An instance s of the parametrized data type Multiset<Type,Compare,Allocator> is a multiset of elements of type Type, represented as a redblack tree (see [CLRS01, Chapter 13] for an excellent introduction to redblack trees). The main difference between Multiset<Type,Compare,Allocator> and STL's multiset is that the latter uses a lessthan functor with a Boolean return type, while our Multiset<Type,Compare,Allocator> class is parameterized by a comparison functor Compare that returns the threevalued Comparison_result (namely it returns either SMALLER, EQUAL, or LARGER). It is thus possible to maintain the underlying redblack 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 threevalued comparison functor can lead to considerable decrease in the running times.
Moreover, Multiset<Type,Compare,Allocator> 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<Type,Compare,Allocator> 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<Type,Compare,Allocator> also allows for lookup of keys whose type may differ from Type, as long as users supply a comparison functor CompareKey, where CompareKey() (key, y) returns the threevalued Comparison_result (key is the lookup key and y is an element of type Type). Indeed, it is very convenient to lookup 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 lookup, as done when using the std::multiset class.
advanced 
Finally, Multiset<Type,Compare,Allocator> introduces the catenate() and split() functions. The first function operates on s and accepts a second set s', such that the maximum element in s is not greater than the minimal element in s', and concatenates s' 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.
advanced 
The Multiset classtemplate has three parameters:
#include <CGAL/Multiset.h>
The assertion and precondition flags for the Multiset class use MULTISET in their names (i.e., CGAL_MULTISET_NO_ASSERTIONS and CGAL_MULTISET_NO_PRECONDITIONS).
In compliance with STL, the types value_type and key_type (both equivalent to Type), reference and const_reference (reference to a valuetype), key_compare and value_compare (both equivalent to Compare), size_type and difference_type are defined as well.
 
creates an an empty set s that uses a default comparison
functor.
 
 
creates an an empty set s that uses the given comparison
functor comp.
 
 
 
creates a set s containing all elements in the range
[first, last), that uses the comparison
functor comp.
 
 
copy constructor.


 assignment operator. 

 swaps the contents of s with those of the other set. 

 erases all elements equivalent to x from the set and returns the number of erased elements. 

 erases the element pointed by position. 

 clears the set (erases all stored elements). 
All methods listed in this section can also accept a Type element as a lookup key. In this case, it is not necessary to supply a CompareKey functor, as the Compare functor will be used by default.
 

 
searches for the an element equivalent to key in the set. If the set contains objects equivalent to key, it returns an iterator pointing to the first one. Otherwise, end() is returned (a const version is also available).  
 

 
returns the number of elements equivalent to key in the set.  
 

 
returns an iterator pointing to the first element in the set that is not less than key. If all set elements are less than key, end() is returned (a const version is also available).  
 

 
returns an iterator pointing to the first element in the set that is greater than key. If no set element is greater than key, end() is returned (a const version is also available).  
 

 
returns the range of set elements equivalent to the given key, namely (lower_bound(key), upper_bound(key)) (a const version is also available).  
 

 
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). 
advanced 

 
replaces the element stored at the given position with x.
 

 
swaps places between the two elements given by pos1 and pos2.
 


concatenates all elements in s_prime into s and clears
s_prime.
All iterators to s and to s_prime remain valid.
 
 

 
splits s such that it contains all elements that are less than
the given key and such that s_prime contains all other elements.
 

 
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()).

advanced 
Multiset uses a proprietary implementation of a redblack tree datastructure. The redblack 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$$n) comparison operations.
On the other hand, the set operations that accept a position iterator (namely insert_before(pos, x), insert_after(pos, x) and erase(pos)) are much more efficient as they can be performed at a constant amortized cost (see [GS78] and [Tar83] for more details). More important, these set operations require no comparison operations. Therefore, it is highly recommended to maintain the set via iterators to the stored elements, whenever possible. The function insert(pos, x) is safer to use, but it takes amortized $$O(min$${d,log$$n}) time, where $$d is the distance between the given position and the true position of x. In addition, it always performs at least two comparison operations.
advanced 
The catenate() and split() functions are also very efficient, and can be performed in $$O(log$$n) time, where $$n is the total number of elements in the sets, and without performing any comparison operations (see [Tar83] 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.
advanced 
The design is derived from the STL multiset classtemplate (see, e.g, [MS96]), where the main differences between the two classes are highlighted in the class definition above.