(d,o)

Argument d is an ordinary Julia dict (or any associative type) used to initialize the container and o is an optional ordering object used for ordering the keys. The default value for o is Forward.

SortedDict(Dict{K,V}(),o)

Construct an empty SortedDict by explicitly specifying the parameters of the type. Ordering argument o is optional and defaults to Forward.

Note that the code snippets in this section are based on the Julia version 0.4.0 Dict-constructor syntax. There are equivalent statements for 0.3.0

Complexity of Sorted Containers

In the list of functions below, the running time of the various operations is provided. In these running times, n denotes the current size (number of items) in the container at the time of the function call, and c denotes the time needed to compare two keys.

Navigating the Containers

m[k]

Argument m is a SortedDict and k is a key. In an expression, this retrieves the value associated with the key (or KeyError if none). On the left-hand side of an assignment, this assigns or reassigns the value associated with the key. (For assigning and reassigning, see also insert! below.) Time: O(c log n)

find(m,k)

Argument m is a SortedDict and argument k is a key. This function returns a token that refers to the item whose key is k, or past-end marker if k is absent. Time: O(c log n)

deref(i)

Argument i is a token. This returns the (key,value) pair pointed to by the token. Time: O(1)

deref_key(i)

Argument i is a token. This returns the key pointed to by the token. Time: O(1)

deref_value(i)

Argument i is a token. This returns the value pointed to by the token. Time: O(1)

startof(m)

Argument m is a SortedDict. This function returns the token of the first item according to the sorted order in the container. If the container is empty, it returns the past-end token. Time: O(log n)

endof(m)

Argument m is a SortedDict. This function returns the token of the last item according to the sorted order in the container. If the container is empty, it returns the before-start token. Time: O(log n)

first(m)

Argument m is a SortedDict. This function returns the first item (a (k,v) pair) according to the sorted order in the container. Thus, first(m) is equivalent to deref(startof(m)). It is an error to call this function on an empty container. Time: O(log n)

last(m)

Argument m is a SortedDict. This function returns the last item (a (k,v) pair) according to the sorted order in the container. Thus, last(m) is equivalent to deref(endof(m)). It is an error to call this function on an empty container. Time: O(log n)

pastendtoken(m)

Argument m is a SortedDict. This function returns the past-end token. Time: O(1)

beforestarttoken(m)

Argument m is a SortedDict. This function returns the before-start token. Time: O(1)

advance(i)

Argument i is a token. This function returns the token of the next entry in the container according to the sort order of the keys. After the last item, this routine returns the past-end token. It is an error to invoke this function if i is the past-end token. If i is the before-start token, then this routine returns the token of the first item in the sort order (i.e., the same token returned by the startof function). Time: O(log n)

regress(i)

Argument i is a token. This function returns the token of the previous entry in the container according to the sort order of the keys. If i indexes the first item, this routine returns the before-start token. It is an error to invoke this function if i is the before-start token. If i is the past-end token, then this routine returns the token of the last item in the sort order (i.e., the same token returned by the endof function). Time: O(log n)

searchsortedfirst(m,k)

Argument m is a SortedDict and k is an element of the key type. This routine returns the token of the first item in the container whose key is greater than or equal to k. If there is no such key, then the past-end token is returned. Time: O(c log n)

searchsortedlast(m,k)

Argument m is a SortedDict and k is an element of the key type. This routine returns the token of the last item in the container whose key is less than or equal to k. If there is no such key, then the before-start token is returned. Time: O(c log n)

searchsortedafter(m,k)

Argument m is a SortedDict and k is an element of the key type. This routine returns the token of the first item in the container whose key is greater than k. If there is no such key, then the past-end token is returned. Time: O(c log n)

Inserting & Deleting in Sorted Containers

empty!(m)

Argument m is a SortedDict. This empties the container. Time: O(1).

insert!(m,k,v)

Argument m is a SortedDict, k is a key and v is the corresponding value. This inserts the (k,v) pair into the container. If the key is already present, SortedDict overwrites the old value. The return value is a pair whose first entry is boolean and indicates whether the insertion was new (i.e., the key was not previously present) and the second entry is the token of the new entry. Time: O(c log n)

delete!(i)

Argument i is a token. This operation deletes the item addressed by i. It is an error to call this on an entry that has already been deleted or on the before-start or past-end tokens. After this operation is complete, i is an invalid token and cannot be used in any further operations. Time: O(log n)

delete!(m,k)

Argument m is a SortedDict and k is a key. This operation deletes the item whose key is k. It is a KeyError if k is not a key of an item in the container. After this operation is complete, any token addressing the deleted item is invalid. Time: O(c log n)

pop!(m,k)

Deletes the item with key k in SortedDict m and returns the value that was associated with k. A KeyError results if k is not in m. Time: O(c log n)

m[st]

If st is a semitoken (extracted from a token for SortedDict m via the semi function below), then m[st] refers to the value field of the (key,value) pair that the full token refers to. This expression may occur on either side of an assignment statement. Time: O(1)

Token Manipulation

semi(i)

Extracts a semitoken from a token. The semitoken is wrapper around an integer (in the current implementation). See the above discussion of semitokens. Time: O(1)

container(i)

Extracts the container from a token. See the above discussion. Time: O(1)

assemble(m,s)

Here, m is a sorted container and s is a semitoken; this function reassembles the complete token. In other words, if i is a valid token, then assemble(container(i), semi(i)) yields i. The validity of the token returned is not checked by this function. Time: O(1)

isless(i1,i2)

Here, i1 and i2 are tokens for the same container; this function determines whether the (k,v) pair addressed by i1 precedes that of i2 in the sorted order. An error is thrown if i1 and i2 refer to different containers. This function compares the tokens by determining their relative position within the tree and without dereferencing them. It is mostly equivalent to lt(o, deref_key(i1), deref_key(i2)) except in the case that either i1 or i2 is the before-start or past-end token, in which case the latter will fail. Which one is more efficient depends on the time-complexity of comparing two keys. Time: O(log n)

isequal(i1,i2)

Here, i1 and i2 are tokens for the same container; this function determines whether they address the same item. An error is thrown if i1 and i2 refer to different containers. Time: O(l)

status(i1)

This function returns 0 if the token i1 is invalid (e.g., refers to a deleted item), 1 if the token is valid and points to data, 2 if the token is the before-start token and 3 if it is the past-end token. Time: O(1)

Iteration Over Sorted Containers

As is standard in Julia, iteration over the containers is implemented via calls to three functions, start, next and done. It is usual practice, however, to call these functions implicitly with a for-loop rather than explicitly, so they are presented here in for-loop notation. Internally, all of these iterations are implemented with tokens that are advanced via the advance operation and start, next and done functions. Each iteration of these loops requires O(log n) operations to advance the token.

The following loops over the entire container m, where m is a SortedDict:

for (k,v) in m
   < body >
end

In this loop, (k,v) takes on successive (key,value) pairs according to the sort order of the key.

There are two ways to iterate over a subrange of a container. The first is the inclusive iteration:

for (k,v) in i1 : i2
  < body >
end

Here, i1 and i2 are tokens that refer to the same container. It is acceptable for i1 to be the past-end token or i2 to be the before-start token (in these cases, the body is not executed). If isless(i2,i1) then the body is not executed.

One can also define a loop that excludes the final item:

for (k,v) in excludelast(i1,i2)
  < body >
end

In this case, all the data addressed by tokens from i1 up to but excluding i2 are executed. The body is not executed at all if !isless(i1,i2). In this setting, either or both can be the past-end token, and i2 can be the before-start token.

Both the excludelast and colon operators return objects that can be saved and used later for iteration. At the time of construction of these object, it is checked that the start and end tokens refer to the same container. The validity of the tokens is not checked until the loop initiates.

One can iterate over just keys or just values:

for k in keys(m)
   < body >
end

for v in values(m)
   < body >
end

The arguments to keys and values may also be ranges of the form i1:i2 or excludelast(i1,i2).

Finally, one can retrieve tokens during any of these iterations:

for (t,(k,v)) in tokens(m)
    < body >
end

for (t,k) in tokens(keys(m))
    < body >
end

for (t,v) in tokens(values(m))
    < body >
end

In each successive iteration, t is a token referring to the current (k,v) pair. In place of m in the above three snippets, one could also use i1:i2 or excludelast(i1,i2).

Note that it is acceptable for the loop body in the above code snippets to invoke delete!(t). This is because the for-loop internal state variable is already advanced to the next token at the beginning of the body, so t is not necessarily referred to in the loop body (unless the user refers to it).

Other Functions

isempty(m)

Returns true if the container is empty (no items). Time: O(1)

length(m)

Returns the length, i.e., number of items, in the container. Time: O(1)

in(p,m)

Returns true if p is in m, where m is a SortedDict and p is a (key,value) pair. Time: O(c log n)

eltype(m)

Returns the (key,value) type for SortedDict. Time: O(1)

orderobject(m)

Returns the order object used to construct the container. Time: O(1)

haskey(m,k)

Returns true if k is present for SortedDict m. Time: O(c log n)

get(m,k,v)

Returns the value associated with key k where m is a SortedDict, or else returns v if k is not in m. Time: O(c log n)

get!(m,k,v)

Returns the value associated with key k where m is a SortedDict, or else returns v if k is not in m, and in the latter case, inserts (k,v) into m. Time: O(c log n)

getkey(m,k,defaultk)

Returns key k where m is a SortedDict, if k is in m else it returns defaultk. If the container uses in its ordering an eq method different from isequal (e.g., case-insensitive ASCII strings illustrated below), then the return value is the actual key stored in the SortedDict that is equivalent to k according to the eq method, which might not be equal to k. Similarly, if the user performs an implicit conversion as part of the call (e.g., the container has keys that are floats, but the k argument to getkey is an Int), then the returned key is the actual stored key rather than k. Time: O(c log n)

isequal(m1,m2)

Checks if two containers are equal in the sense that they contain the same items; the keys are compared using the eq method, while the values are compared with the isequal function. Note that isequal in this sense does not imply any correspondence between semitokens for items in m1 with those for m2. If the equality-testing method associated with the keys and values implies hash-equivalence, then isequal of the entire containers implies hash-equivalence of the containers. Time: O(cn + n log n)

packcopy(m)

This returns a copy of m in which the data is packed. When deletions take place, the previously allocated memory is not returned. This function can be used to reclaim memory after many deletions. Time: O(cn log n)

deepcopy(m)

This returns a copy of m in which the data is deep-copied, i.e., the keys and values are replicated if they are mutable types. A semitoken for the original m can be composed with the deep-copy output to make a valid token for the copy because this operation preserves the relative positions of the data in memory. Time O(maxn), where maxn denotes the maximum size that m has attained in the past.

packdeepcopy(m)

This returns a packed copy of m in which the keys and values are deep-copied. This function can be used to reclaim memory after many deletions. Time: O(cn log n)

merge(s, t...)

This returns a SortedDict that results from merging SortedDicts s, t, etc., which all must have the same key-value-ordering types. In the case of keys duplicated among the arguments, the rightmost argument that owns the key gets its value stored. Time: O(cN log N), where N is the total size of all the arguments.

merge!(s, t...)

This updates s by merging SortedDicts t, etc. into s. These must all must have the same key-value types. In the case of keys duplicated among the arguments, the rightmost argument that owns the key gets its value stored. Time: O(cN log N), where N is the total size of all the arguments.

Ordering of keys

As mentioned earlier, the default ordering of keys uses isless and isequal functions. If the default ordering is used, it is a requirement of the container that isequal(a,b) is true if and only if !isless(a,b) and !isless(b,a) are both true. This relationship between isequal and isless holds for common built-in types, but it may not hold for all types, especially user-defined types. If it does not hold for a certain type, then a custom ordering argument must be defined as discussed in the next few paragraphs.

The name for the default ordering (i.e., using isless and isequal) is Forward. Another possible choice is Reverse, which reverses the usual sorted order. This name must be imported import Base.Reverse if it is used.

As an example of a custom ordering, suppose the keys are of type ASCIIString, and the user wishes to order the keys ignoring case: APPLE, berry and Cherry would appear in that order, and APPLE and aPPlE would be indistinguishable in this ordering.

The simplest approach is to define an ordering object of the form Lt(my_isless), where Lt is a built-in type (see ordering.jl) and my_isless is the user's comparison function. In the above example, the ordering object would be:

Lt((x,y) -> isless(lowercase(x),lowercase(y)))

The ordering object is the second argument to the SortedDict constructor (see above for constructor syntax).

This approach suffers from a performance hit (10%-50% depending on the container) because the compiler cannot inline or compute the correct dispatch for the function in parentheses, so the dispatch takes place at run-time. A more complicated but higher-performance method to implement a custom ordering is as follows. First, the user creates a singleton type that is a subtype of Ordering as follows:

immutable CaseInsensitive <: Ordering
end

Next, the user defines a method named lt for less-than in this ordering:

lt(::CaseInsensitive, a, b) = isless(lowercase(a), lowercase(b))

The first argument to lt is an object of the CaseInsensitive type (there is only one such object since it is a singleton type). The container also needs an equal-to function; the default is:

eq(o::Ordering, a, b) = !lt(o, a, b) && !lt(o, b, a)

For a further slight performance boost, the user can also customize this function with a more efficient implementation. In the above example, an appropriate customization would be:

eq(::CaseInsensitive, a, b) = isequal(lowercase(a), lowercase(b))

Finally, the user specifies the unique element of CaseInsensitive, namely the object CaseInsensitive(), as the ordering object to the SortedDict constructor.

For the above code to work, the module must make the following declarations, typically near the beginning:

import Base.Ordering
import Base.lt
import DataStructures.eq

Cautionary note on mutable keys

As with ordinary Dicts, keys for SortedDict can be either mutable or immutable. In the case of mutable keys, it is important that the keys not be mutated once they are in the SortedDict else the indexing structure will be corrupted. (The same restriction applies to Dict.) For example, suppose a SortedDict m is defined in which the keys are of type Array{Int,1}. (For this to be possible, the user must provide an isless function or order object for Array{Int,1} since none is built into Julia.) Suppose the values of m are of type Int. Then the following sequence of statements leave m in a corrupted state:

a = [1,2,3]
m[a] = 19
b = [4,5,6]
m[b] = 20
a[1] = 7

Performance of Sorted Containers

Timing tests indicate that the code is about 1.5 to 2 times slower than equivalent C++ code that uses the C++ standard library container map. and compiled with /O2 optimization. These tests were conducted on a Windows 8.1 64-bit machine with the Microsoft Visual Studio 12.0 compiler.

There is a minor performance issue as follows: the container may hold onto a small number of keys and values even after the data records containing those keys and values have been deleted. This may cause a memory drain in the case of large keys and values. It may also lead to a delay in the invocation of finalizers. All keys and values are released completely by the empty! function.