#include <CGAL/Multiset.h>

Definition

An instance s of the parametrized data type Multiset is a multi-set of elements of type Type, represented as a red-black tree (see [[3] Chapter 13 for an excellent introduction to red-black trees). The main difference between Multiset and the STL std::multiset is that the latter uses a less-than functor with a Boolean return type, while our Multiset class is parameterized by a comparison functor Compare that returns the three-valued Comparison_result (namely it returns either SMALLER, EQUAL, or LARGER). It is thus possible to maintain the underlying red-black tree with less invocations of the comparison functor. This leads to a speedup of about 5% even if we maintain a set of integers. When each comparison of two elements of type Type is an expensive operation (for example, when they are geometric entities represented using exact arithmetic), the usage of a three-valued comparison functor can lead to considerable decrease in the running times.

Moreover, Multiset allows the insertion of an element into the set given its exact position, and not just using an insertion hint, as done by std::multiset. This can further reduce the running times, as additional comparison operations can be avoided.

In addition, the Multiset guarantees that the order of elements sent to the comparison functor is fixed. For example, if we insert a new element x into the set (or erase an element from the set), then we always invoke Compare() (x, y) (and never Compare() (y, x)), where y is an element already stored in the set. This behavior, not supported by std::multiset, is sometimes crucial for designing more efficient comparison predicates.

Multiset also allows for look-up of keys whose type may differ from Type, as long as users supply a comparison functor CompareKey, where CompareKey() (key, y) returns the three-valued Comparison_result (key is the look-up key and y is an element of type Type). Indeed, it is very convenient to look-up equivalent objects in the set given just by their key. We note however that it is also possible to use a key of type Type and to employ the default Compare functor for the look-up, as done when using the std::multiset class.

Warning: Finally, Multiset introduces the catenate() and split() functions. The first function operates on s and accepts a second set s2, such that the maximum element in s is not greater than the minimal element in s2, and concatenates s2 to s. The second function splits s into two sets, one containing all the elements that are less than a given key, and the other contains all elements greater than (or equal to) this key.

Template Parameters

Type	the type of the stored elements.
Compare	the comparison-functor type. This type should provide the following operator for comparing two `Type` elements, namely: `Comparison_result operator() (const Type& t1, const Type& t2) const;` The `CGAL::Compare<Type>` functor is used by default. In this case, `Type` must support an equality operator (`operator==`) and a less-than operator (`operator<`).
Allocator	the allocator type. `CGAL_ALLOCATOR` is used by default.

Implementation

Multiset uses a proprietary implementation of a red-black tree data-structure. The red-black tree invariants guarantee that the height of a tree containing \( n\) elements is \( O(\log{n})\) (more precisely, it is bounded by \( 2 \log_{2}{n}\)). As a consequence, all methods that accept an element and need to locate it in the tree (namely insert(x), erase(x), find(x), count(x), lower_bound(x) , upper_bound(x), find_lower(x) and equal_range(x)) take \( O(\log{n})\) time and perform \( O(\log{n})\) comparison operations.

On the other hand, the set operations that accept a position iterator (namely insert_before(pos, x), insert_after(pos, x) and erase(pos)) are much more efficient as they can be performed at a constant amortized cost (see [4] and [6] for more details). More important, these set operations require no comparison operations. Therefore, it is highly recommended to maintain the set via iterators to the stored elements, whenever possible. The function insert(pos, x) is safer to use, but it takes amortized \( O(\min\{d,\log{n}\})\) time, where \( d\) is the distance between the given position and the true position of x. In addition, it always performs at least two comparison operations.

The catenate() and split() functions are also very efficient, and can be performed in \( O(\log{n})\) time, where \( n\) is the total number of elements in the sets, and without performing any comparison operations (see [6] for the details). Note however that the size of two sets resulting from a split operation is initially unknown, as it is impossible to compute it in less than linear time. Thus, the first invocation of size() on such a set takes linear time, and not constant time.

The design is derived from the STL multiset class-template (see, e.g, [5]), where the main differences between the two classes are highlighted in the class definition above.

Types
In compliance with STL, the types `value_type` and `key_type` (both equivalent to `Type`), `reference` and `const_reference` (reference to a value-type), `key_compare` and `value_compare` (both equivalent to `Compare`), `size_type` and `difference_type` are defined as well.
typedef unspecified_type	iterator

typedef unspecified_type	const_iterator
	bi-directional iterators for the elements stored in the set.

typedef unspecified_type	reverse_iterator

typedef unspecified_type	const_reverse_iterator
	reverse bi-directional iterators for the elements stored in the set.

Creation
	Multiset ()
	creates an an empty set `s` that uses a default comparison functor.

	Multiset (const Compare &comp)
	creates an an empty set `s` that uses the given comparison functor `comp`.

template<class InputIterator >
	Multiset (InputIterator first, InputIterator last, const Compare &comp=Compare())
	creates a set `s` containing all elements in the range `[first, last)`, that uses the comparison functor `comp`.

	Multiset (const Multiset< Type, Compare, Allocator > &other)
	copy constructor.

const Multiset< Type, Compare, Allocator > &	operator= (const Multiset< Type, Compare, Allocator > &other)
	assignment operator.

void	swap (Multiset< Type, Compare, Allocator > &other)
	swaps the contents of `s` with those of the other set.

Access Member Functions
Compare	key_comp () const
	the comparison functor used.

Compare	value_comp () const
	the comparison functor used (same as above). More...

bool	empty ()
	returns `true` if the set is empty, `false` otherwise.

size_t	size ()
	returns the number of elements stored in the set.

size_t	max_size ()
	returns the maximal number of elements the set can store (same as `size()`).

iterator	begin ()
	returns an iterator pointing to the first element stored in the set (a `const` version is also available).

iterator	end ()
	returns an iterator pointing beyond the last element stored in the set (a `const` version is also available).

reverse_iterator	rbegin ()
	returns a reverse iterator pointing beyond the last element stored in the set (a `const` version is also available).

reverse_iterator	rend ()
	returns a reverse iterator pointing to the first element stored in the set (a `const` version is also available).

Comparison Operations
bool	operator== (const Multiset< Type, Compare, Allocator > &other) const
	returns `true` if the sequences of elements in the two sets are pairwise equal (using the comparison functor).

bool	operator< (const Multiset< Type, Compare, Allocator > &other) const
	returns `true` if the element sequence in `s` is lexicographically smaller than the element sequence of `other`.

Insertion Methods
iterator	insert (const Type &x)
	inserts the element `x` into the set and returns an iterator pointing to the newly inserted element.

template<class InputIterator >
void	insert (InputIterator first, InputIterator last)
	inserts all elements in the range `[first, last)` into the set.

iterator	insert (iterator position, const Type &x)
	inserts the element `x` with a given iterator used as a hint for the position of the new element. More...

iterator	insert_before (iterator position, const Type &x)
	inserts the element `x` as the predecessor of the element at the given position. More...

iterator	insert_after (iterator position, const Type &x)
	inserts the element `x` as the successor of the element at the given position. More...

Removal Methods
size_t	erase (const Type &x)
	erases all elements equivalent to `x` from the set and returns the number of erased elements.

void	erase (iterator position)
	erases the element pointed by `position`.

void	clear ()
	clears the set (erases all stored elements).

Look-up Methods
All methods listed in this section can also accept a `Type` element as a look-up key. In this case, it is not necessary to supply a `CompareKey` functor, as the `Compare` functor will be used by default.
template<class Key , class CompareKey >
iterator	find (const Key &key, const CompareKey &comp_key)
	searches for the an element equivalent to `key` in the set. More...

template<class Key , class CompareKey >
size_t	count (const Key &key, const CompareKey &comp_key) const
	returns the number of elements equivalent to `key` in the set.

template<class Key , class CompareKey >
iterator	lower_bound (const Key &key, const CompareKey &comp_key)
	returns an iterator pointing to the first element in the set that is not less than `key`. More...

template<class Key , class CompareKey >
iterator	upper_bound (const Key &key, const CompareKey &comp_key)
	returns an iterator pointing to the first element in the set that is greater than `key`. More...

template<class Key , class CompareKey >
std::pair< iterator, iterator >	equal_range (const Key &key, const CompareKey &comp_key)
	returns the range of set elements equivalent to the given key, namely `(lower_bound(key), upper_bound(key))` (a `const` version is also available).

template<class Key , class CompareKey >
std::pair< iterator, bool >	find_lower (const Key &key, const CompareKey &comp_key)
	returns a pair comprised of `lower_bound(key)` and a Boolean flag indicating whether this iterator points to an element equivalent to the given key (a `const` version is also available).

Special Operations
void	replace (iterator position, const Type &x)
	replaces the element stored at the given position with `x`. More...

void	swap (iterator pos1, iterator pos2)
	swaps places between the two elements given by `pos1` and `pos2`. More...

void	catenate (Self &s_prime)
	concatenates all elements in `s_prime` into `s` and clears `s_prime`. More...

template<class Key , class CompareKey >
void	split (Key key, CompareKey comp_key, Self &s_prime)
	splits `s` such that it contains all elements that are less than the given `key` and such that `s_prime` contains all other elements. More...

void	split (iterator position, Self &s_prime)
	splits `s` such that it contains all set elements in the range `[begin, position)` and such that `s_prime` contains all elements in the range `[position, end())`. More...