[/============================================================================
  Boost.Geometry Index

  Copyright (c) 2011-2012 Adam Wulkiewicz.

  Use, modification and distribution is subject to the Boost Software License,
  Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt)
=============================================================================/]

[section Introduction]

__rtree__ is a tree data structure used for spatial searching. It was proposed by
Antonin Guttman in 1984 [footnote Guttman, A. (1984). /R-Trees: A Dynamic Index Structure for Spatial Searching/]
as an expansion of B-tree for multi-dimensional data. It may be used to store points or volumetric data in order to
perform a spatial query later. This query may return objects that are inside some area or are close to some point in space
[footnote Cheung, K.; Fu, A. (1998). /Enhanced Nearest Neighbour Search on the R-tree/].

The __rtree__ structure is presented on the image below. Each __rtree__'s node store a box describing the space occupied by
its children nodes. At the bottom of the structure, there are leaf-nodes which contains values
(geometric objects representations). 

[$img/index/rtree/rstar.png]

The __rtree__ is a self-balanced data structure. The key part of balancing algorithm is node splitting algorithm
[footnote Greene, D. (1989). /An implementation and performance analysis of spatial data access methods/]
[footnote Beckmann, N.; Kriegel, H. P.; Schneider, R.; Seeger, B. (1990). /The R*-tree: an efficient and robust access method for points and rectangles/].
Each algorithm produces different splits so the internal structure of a tree may be different for each one of them.
In general more complex algorithms analyses elements better and produces less overlapping nodes. In the searching process less nodes must be traversed
in order to find desired objects. On the other hand more complex analysis takes more time. In general faster inserting will result in slower searching
and vice versa. The performance of the R-tree depends on balancing algorithm, parameters and data inserted into the container.
Example structures of trees created by use of three different algorithms and operations time are presented below. Data used in benchmark was random,
non-overlapping boxes.

[table
[[]                         [linear algorithm] [quadratic algorithm] [R*-tree]]
[[*Example structure*]      [[$img/index/rtree/linear.png]] [[$img/index/rtree/quadratic.png]] [[$img/index/rtree/rstar.png]]]
[[*1M Values inserts*]      [1.65s] [2.51s] [4.96s]]
[[*100k spatial queries*]   [0.87s] [0.25s] [0.09s]]
[[*100k knn queries*]       [3.25s] [1.41s] [0.51s]]
]

[heading Implementation details]

Key features of this implementation of the __rtree__ are:

* capable to store arbitrary __value__ type,
* three different creation algorithms - linear, quadratic or rstar,
* parameters (including maximal and minimal number of elements) may be passed as compile- or run-time parameters,
* advanced queries - e.g. search for 5 nearest values to some point and intersecting some region but not within the other one,
* C++11 conformant: move semantics, stateful allocators,
* capable to store __value__ type with no default constructor.

[heading Dependencies]

R-tree depends on *Boost.Move*, *Boost.Container*, *Boost.Tuple*, *Boost.Utility*, *Boost.MPL*.

[heading Contributors]

The spatial index was originally started by Federico J. Fernandez during the Google Summer of Code 2008 program, mentored by Hartmut Kaiser.

[heading Spatial thanks]

I'd like to thank Barend Gehrels, Bruno Lalande, Mateusz Łoskot, Lucanus J. Simonson for their support and ideas.

[endsect]