## CGAL::Triangulation_hierarchy_3<Tr>

### Definition

The class Triangulation_hierarchy_3 implements a triangulation augmented with a data structure which allows fast point location queries. As proved in [Dev02], this structure has an optimal behavior when it is built for Delaunay triangulations. It can however be used for other triangulations.

#include <CGAL/Triangulation_hierarchy_3.h>

### Parameters

It is templated by a parameter which must be instantiated by one of the CGAL triangulation classes. In the current implementation, only Delaunay_triangulation_3 is supported for Tr.

Tr::Vertex has to be a model of the concept TriangulationHierarchyVertexBase_3.
Tr::Geom_traits has to be a model of the concept DelaunayTriangulationTraits_3.

### Inherits From

Tr

Triangulation_hierarchy_3<Tr> offers exactly the same functionalities as Tr. Most of them (point location, insertion, removal...) are overloaded to improve their efficiency by using the hierarchic structure.

Note that, since the algorithms that are provided are randomized, the running time of constructing a triangulation with a hierarchy may be improved when shuffling the data points.

However, the I/O operations are not overloaded. So, writing a hierarchy into a file will lose the hierarchic structure and reading it from the file will result in an ordinary triangulation whose efficiency will be the same as Tr.

### Implementation

The data structure is a hierarchy of triangulations. The triangulation at the lowest level is the original triangulation where operations and point location are to be performed. Then at each succedding level, the data structure stores a triangulation of a small random sample of the vertices of the triangulation at the preceeding level. Point location is done through a top-down nearest neighbor query. The nearest neighbor query is first performed naively in the top level triangulation. Then, at each following level, the nearest neighbor at that level is found through a linear walk performed from the nearest neighbor found at the preceeding level. Because the number of vertices in each triangulation is only a small fraction of the number of vertices of the preceeding triangulation the data structure remains small and achieves fast point location queries on real data.

### See Also

CGAL::Triangulation_hierarchy_vertex_base_3
CGAL::Delaunay_triangulation_3