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      <td style="background-color: rgb(238, 238, 238);" nowrap="1" valign="top">
      <div style="padding: 5px;" align="center"> <img src="images/boost.png" border="0" height="86" width="277"><a title="www.boost.org home page" tabindex="2" style="border: medium none ;" href="http://www.boost.org/"> </a></div>
      <div style="margin: 5px;">
      <h3 class="navbar">Contents</h3>
      <ul>
        <li><a href="index.htm">Boost.Polygon Main Page</a></li>
        <li><a href="gtl_design_overview.htm">Design Overview</a></li>
        <li><a href="gtl_isotropy.htm">Isotropy</a></li>
        <li><a href="gtl_coordinate_concept.htm">Coordinate Concept</a></li>
        <li><a href="gtl_interval_concept.htm">Interval Concept</a></li>
        <li><a href="gtl_point_concept.htm">Point Concept</a></li>
        <li><a href="gtl_segment_concept.htm">Segment Concept</a></li>
        <li><a href="gtl_rectangle_concept.htm">Rectangle Concept</a></li>
        <li><a href="gtl_polygon_90_concept.htm">Polygon 90 Concept</a></li>
        <li><a href="gtl_polygon_90_with_holes_concept.htm">Polygon 90
With Holes Concept</a></li>
        <li><a href="gtl_polygon_45_concept.htm">Polygon 45 Concept</a></li>
        <li><a href="gtl_polygon_45_with_holes_concept.htm">Polygon 45
With Holes Concept</a></li>
        <li><a href="gtl_polygon_concept.htm">Polygon Concept</a></li>
        <li><a href="gtl_polygon_with_holes_concept.htm">Polygon With
Holes Concept</a></li>
        <li><a href="gtl_polygon_90_set_concept.htm">Polygon 90 Set
Concept</a></li>
        <li><a href="gtl_polygon_45_set_concept.htm">Polygon 45 Set
Concept</a></li>
        <li><a href="gtl_polygon_set_concept.htm">Polygon Set Concept</a></li>
        <li><a href="gtl_connectivity_extraction_90.htm">Connectivity
Extraction 90</a></li>
        <li><a href="gtl_connectivity_extraction_45.htm">Connectivity
Extraction 45</a></li>
        <li><a href="gtl_connectivity_extraction.htm">Connectivity
Extraction</a></li>
        <li><a href="gtl_property_merge_90.htm">Property Merge 90</a></li>
        <li><a href="gtl_property_merge_45.htm">Property Merge 45</a></li>
        <li><a href="gtl_property_merge.htm">Property Merge</a></li>
        <li><a href="voronoi_main.htm">Voronoi Main Page<br>
          </a></li>
        <li>Voronoi Benchmark<br>
        </li>
        <li><a href="voronoi_builder.htm">Voronoi Builder</a></li>
        <li><a href="voronoi_diagram.htm">Voronoi Diagram</a></li>
      </ul>
      <h3 class="navbar">Other Resources</h3>
      <ul>
        <li><a href="GTL_boostcon2009.pdf">GTL Boostcon 2009 Paper</a></li>
        <li><a href="GTL_boostcon_draft03.pdf">GTL Boostcon 2009
Presentation</a></li>
        <li><a href="analysis.htm">Performance Analysis</a></li>
        <li><a href="gtl_tutorial.htm">Layout Versus Schematic Tutorial</a></li>
        <li><a href="gtl_minkowski_tutorial.htm">Minkowski Sum Tutorial</a></li>
        <li><a href="voronoi_basic_tutorial.htm">Voronoi Basic Tutorial</a></li>
        <li><a href="voronoi_advanced_tutorial.htm">Voronoi Advanced
Tutorial</a></li>
      </ul>
      </div>
      <h3 class="navbar">Polygon Sponsor</h3>
      <div style="padding: 5px;" align="center"> <img src="images/intlogo.gif" border="0" height="51" width="127"><a title="www.adobe.com home page" tabindex="2" style="border: medium none ;" href="http://www.adobe.com/"> </a></div>
      </td>
      <td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%"><!-- End Header --> <br>
      <h1>Voronoi Benchmark</h1>
      <h2>Benchmark Details</h2>
From the topological perspective Delaunay triangulation is a dual
data structure to the Voronoi diagram, thus libraries that provide
Delaunay triangulation construction routines were also included into
the benchmark. However, from the computation perspective Voronoi
diagram contains more information as it embeds information regarding
the coordinates of the centers of the inscribed circles tangent to the
three or more input geometries.<br>


The benchmark consists of the two parts:<br>
      <ol>
<li>construction of the Voronoi diagram / Delaunay triangulation of a set of
random points uniformly distributed over 32-bit integer grid (<a href="../benchmark/voronoi_benchmark_points.cpp"><span style="text-decoration: underline;">voronoi_benchmark_points.cpp</span></a>)</li><li>construction of the Voronoi diagram / Delaunay triangulation of a set of
random non-intersecting
segments uniformly distributed over 32-bit integer grid (<a href="../benchmark/voronoi_benchmark_segments.cpp">voronoi_benchmark_segments.cpp</a>).</li>
      </ol>


      <h2>Libraries</h2>
      <table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="2">
        <tbody>
          <tr>
            <td style="vertical-align: top;"><span style="font-weight: bold;">Library</span><br>
            </td>
            <td style="vertical-align: top; font-weight: bold;">Boost.Polygon<br>
            </td>
            <td style="vertical-align: top; font-weight: bold;">CGAL<br>
            </td>
            <td style="vertical-align: top; font-weight: bold;">SHull<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Official page<br>
            </td>
            <td style="vertical-align: top;">www.boost.org/libs/polygon&#8206;<br>
            </td>
            <td style="vertical-align: top;">http://www.cgal.org<br>
            </td>
            <td style="vertical-align: top;">http://www.s-hull.org<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Algorithm<br>
            </td>
            <td style="vertical-align: top;">sweep-line<br>
            </td>
            <td style="vertical-align: top;">incremental<br>
            </td>
            <td style="vertical-align: top;">sweep-hull<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Supported input geometries<br>
            </td>
            <td style="vertical-align: top;">points and segments<br>
            </td>
            <td style="vertical-align: top;">points and segments<br>
            </td>
            <td style="vertical-align: top;">points<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Output data structure<br>
            </td>
            <td style="vertical-align: top;">Voronoi diagram<br>
            </td>
            <td style="vertical-align: top;">Delaunay triangulation<br>
            </td>
            <td style="vertical-align: top;">Delaunay triangulation<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Complexity<br>
            </td>
            <td style="vertical-align: top;">O(N log N)<br>
            </td>
            <td style="vertical-align: top;">O(N log N)</td>
            <td style="vertical-align: top;">O(N log N)</td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Memory usage<br>
            </td>
            <td style="vertical-align: top;">O(N)<br>
            </td>
            <td style="vertical-align: top;">O(N)<br>
            </td>
            <td style="vertical-align: top;">O(N)<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Robustness policies<br>
            </td>
            <td style="vertical-align: top;">lazy arithmetic, exact predicates, topology analysis<br>
            </td>
            <td style="vertical-align: top;">lazy arithmetic, exact predicates, topology analysis<br>
            </td>
            <td style="vertical-align: top;">non-robust<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">Output coordinates precision<br>
            </td>
            <td style="vertical-align: top;">128 machine epsilon<br>
            </td>
            <td style="vertical-align: top;">no output coordinates<br>
            </td>
            <td style="vertical-align: top;">no output coordinates<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top;">External dependencies<br>
            </td>
            <td style="vertical-align: top;">No<br>
            </td>
            <td style="vertical-align: top;">Boost, GMP, MPFR<br>
            </td>
            <td style="vertical-align: top;">No<br>
            </td>
          </tr>
        </tbody>
      </table>
      <br>

The other known libraries (<a href="https://github.com/aewallin/openvoronoi">OpenVoronoi</a>,
      <a href="http://www.cs.cmu.edu/%7Equake/triangle.html">Triangle</a>, <a href="http://www.cosy.sbg.ac.at/%7Eheld/projects/vroni/vroni.html">Vroni</a>) will be considered for the inclusion into the benchmark in the future.<br>
<ul>
        <ul>

        
        </ul>

        
        <ul>

        
        </ul>

      </ul>
      <h2>System Configuration<br>
      </h2>


Hardware: Intel i5-7600 2.8 GHz, 16GiB RAM.<br>
OS: Ubuntu 13.04 64-bit.<br>
Compiler: GCC 4.7.3.<br>
Libraries and dependencies: Boost 1.54.0, CGAL-4.3-beta1, GMP 5.1.4, MPFR 3.1.2, S-Hull.<br>
      <h2>Benchmark Results</h2>
      <h3>Random Points Benchmark</h3><img style="border: 1px solid ; width: 500px; height: 350px;" alt="Benchmark Points Chart" src="../benchmark/benchmark_results/plots/benchmark_points.png"><br>

      <table style="text-align: left; width: 100%; margin-left: auto; margin-right: auto;" border="1" cellpadding="2" cellspacing="2">
        <tbody>
          <tr>
            <td colspan="2" rowspan="1" style="vertical-align: top; text-align: center;">Test specification<br>
            </td>
            <td colspan="6" rowspan="1" style="vertical-align: top; text-align: center;">Average construction
time (in secs)<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: center;">Number
of Points<br>
            </td>
            <td style="vertical-align: top; text-align: center;">Number
of Runs<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: center;">Boost
Linux-64<br>
            </td>
            <td style="vertical-align: top; text-align: center;">CGAL
Linux-64<br>
            </td>
            <td style="vertical-align: top; text-align: center;">S-Hull
Linux-64<br>
            </td>
          </tr>
          
          <tr>
            <td style="vertical-align: top; text-align: right;">100<br>
            </td>
            <td style="vertical-align: top; text-align: right;">10000<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">0.000206<br>
            </td>
            <td style="vertical-align: top; text-align: right;">&nbsp;0.000073<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.000243<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">1000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">1000<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">0.002250<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.000753<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.002184<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">10000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">100<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">0.024100<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.007917<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.030552<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">100000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">10<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">0.292000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.084336<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.451913<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">1000000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">1<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">3.470000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.902300<br>
            </td>
            <td style="vertical-align: top; text-align: right;">6.603814<br>
            </td>
          </tr>
        </tbody>
      </table>
      <br>
<h3>Random Segments Benchmark</h3><img style="border: 1px solid ; width: 500px; height: 350px;" alt="Benchmark Segment Chart" src="../benchmark/benchmark_results/plots/benchmark_segments.png"><br>

      <table style="text-align: left; width: 100%;" border="1" cellpadding="2" cellspacing="2">
        <tbody>
          <tr>
            <td colspan="2" rowspan="1" style="vertical-align: top; text-align: center;">Test specification<br>
            </td>
            <td colspan="4" rowspan="1" style="vertical-align: top; text-align: center;">Average construction
time (in secs)</td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: center;">Number
of Segments<br>
            </td>
            <td style="vertical-align: top; text-align: center;">Number
of Runs<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: center;">Boost
Linux-64<br>
            </td>
            <td style="vertical-align: top; text-align: center;">CGAL
Linux-64<br>
            </td>
          </tr>
          
          <tr>
            <td style="vertical-align: top; text-align: right;">100<br>
            </td>
            <td style="vertical-align: top; text-align: right;">10000<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">0.002284<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.002985<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">1000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">1000<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">0.023240<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.032180<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">10000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">100<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">&nbsp;0.237700<br>
            </td>
            <td style="vertical-align: top; text-align: right;">0.337400<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">100000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">10<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">2.488000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">3.633000<br>
            </td>
          </tr>
          <tr>
            <td style="vertical-align: top; text-align: right;">1000000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">1<br>
            </td>
            
            
            <td style="vertical-align: top; text-align: right;">25.00000<br>
            </td>
            <td style="vertical-align: top; text-align: right;">39.090000<br>
            </td>
          </tr>
        </tbody>
      </table>
      <br><span style="font-weight: bold;"></span></td>
    </tr>
    <tr>
      <td style="background-color: rgb(238, 238, 238);" nowrap="1" valign="top">&nbsp;</td>
      <td style="padding-left: 10px; padding-right: 10px; padding-bottom: 10px;" valign="top" width="100%">
      <table class="docinfo" id="table2" frame="void" rules="none">
        <colgroup> <col class="docinfo-name"><col class="docinfo-content"> </colgroup> <tbody valign="top">
          <tr>
            <th class="docinfo-name">Copyright:</th>
            <td>Copyright © Andrii Sydorchuk 2010-2012.</td>
          </tr>
          <tr class="field">
            <th class="docinfo-name">License:</th>
            <td class="field-body">Distributed under the Boost Software
License, Version 1.0. (See accompanying file <tt class="literal"><span class="pre">LICENSE_1_0.txt</span></tt> or copy at <a class="reference" target="_top" href="http://www.boost.org/LICENSE_1_0.txt">
http://www.boost.org/LICENSE_1_0.txt</a>)</td>
          </tr>
        </tbody>
      </table>
      </td>
    </tr>
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</table>

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