\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - STL Extensions for CGAL

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CGAL::Multiset< Type, Compare, Allocator > Class Template Reference
STL Extensions for CGAL Reference

`#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.

Assertions

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`).

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...

## Member Function Documentation

template<typename Type , typename Compare , typename Allocator >

void CGAL::Multiset< Type, Compare, Allocator >::catenate | ( | Self & | s_prime) |  |

concatenates all elements in `s_prime` into `s` and clears `s_prime`.

All iterators to `s` and to `s_prime` remain valid.

Precondition
  The maximal element in `s` is not greater than the minimal element in `s_prime`.

template<typename Type , typename Compare , typename Allocator >
template<class Key , class CompareKey >

iterator CGAL::Multiset< Type, Compare, Allocator >::find | ( | const Key & | key,
|  | const CompareKey & | comp_key
| ) |  |

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).

template<typename Type , typename Compare , typename Allocator >

iterator CGAL::Multiset< Type, Compare, Allocator >::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.

It Returns an iterator pointing to the newly inserted element.

template<typename Type , typename Compare , typename Allocator >

iterator CGAL::Multiset< Type, Compare, Allocator >::insert_after | ( | iterator | position,
|  | const Type & | x
| ) |  |

inserts the element `x` as the successor of the element at the given position.

Precondition
  The operation does not violate the set order - that is, `x` is not less than the element pointed by `position` and not greater than its current successor.

template<typename Type , typename Compare , typename Allocator >

iterator CGAL::Multiset< Type, Compare, Allocator >::insert_before | ( | iterator | position,
|  | const Type & | x
| ) |  |

inserts the element `x` as the predecessor of the element at the given position.

Precondition
  The operation does not violate the set order - that is, `x` is not greater than the element pointed by `position` and not less than its current predecessor.

template<typename Type , typename Compare , typename Allocator >
template<class Key , class CompareKey >

iterator CGAL::Multiset< Type, Compare, Allocator >::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`.

If all set elements are less than `key`, `end()` is returned (a `const` version is also available).

template<typename Type , typename Compare , typename Allocator >

void CGAL::Multiset< Type, Compare, Allocator >::replace | ( | iterator | position,
|  | const Type & | x
| ) |  |

replaces the element stored at the given position with `x`.

Precondition
  The operation does not violate the set order - that is, `x` is not less that `position`'s predecessor and not greater than its successor.

template<typename Type , typename Compare , typename Allocator >
template<class Key , class CompareKey >

void CGAL::Multiset< Type, Compare, Allocator >::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.

Precondition
  `s_prime` is initially empty.

template<typename Type , typename Compare , typename Allocator >

void CGAL::Multiset< Type, Compare, Allocator >::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())`.

Precondition
  `s_prime` is initially empty.

template<typename Type , typename Compare , typename Allocator >

void CGAL::Multiset< Type, Compare, Allocator >::swap | ( | iterator | pos1,
|  | iterator | pos2
| ) |  |

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

Precondition
  The operation does not violate the set order - that is, `pos1` and `pos2` store equivalent elements.

template<typename Type , typename Compare , typename Allocator >
template<class Key , class CompareKey >

iterator CGAL::Multiset< Type, Compare, Allocator >::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`.

If no set element is greater than `key`, `end()` is returned (a `const` version is also available).

template<typename Type , typename Compare , typename Allocator >

Compare CGAL::Multiset< Type, Compare, Allocator >::value_comp | ( | ) |  | const

the comparison functor used (same as above).

Both functions have a non-const version that return a reference to the comparison functor.