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26 <div class="titlepage"><div><div><h3 class="title">
27 <a name="geometry.spatial_indexes.introduction"></a><a class="link" href="introduction.html" title="Introduction">Introduction</a>
28 </h3></div></div></div>
30 The Boost.Geometry.Index is intended to gather data structures called spatial
31 indexes which may be used to accelerate searching for objects in space. In
32 general, spatial indexes stores geometric objects' representations and allows
33 searching for objects occupying some space or close to some point in space.
36 Currently, only one spatial index is implemented - R-tree.
39 <a name="geometry.spatial_indexes.introduction.h0"></a>
40 <span class="phrase"><a name="geometry.spatial_indexes.introduction.r_tree"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.r_tree">R-tree</a>
43 R-tree is a tree data structure used for spatial searching. It was proposed
44 by Antonin Guttman in 1984 <a href="#ftn.geometry.spatial_indexes.introduction.f0" class="footnote" name="geometry.spatial_indexes.introduction.f0"><sup class="footnote">[1]</sup></a> as an expansion of B-tree for multi-dimensional data. It may
45 be used to store points or volumetric data in order to perform a spatial
46 query. This query may for example return objects that are inside some area
47 or are close to some point in space <a href="#ftn.geometry.spatial_indexes.introduction.f1" class="footnote" name="geometry.spatial_indexes.introduction.f1"><sup class="footnote">[2]</sup></a>. It's possible to insert new objects or to remove the ones already
51 The R-tree structure is presented on the image below. Each R-tree's node
52 store a box describing the space occupied by its children nodes. At the bottom
53 of the structure, there are leaf-nodes which contains values (geometric objects
57 <span class="inlinemediaobject"><img src="../../img/index/rtree/rstar.png" alt="rstar"></span>
60 The R-tree is a self-balanced data structure. The key part of balancing algorithm
61 is node splitting algorithm <a href="#ftn.geometry.spatial_indexes.introduction.f2" class="footnote" name="geometry.spatial_indexes.introduction.f2"><sup class="footnote">[3]</sup></a> <a href="#ftn.geometry.spatial_indexes.introduction.f3" class="footnote" name="geometry.spatial_indexes.introduction.f3"><sup class="footnote">[4]</sup></a>. Each algorithm produces different splits so the internal structure
62 of a tree may be different for each one of them. In general, more complex
63 algorithms analyses elements better and produces less overlapping nodes.
64 In the searching process less nodes must be traversed in order to find desired
65 objects. On the other hand more complex analysis takes more time. In general
66 faster inserting will result in slower searching and vice versa. The performance
67 of the R-tree depends on balancing algorithm, parameters and data inserted
71 Additionally there are also algorithms creating R-tree containing some, number
72 of objects. This technique is called bulk loading and is done by use of packing
73 algorithm <a href="#ftn.geometry.spatial_indexes.introduction.f4" class="footnote" name="geometry.spatial_indexes.introduction.f4"><sup class="footnote">[5]</sup></a> <a href="#ftn.geometry.spatial_indexes.introduction.f5" class="footnote" name="geometry.spatial_indexes.introduction.f5"><sup class="footnote">[6]</sup></a>. This method is faster and results in R-trees with better internal
74 structure. This means that the query performance is increased.
77 The examples of structures of trees created by use of different algorithms
78 and exemplary operations times are presented below.
80 <div class="informaltable"><table class="table">
116 <span class="bold"><strong>Example structure</strong></span>
121 <span class="inlinemediaobject"><img src="../../img/index/rtree/linear.png" alt="linear"></span>
126 <span class="inlinemediaobject"><img src="../../img/index/rtree/quadratic.png" alt="quadratic"></span>
131 <span class="inlinemediaobject"><img src="../../img/index/rtree/rstar.png" alt="rstar"></span>
136 <span class="inlinemediaobject"><img src="../../img/index/rtree/bulk.png" alt="bulk"></span>
143 <span class="bold"><strong>1M Values inserts</strong></span>
170 <span class="bold"><strong>100k spatial queries</strong></span>
197 <span class="bold"><strong>100k knn queries</strong></span>
224 The configuration of the machine used for testing was: <span class="emphasis"><em>Intel(R)
225 Core(TM) i7 870 @ 2.93GHz, 8GB RAM, MS Windows 7 x64</em></span>. The code
226 was compiled with optimization for speed (<code class="computeroutput"><span class="identifier">O2</span></code>).
229 The performance of the R-tree for different values of Max parameter and Min=0.5*Max
230 is presented in the table below. In the two upper figures you can see the
231 performance of the R-tree storing random, relatively small, non-overlapping,
232 2d boxes. In the lower ones, the performance of the R-tree also storing random,
233 2d boxes, but this time quite big and possibly overlapping. As you can see,
234 the R-tree performance is different in both cases.
236 <div class="informaltable"><table class="table">
260 <span class="bold"><strong>non overlapping</strong></span>
265 <span class="inlinemediaobject"><img src="../../img/index/rtree/build_non_ovl.png" alt="build_non_ovl"></span>
270 <span class="inlinemediaobject"><img src="../../img/index/rtree/query_non_ovl.png" alt="query_non_ovl"></span>
277 <span class="bold"><strong>overlapping</strong></span>
282 <span class="inlinemediaobject"><img src="../../img/index/rtree/build_ovl.png" alt="build_ovl"></span>
287 <span class="inlinemediaobject"><img src="../../img/index/rtree/query_ovl.png" alt="query_ovl"></span>
294 <a name="geometry.spatial_indexes.introduction.h1"></a>
295 <span class="phrase"><a name="geometry.spatial_indexes.introduction.implementation_details"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.implementation_details">Implementation
299 Key features of this implementation of the R-tree are:
301 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
302 <li class="listitem">
303 capable to store arbitrary Value type,
305 <li class="listitem">
306 three different balancing algorithms - linear, quadratic or rstar,
308 <li class="listitem">
309 creation using packing algorithm,
311 <li class="listitem">
312 parameters (including maximal and minimal number of elements) may be
313 passed as compile- or run-time parameters, in compile-time version nodes
314 elements are stored in static-size containers,
316 <li class="listitem">
317 advanced queries, e.g. search for 5 nearest Values to some point and
318 intersecting some Geometry but not within the other one,
320 <li class="listitem">
321 iterative queries by use of iterators,
323 <li class="listitem">
324 C++11 conformant - move semantics, stateful allocators,
326 <li class="listitem">
327 capable to store Value type with no default constructor,
329 <li class="listitem">
330 in-memory storage by use of the default std::allocator<>,
332 <li class="listitem">
333 other storage options - shared memory and mapped file by use of Boost.Interprocess
338 <a name="geometry.spatial_indexes.introduction.h2"></a>
339 <span class="phrase"><a name="geometry.spatial_indexes.introduction.dependencies"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.dependencies">Dependencies</a>
342 R-tree depends on Boost.Container, Boost.Core, Boost.Move, Boost.MPL, Boost.Range,
346 <a name="geometry.spatial_indexes.introduction.h3"></a>
347 <span class="phrase"><a name="geometry.spatial_indexes.introduction.contributors"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.contributors">Contributors</a>
350 The spatial index was originally started by Federico J. Fernandez during
351 the Google Summer of Code 2008 program, mentored by Hartmut Kaiser.
354 <a name="geometry.spatial_indexes.introduction.h4"></a>
355 <span class="phrase"><a name="geometry.spatial_indexes.introduction.spatial_thanks"></a></span><a class="link" href="introduction.html#geometry.spatial_indexes.introduction.spatial_thanks">Spatial thanks</a>
358 I'd like to thank Barend Gehrels, Bruno Lalande, Mateusz Łoskot, Lucanus
359 J. Simonson for their support and ideas.
361 <div class="footnotes">
362 <br><hr style="width:100; text-align:left;margin-left: 0">
363 <div id="ftn.geometry.spatial_indexes.introduction.f0" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f0" class="para"><sup class="para">[1] </sup></a>
364 Guttman, A. (1984). <span class="emphasis"><em>R-Trees: A Dynamic Index Structure for Spatial
365 Searching</em></span>
367 <div id="ftn.geometry.spatial_indexes.introduction.f1" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f1" class="para"><sup class="para">[2] </sup></a>
368 Cheung, K.; Fu, A. (1998). <span class="emphasis"><em>Enhanced Nearest Neighbour Search
369 on the R-tree</em></span>
371 <div id="ftn.geometry.spatial_indexes.introduction.f2" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f2" class="para"><sup class="para">[3] </sup></a>
372 Greene, D. (1989). <span class="emphasis"><em>An implementation and performance analysis
373 of spatial data access methods</em></span>
375 <div id="ftn.geometry.spatial_indexes.introduction.f3" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f3" class="para"><sup class="para">[4] </sup></a>
376 Beckmann, N.; Kriegel, H. P.; Schneider, R.; Seeger, B. (1990). <span class="emphasis"><em>The
377 R*-tree: an efficient and robust access method for points and rectangles</em></span>
379 <div id="ftn.geometry.spatial_indexes.introduction.f4" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f4" class="para"><sup class="para">[5] </sup></a>
380 Leutenegger, Scott T.; Edgington, Jeffrey M.; Lopez, Mario A. (1997).
381 <span class="emphasis"><em>STR: A Simple and Efficient Algorithm for R-Tree Packing</em></span>
383 <div id="ftn.geometry.spatial_indexes.introduction.f5" class="footnote"><p><a href="#geometry.spatial_indexes.introduction.f5" class="para"><sup class="para">[6] </sup></a>
384 Garcia, Yvan J.; Lopez, Mario A.; Leutenegger, Scott T. (1997). <span class="emphasis"><em>A
385 Greedy Algorithm for Bulk Loading R-trees</em></span>
389 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
390 <td align="left"></td>
391 <td align="right"><div class="copyright-footer">Copyright © 2009-2017 Barend
392 Gehrels, Bruno Lalande, Mateusz Loskot, Adam Wulkiewicz, Oracle and/or its
394 Distributed under the Boost Software License, Version 1.0. (See accompanying
395 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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