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

CGAL 4.8 - 2D Segment Delaunay Graphs

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CGAL::Segment_Delaunay_graph_hierarchy_2< Gt, STag, DS > Class Template Reference
2D Segment Delaunay Graphs Reference

`#include <CGAL/Segment_Delaunay_graph_hierarchy_2.h>`

## Inherits from

CGAL::Segment_Delaunay_graph_2< Gt, DS >.

## Definition

We provide an alternative to the class `Segment_Delaunay_graph_2<Gt,DS>` for the incremental construction of the segment Delaunay graph.

The `Segment_Delaunay_graph_hierarchy_2` class maintains a hierarchy of Delaunay graphs. There are two possibilities as to how this hierarchy is constructed.

In the first case the bottom-most level of the hierarchy contains the full segment Delaunay graph. The upper levels of the hierarchy contain only points that are either point sites or endpoints of segment sites in the bottom-most Delaunay graph. A point that is in level \( i\) (either as an individdual point or as the endpoint of a segment), is inserted in level \( i+1\) with probability \( 1/\alpha\) where \( \alpha>1\) is some constant. In the second case the upper levels of the hierarchy contains not only points but also segments. A site that is in level \( i\), is in level \( i+1\) with probability \( 1/\beta\) where \( \beta > 1\) is some constant.

The difference between the `Segment_Delaunay_graph_2<Gt,DS>` class and the `Segment_Delaunay_graph_hierarchy_2` class (both versions of it) is on how the nearest neighbor location is done. Given a point \( p\) the location is done as follows: at the top most level we find the nearest neighbor of \( p\) as in the `Segment_Delaunay_graph_2<Gt,DS>` class. At every subsequent level \( i\) we use the nearest neighbor found at level \( i+1\) to find the nearest neighbor at level \( i\). This is a variant of the corresponding hierarchy for points found in [3]. The details are described in [4].

The class has three template parameters. The first and third have essentially the same semantics as in the `Segment_Delaunay_graph_2<Gt,DS>` class.

Template Parameters

  Gt | must be a model of the `SegmentDelaunayGraphTraits_2` concept.
  STag | The second template parameter controls whether or not segments are added in the upper levels of the hierarchy. It's possible values are `Tag_true` and `Tag_false`. If it is set to `Tag_true`, segments are also inserted in the upper levels of the hierarchy. The value `Tag_false` indicates that only points are to be inserted in the upper levels of the hierarchy. The default value for the second template parameter is `Tag_false`.
  DS | must be a model of the `SegmentDelaunayGraphDataStructure_2` concept. However, the vertex base class that is to be used in the segment Delaunay graph data structure must be a model of the `SegmentDelaunayGraphHierarchyVertexBase_2` concept. The third template parameter defaults to `Triangulation_data_structure_2< Segment_Delaunay_graph_hierarchy_vertex_base_2< Segment_Delaunay_graph_vertex_base_2<Gt> >, Triangulation_face_base_2<Gt> >`.

The `Segment_Delaunay_graph_hierarchy_2` class derives publicly from the `Segment_Delaunay_graph_2<Gt,DS>` class. The interface is the same with its base class. In the sequel only additional types and methods defined are documented.

Is Model Of:

  `DefaultConstructible`

  `CopyConstructible`

  `Assignable`

See Also
  `SegmentDelaunayGraphDataStructure_2`
  `SegmentDelaunayGraphTraits_2`
  `SegmentDelaunayGraphHierarchyVertexBase_2`
  `CGAL::Segment_Delaunay_graph_2<Gt,DS>`
  `CGAL::Triangulation_data_structure_2<Vb,Fb>`
  `CGAL::Segment_Delaunay_graph_traits_2<K,MTag>`
  `CGAL::Segment_Delaunay_graph_traits_without_intersections_2<K,MTag>`
  `CGAL::Segment_Delaunay_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM>`
  `CGAL::Segment_Delaunay_graph_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM>`
  `CGAL::Segment_Delaunay_graph_hierarchy_vertex_base_2<Vbb>`

Examples:
  Segment_Delaunay_graph_2/sdg-filtered-traits.cpp, and Segment_Delaunay_graph_2/sdg-info-set.cpp.

## Related Functions
(Note that these are not member functions.)
std::ostream & | operator<< (std::ostream &os, Segment_Delaunay_graph_hierarchy_2< Gt, STag, DS > svdh)
| Writes the current state of the segment Delaunay graph hierarchy to an output stream. More...
std::istream & | operator>> (std::istream &is, Segment_Delaunay_graph_hierarchy_2< Gt, STag, DS > svdh)
| Reads the state of the segment Delaunay graph hierarchy from an input stream.

## Types
`Segment_Delaunay_graph_hierarchy_2` introduces the following types in addition to those introduced by its base class `Segment_Delaunay_graph_2<Gt,DS>`.
typedef STag | Segments_in_hierarchy_tag
| A type for the `STag` template parameter.
typedef CGAL::Segment_Delaunay_graph_2 < Gt, DS > | Base
| A type for the base class.

## Creation
In addition to the default and copy constructors, the following constructors are defined:
| Segment_Delaunay_graph_hierarchy_2 (Gt gt=Gt())
| Creates a hierarchy of segment Delaunay graphs using `gt` as geometric traits.
template<class Input_iterator >
| Segment_Delaunay_graph_hierarchy_2 (Input_iterator first, Input_iterator beyond, Gt gt=Gt())
| Creates a segment Delaunay graph hierarchy using `gt` as geometric traits and inserts all sites in the range [`first`, `beyond`). More...

## Additional Inherited Members
Public Types inherited from CGAL::Segment_Delaunay_graph_2< Gt, DS >
typedef Gt | Geom_traits
| A type for the geometric traits.
typedef DS | Data_structure
| A type for the underlying data structure.
typedef Data_structure | Triangulation_data_structure
| This type has been added so that the `Segment_Delaunay_graph_2` class is a model of the `DelaunayGraph_2` concept.
typedef DS::size_type | size_type
| Size type (an unsigned integral type)
typedef Gt::Point_2 | Point_2
| A type for the point defined in the geometric traits.
typedef Gt::Site_2 | Site_2
| A type for the segment Delaunay graph site, defined in the geometric traits.
typedef unspecified_type | Point_container
| A type for the container of points.
typedef Point_container::iterator | Point_handle
| A handle for points in the point container.
typedef DS::Edge | Edge
| The edge type. More...
typedef DS::Vertex | Vertex
| A type for a vertex.
typedef DS::Face | Face
| A type for a face.
typedef DS::Vertex_handle | Vertex_handle
| A type for a handle to a vertex.
typedef DS::Face_handle | Face_handle
| A type for a handle to a face.
typedef DS::Vertex_circulator | Vertex_circulator
| A type for a circulator over vertices incident to a given vertex.
typedef DS::Face_circulator | Face_circulator
| A type for a circulator over faces incident to a given vertex.
typedef DS::Edge_circulator | Edge_circulator
| A type for a circulator over edges incident to a given vertex.
typedef DS::Vertex_iterator | All_vertices_iterator
| A type for an iterator over all vertices.
typedef DS::Face_iterator | All_faces_iterator
| A type for an iterator over all faces.
typedef DS::Edge_iterator | All_edges_iterator
| A type for an iterator over all edges.
typedef unspecified_type | Finite_vertices_iterator
| A type for an iterator over finite vertices.
typedef unspecified_type | Finite_faces_iterator
| A type for an iterator over finite faces.
typedef unspecified_type | Finite_edges_iterator
| A type for an iterator over finite edges.
typedef unspecified_type | Input_sites_iterator
| A type for a bidirectional iterator over all input sites.
typedef unspecified_type | Output_sites_iterator
| A type for a bidirectional iterator over all output sites (the sites in the Delaunay graph).
Public Member Functions inherited from CGAL::Segment_Delaunay_graph_2< Gt, DS >
Input_sites_iterator | input_sites_begin ()
| Starts at an arbitrary input site.
Input_sites_iterator | input_sites_end ()
| Past-the-end iterator.
Output_sites_iterator | output_sites_begin ()
| Starts at an arbitrary output site.
Output_sites_iterator | output_sites_end ()
| Past-the-end iterator.
| Segment_Delaunay_graph_2 (Gt gt=Gt())
| Creates the segment Delaunay graph using `gt` as geometric traits.
template<class Input_iterator >
| Segment_Delaunay_graph_2 (Input_iterator first, Input_iterator beyond, Gt gt=Gt())
| Creates the segment Delaunay graph using `gt` as geometric traits and inserts all sites in the range [`first`, `beyond`). More...
Geom_traits | geom_traits ()
| Returns a reference to the segment Delaunay graph traits object.
int | dimension ()
| Returns the dimension of the segment Delaunay graph. More...
size_type | number_of_vertices ()
| Returns the number of finite vertices of the segment Delaunay graph.
size_type | number_of_faces ()
| Returns the number of faces (both finite and infinite) of the segment Delaunay graph.
size_type | number_of_input_sites ()
| Return the number of input sites.
size_type | number_of_output_sites ()
| Return the number of output sites. More...
Face_handle | infinite_face ()
| Returns a face incident to the `infinite_vertex`.
Vertex_handle | infinite_vertex ()
| Returns the `infinite_vertex`.
Vertex_handle | finite_vertex ()
| Returns a vertex distinct from the `infinite_vertex`. More...
Data_structure | data_structure ()
| Returns a reference to the segment Delaunay graph data structure object.
Data_structure | tds ()
| Same as `data_structure()`. More...
Point_container | point_container ()
| Returns a reference to the point container object.
Finite_vertices_iterator | finite_vertices_begin ()
| Starts at an arbitrary finite vertex.
Finite_vertices_iterator | finite_vertices_end ()
| Past-the-end iterator.
Finite_edges_iterator | finite_edges_begin ()
| Starts at an arbitrary finite edge.
Finite_edges_iterator | finite_edges_end ()
| Past-the-end iterator.
Finite_faces_iterator | finite_faces_begin ()
| Starts at an arbitrary finite face.
Finite_faces_iterator | finite_faces_end () const
| Past-the-end iterator.
All_vertices_iterator | all_vertices_begin ()
| Starts at an arbitrary vertex.
All_vertices_iterator | all_vertices_end ()
| Past-the-end iterator.
All_edges_iterator | all_edges_begin ()
| Starts at an arbitrary edge.
All_edges_iterator | all_edges_end ()
| Past-the-end iterator.
All_faces_iterator | all_faces_begin ()
| Starts at an arbitrary face.
All_faces_iterator | all_faces_end ()
| Past-the-end iterator.
Face_circulator | incident_faces (Vertex_handle v)
| Starts at an arbitrary face incident to `v`.
Face_circulator | incident_faces (Vertex_handle v, Face_handle f)
| Starts at face `f`. More...
Edge_circulator | incident_edges (Vertex_handle v)
| Starts at an arbitrary edge incident to `v`.
Edge_circulator | incident_edges (Vertex_handle v, Face_handle f)
| Starts at the first edge of `f` incident to `v`, in counterclockwise order around `v`. More...
Vertex_circulator | incident_vertices (Vertex_handle v)
| Starts at an arbitrary vertex incident to `v`.
Vertex_circulator | incident_vertices (Vertex_handle v, Face_handle f)
| Starts at the first vertex of `f` adjacent to `v` in counterclockwise order around `v`. More...
bool | is_infinite (Vertex_handle v) const
| `true`, iff `v` is the `infinite_vertex`.
bool | is_infinite (Face_handle f) const
| `true`, iff face `f` is infinite.
bool | is_infinite (Face_handle f, int i) const
| `true`, iff edge `(f,i)` is infinite.
bool | is_infinite (Edge e) const
| `true`, iff edge `e` is infinite.
bool | is_infinite (Edge_circulator ec) const
| `true`, iff edge `*ec` is infinite.
template<class Input_iterator >
size_type | insert (Input_iterator first, Input_iterator beyond)
| Inserts the sites in the range [`first`,`beyond`). More...
template<class Input_iterator >
size_type | insert (Input_iterator first, Input_iterator beyond, Tag_false)
| Same as the previous method. More...
template<class Input_iterator >
size_type | insert (Input_iterator first, Input_iterator beyond, Tag_true)
| Decomposes the range [first,beyond) into a range of input points and a range of input segments that are respectively passed to `insert_segments()` and `insert_points()`. More...
template<class PointIterator >
std::size_t | insert_points (PointIterator first, PointIterator beyond)
| Inserts the points in the range [first,beyond) as sites. More...
template<class SegmentIterator >
std::size_t | insert_segments (SegmentIterator first, SegmentIterator beyond)
| Inserts the segments in the range [first,beyond) as sites. More...
template<class PointIterator , class IndicesIterator >
std::size_t | insert_segments (PointIterator points_first, PointIterator points_beyond, IndicesIterator indices_first, IndicesIterator indices_beyond)
| Same as above except that each segment is given as a pair of indices of the points in the range [points_first, points_beyond). More...
Vertex_handle | insert (Point_2 p)
| Inserts the point `p` in the segment Delaunay graph. More...
Vertex_handle | insert (Point_2 p, Vertex_handle vnear)
| Inserts `p` in the segment Delaunay graph using the site associated with `vnear` as an estimate for the nearest neighbor of `p`. More...
Vertex_handle | insert (Point_2 p1, Point_2 p2)
| Inserts the closed segment with endpoints `p1` and `p2` in the segment Delaunay graph. More...
Vertex_handle | insert (Point_2 p1, Point_2 p2, Vertex_handle vnear)
| Inserts the segment whose endpoints are `p1` and `p2` in the segment Delaunay graph using the site associated with `vnear` as an estimate for the nearest neighbor of `p1`. More...
Vertex_handle | insert (Site_2 s)
| Inserts the site `s` in the segment Delaunay graph. More...
Vertex_handle | insert (Site_2 s, Vertex_handle vnear)
| Inserts `s` in the segment Delaunay graph using the site associated with `vnear` as an estimate for the nearest neighbor of `s`, if `s` is a point, or the first endpoint of `s`, if `s` is a segment. More...
Vertex_handle | nearest_neighbor (Point_2 p)
| Finds the nearest neighbor of the point `p`. More...
Vertex_handle | nearest_neighbor (Point_2 p, Vertex_handle vnear)
| Finds the nearest neighbor of the point `p` using the site associated with `vnear` as an estimate for the nearest neighbor of `p`. More...
template<class Stream >
Stream & | draw_dual (Stream &str)
| Draws the segment Voronoi diagram to the stream `str`. More...
template<class Stream >
Stream & | draw_skeleton (Stream &str)
| Draws the segment Voronoi diagram to the stream `str`, except the edges of the diagram corresponding to a segment and its endpoints. More...
template<class Stream >
Stream & | draw_dual_edge (Edge e, Stream &str)
| Draws the edge `e` of the segment Voronoi diagram to the stream `str`. More...
template<class Stream >
Stream & | draw_dual_edge (Edge_circulator ec, Stream &str)
| Draws the edge `*ec` of the segment Voronoi diagram to the stream `str`. More...
template<class Stream >
Stream & | draw_dual_edge (All_edges_iterator eit, Stream &str)
| Draws the edge `*eit` of the segment Voronoi diagram to the stream `str`. More...
template<class Stream >
Stream & | draw_dual_edge (Finite_edges_iterator eit, Stream &str)
| Draws the edge `*eit` of the segment Voronoi diagram to the stream `str`. More...
void | file_output (std::ostream &os)
| Writes the current state of the segment Delaunay graph to an output stream. More...
void | file_input (std::istream &is)
| Reads the state of the segment Delaunay graph from an input stream.
std::ostream & | operator<< (std::ostream &os, Segment_Delaunay_graph_2< Gt, DS > sdg)
| Writes the current state of the segment Delaunay graph to an output stream.
std::istream & | operator>> (std::istream &is, Segment_Delaunay_graph_2< Gt, DS > sdg)
| Reads the state of the segment Delaunay graph from an input stream.
bool | is_valid (bool verbose=false, int level=1)
| Checks the validity of the segment Delaunay graph. More...
void | clear ()
| Clears all contents of the segment Delaunay graph.
void | swap (Segment_Delaunay_graph_2< Gt, DS > other)
| The segment Delaunay graphs `other` and `*this` are swapped. More...

## Constructor & Destructor Documentation

template<typename Gt , typename STag , typename DS >
template<class Input_iterator >

CGAL::Segment_Delaunay_graph_hierarchy_2< Gt, STag, DS >::Segment_Delaunay_graph_hierarchy_2 | ( | Input_iterator | first,
|  | Input_iterator | beyond,
|  | Gt | gt = `Gt()`
| ) |  |

Creates a segment Delaunay graph hierarchy using `gt` as geometric traits and inserts all sites in the range [`first`, `beyond`).

`Input_iterator` must be a model of `InputIterator`. The value type of `Input_iterator` must be either `Point_2` or `Site_2`.

## Friends And Related Function Documentation

template<typename Gt , typename STag , typename DS >

std::ostream & operator<< | ( | std::ostream & | os,
|  | Segment_Delaunay_graph_hierarchy_2< Gt, STag, DS > | svdh
| ) |  |
| related

Writes the current state of the segment Delaunay graph hierarchy to an output stream.

In particular, all sites in the diagram are written to the stream (represented through appropriate input sites), as well as the underlying combinatorial hierarchical data structure.

Examples:
  Segment_Delaunay_graph_2/sdg-red-blue-info.cpp.