Abdelrahman Said
1017 lines
37 KiB
HTML
1017 lines
37 KiB
HTML
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<title>Chapter 14. Spatial graphs</title>
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<link rel="chapter" href="igraph-Data-structures.html" title="Chapter 7. Data structure library: vector, matrix, other data types">
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<link rel="chapter" href="igraph-Random.html" title="Chapter 8. Random numbers">
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<link rel="chapter" href="igraph-Iterators.html" title="Chapter 9. Vertex and edge selectors and sequences, iterators">
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<link rel="chapter" href="igraph-Attributes.html" title="Chapter 10. Graph, vertex and edge attributes">
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<link rel="chapter" href="igraph-Generators.html" title="Chapter 11. Deterministic graph generators">
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<link rel="chapter" href="igraph-Games.html" title='Chapter 12. Stochastic graph generators ("games")'>
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<link rel="chapter" href="igraph-Bipartite.html" title="Chapter 13. Bipartite, i.e. two-mode graphs">
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<link rel="chapter" href="igraph-Spatial.html" title="Chapter 14. Spatial graphs">
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<link rel="chapter" href="igraph-Operators.html" title="Chapter 15. Graph operators">
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<link rel="chapter" href="igraph-Visitors.html" title="Chapter 16. Graph visitors">
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<link rel="chapter" href="igraph-Structural.html" title="Chapter 17. Structural properties of graphs">
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<link rel="chapter" href="igraph-Cycles.html" title="Chapter 18. Graph cycles">
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<link rel="chapter" href="igraph-Cliques.html" title="Chapter 19. Cliques and independent vertex sets">
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<link rel="chapter" href="igraph-Motifs.html" title="Chapter 20. Graph motifs, dyad census and triad census">
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<link rel="chapter" href="igraph-Isomorphism.html" title="Chapter 21. Graph isomorphism">
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<link rel="chapter" href="igraph-Coloring.html" title="Chapter 22. Graph coloring">
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<link rel="chapter" href="igraph-Flows.html" title="Chapter 23. Maximum flows, minimum cuts and related measures">
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<link rel="chapter" href="igraph-Separators.html" title="Chapter 24. Vertex separators">
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<link rel="chapter" href="igraph-Community.html" title="Chapter 25. Detecting community structure">
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<link rel="chapter" href="igraph-Graphlets.html" title="Chapter 26. Graphlets">
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<link rel="chapter" href="igraph-HRG.html" title="Chapter 27. Hierarchical random graphs">
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<link rel="chapter" href="igraph-Embedding.html" title="Chapter 28. Embedding of graphs">
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<link rel="chapter" href="igraph-Layout.html" title="Chapter 29. Generating layouts for graph drawing">
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<link rel="chapter" href="igraph-Processes.html" title="Chapter 30. Processes on graphs">
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<link rel="chapter" href="igraph-Foreign.html" title="Chapter 31. Reading and writing graphs from and to files">
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<link rel="chapter" href="igraph-Linalg.html" title="Chapter 32. Using BLAS, LAPACK and ARPACK for igraph matrices and graphs">
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<link rel="chapter" href="igraph-Nongraph.html" title="Chapter 33. Non-graph related functions">
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<link rel="chapter" href="igraph-Advanced.html" title="Chapter 34. Advanced igraph programming">
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<link rel="chapter" href="igraph-Glossary.html" title="Chapter 35. Glossary">
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<link rel="chapter" href="igraph-Licenses.html" title="Chapter 36. Licenses for igraph and this manual">
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</div></div>
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<div class="chapter">
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<div class="titlepage"><div><div><h1 class="title">
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<a name="igraph-Spatial"></a>Chapter 14. Spatial graphs</h1></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="igraph-Spatial.html#spatial-helpers">1. Metrics</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#spatial-generators">2. Spatial graph generators</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#spatial-properties">3. Properties of spatial graphs</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#nongraph-spatial">4. Non-graph related spatial processing</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="spatial-helpers"></a>1. Metrics</h2></div></div></div>
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<div class="toc"><dl class="toc"><dt><span class="section"><a href="igraph-Spatial.html#igraph_metric_t">1.1. <code class="function">igraph_metric_t</code> — Metric functions for use with spatial computation.</a></span></dt></dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="igraph_metric_t"></a>1.1. <code class="function">igraph_metric_t</code> — Metric functions for use with spatial computation.</h3></div></div></div>
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<a class="indexterm" name="id-1.15.2.2.2"></a><p>
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</p>
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<pre class="programlisting">
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typedef enum {
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IGRAPH_METRIC_EUCLIDEAN = 0,
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IGRAPH_METRIC_L2 = IGRAPH_METRIC_EUCLIDEAN,
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IGRAPH_METRIC_MANHATTAN = 1,
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IGRAPH_METRIC_L1 = IGRAPH_METRIC_MANHATTAN
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} igraph_metric_t;
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</pre>
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<p>
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</p>
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<p>
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</p>
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<p><b>Values: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><code class="constant">IGRAPH_METRIC_EUCLIDEAN</code>:</span></p></td>
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<td><p>
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The Euclidean distance, i.e. L2 metric.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><code class="constant">IGRAPH_METRIC_MANHATTAN</code>:</span></p></td>
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<td><p>
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The Manhattan distance, i.e. L1 metric.</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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</div>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="spatial-generators"></a>2. Spatial graph generators</h2></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="igraph-Spatial.html#igraph_delaunay_graph">2.1. <code class="function">igraph_delaunay_graph</code> — Computes the Delaunay graph of a spatial point set.</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#igraph_nearest_neighbor_graph">2.2. <code class="function">igraph_nearest_neighbor_graph</code> — Computes the nearest neighbor graph for a spatial point set.</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#igraph_gabriel_graph">2.3. <code class="function">igraph_gabriel_graph</code> — The Gabriel graph of a point set.</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#igraph_relative_neighborhood_graph">2.4. <code class="function">igraph_relative_neighborhood_graph</code> — The relative neighborhood graph of a point set.</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#igraph_lune_beta_skeleton">2.5. <code class="function">igraph_lune_beta_skeleton</code> — The lune based β-skeleton of a spatial point set.</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#igraph_circle_beta_skeleton">2.6. <code class="function">igraph_circle_beta_skeleton</code> — The circle based β-skeleton of a 2D spatial point set.</a></span></dt>
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<dt><span class="section"><a href="igraph-Spatial.html#igraph_beta_weighted_gabriel_graph">2.7. <code class="function">igraph_beta_weighted_gabriel_graph</code> — A Gabriel graph, with edges weighted by the β value at which it disappears.</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="igraph_delaunay_graph"></a>2.1. <code class="function">igraph_delaunay_graph</code> — Computes the Delaunay graph of a spatial point set.</h3></div></div></div>
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<a class="indexterm" name="id-1.15.3.2.2"></a><p>
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</p>
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<div class="informalexample"><pre class="programlisting">
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igraph_error_t igraph_delaunay_graph(igraph_t *graph, const igraph_matrix_t *points);
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</pre></div>
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<p>
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</p>
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<p>
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</p>
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<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
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<h3 class="title">Warning</h3>
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<p>This function is experimental and its signature is not considered final yet.
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We reserve the right to change the function signature without changing the
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major version of igraph. Use it at your own risk.</p>
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</div>
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<p>This function constructs the graph corresponding to the Delaunay triangulation
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of an n-dimensional spatial point set.
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</p>
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<p>
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The current implementation uses Qhull.
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</p>
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<p>
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Reference:
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</p>
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<p>
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Barber, C. Bradford, David P. Dobkin, and Hannu Huhdanpaa.
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The Quickhull Algorithm for Convex Hulls.
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ACM Transactions on Mathematical Software 22, no. 4 (1996): 469–83.
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<a class="ulink" href="https://doi.org/10.1145/235815.235821" target="_top">https://doi.org/10.1145/235815.235821</a>.
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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A pointer to the graph that will be created.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
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<td><p>
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A matrix containing the points that will be used to create the graph.
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Each row is a point, dimensionality is inferred from the column count.
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There must not be duplicate points.
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</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
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Error code.
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</p></td>
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</tr></tbody>
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</table></div>
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<p>
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Time complexity: According to Theorem 3.2 in the Qhull paper,
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O(n log n) for d <= 3 and O(n^floor(d/2) / floor(d/2)!) where
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n is the number of points and d is the dimensionality of the point set.
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="igraph_nearest_neighbor_graph"></a>2.2. <code class="function">igraph_nearest_neighbor_graph</code> — Computes the nearest neighbor graph for a spatial point set.</h3></div></div></div>
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<a class="indexterm" name="id-1.15.3.3.2"></a><p>
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</p>
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<div class="informalexample"><pre class="programlisting">
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igraph_error_t igraph_nearest_neighbor_graph(igraph_t *graph,
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const igraph_matrix_t *points,
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igraph_metric_t metric,
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igraph_int_t k,
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igraph_real_t cutoff,
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igraph_bool_t directed);
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</pre></div>
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<p>
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</p>
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<p>
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</p>
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<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
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<h3 class="title">Warning</h3>
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<p>This function is experimental and its signature is not considered final yet.
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We reserve the right to change the function signature without changing the
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major version of igraph. Use it at your own risk.</p>
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</div>
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<p>This function constructs the <em class="parameter"><code>k</code></em> nearest neighbor graph of a given point
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set. Each point is connected to at most <em class="parameter"><code>k</code></em> spatial neighbors within a
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radius of <em class="parameter"><code>cutoff</code></em>.
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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A pointer to the graph that will be created.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
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<td><p>
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A matrix containing the points that will be used to create
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the graph. Each row is a point, dimensionality is inferred from the
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column count.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>metric</code></em>:</span></p></td>
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<td><p>
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The distance metric to use. See <a class="link" href="igraph-Spatial.html#igraph_metric_t" title="1.1. igraph_metric_t — Metric functions for use with spatial computation."><code class="function">igraph_metric_t</code></a>.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>k</code></em>:</span></p></td>
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<td><p>
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At most how many neighbors will be added for each vertex, set to
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a negative value to ignore.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>cutoff</code></em>:</span></p></td>
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<td><p>
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Maximum distance at which connections will be made, set to a
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negative value or <code class="constant">IGRAPH_INFINITY</code> to ignore.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>directed</code></em>:</span></p></td>
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<td><p>
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Whether to create a directed graph.
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</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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||
</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
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Error code.
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</p></td>
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</tr></tbody>
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</table></div>
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<p>
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Time complexity: O(n log(n)) where n is the number of points.
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="igraph_gabriel_graph"></a>2.3. <code class="function">igraph_gabriel_graph</code> — The Gabriel graph of a point set.</h3></div></div></div>
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<a class="indexterm" name="id-1.15.3.4.2"></a><p>
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||
</p>
|
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<div class="informalexample"><pre class="programlisting">
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||
igraph_error_t igraph_gabriel_graph(igraph_t *graph, const igraph_matrix_t *points);
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</pre></div>
|
||
<p>
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</p>
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<p>
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||
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||
|
||
|
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</p>
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<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
|
||
<h3 class="title">Warning</h3>
|
||
<p>This function is experimental and its signature is not considered final yet.
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||
We reserve the right to change the function signature without changing the
|
||
major version of igraph. Use it at your own risk.</p>
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</div>
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<p>In the Gabriel graph of a point set, two points A and B are connected if
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there is no other point C within the closed ball of which AB is a diameter.
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The Gabriel graph is connected, and in 2D it is planar. igraph supports
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computing the Gabriel graph of arbitrary dimensional point sets.
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</p>
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<p>
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The Gabriel graph is a special case of lune-based and circle-based β-skeletons
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with <code class="literal">β=1</code>.
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</p>
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<p><b>Arguments: </b>
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</p>
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<div class="variablelist"><table border="0" class="variablelist">
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<colgroup>
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<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
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<td><p>
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A pointer to the graph to be created.
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</p></td>
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</tr>
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<tr>
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<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
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<td><p>
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The point set that will be used. Each row is a point,
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dimensionality is inferred from column count.</p></td>
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</tr>
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</tbody>
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</table></div>
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<p>
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</p>
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<p><b>Returns: </b></p>
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||
<div class="variablelist"><table border="0" class="variablelist">
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||
<colgroup>
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||
<col align="left" valign="top">
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<col>
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</colgroup>
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<tbody><tr>
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<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
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<td><p>
|
||
Error Code.
|
||
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p></p>
|
||
<p><b>See also: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
The Gabriel graph is a special case of
|
||
<a class="link" href="igraph-Spatial.html#igraph_lune_beta_skeleton" title="2.5. igraph_lune_beta_skeleton — The lune based β-skeleton of a spatial point set."><code class="function">igraph_lune_beta_skeleton()</code></a> and <a class="link" href="igraph-Spatial.html#igraph_circle_beta_skeleton" title="2.6. igraph_circle_beta_skeleton — The circle based β-skeleton of a 2D spatial point set."><code class="function">igraph_circle_beta_skeleton()</code></a>
|
||
where <code class="literal">β = 1</code>.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time Complexity: Around O(n^floor(d/2) log n), where n is the number of points
|
||
and d is the dimensionality of the point set.
|
||
|
||
|
||
</p>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h3 class="title">
|
||
<a name="igraph_relative_neighborhood_graph"></a>2.4. <code class="function">igraph_relative_neighborhood_graph</code> — The relative neighborhood graph of a point set.</h3></div></div></div>
|
||
<a class="indexterm" name="id-1.15.3.5.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_relative_neighborhood_graph(igraph_t *graph, const igraph_matrix_t *points);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
</p>
|
||
<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
|
||
<h3 class="title">Warning</h3>
|
||
<p>This function is experimental and its signature is not considered final yet.
|
||
We reserve the right to change the function signature without changing the
|
||
major version of igraph. Use it at your own risk.</p>
|
||
</div>
|
||
<p>The relative neighborhood graph is constructed from a set of points in space.
|
||
Two points A and B are connected if and only if there is no other point C so
|
||
that AC < AB and BC < AB, with the inequalities being strict.
|
||
|
||
</p>
|
||
<p>
|
||
Most authors define the relative neighborhood graph to coincide with a
|
||
lune-based β-skeleton for <code class="literal">β = 2</code>. In igraph, there is a subtle
|
||
difference: the <code class="literal">β = 2</code> skeleton connects points A and B when there
|
||
is no point C so that AC <= AB and BC <= AB. Therefore, three points
|
||
forming an equilateral triangle are connected in the relative neighborhood graph,
|
||
but disconnected in the <code class="literal">β = 2</code> skeleton.
|
||
|
||
</p>
|
||
<p>
|
||
With these definitions, the relative neighborhood graph is always connected,
|
||
while the <code class="literal">β = 2</code> skeleton is always triangle-free.
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
A pointer to the graph that will be created.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
|
||
<td><p>
|
||
The point set that will be used, each row is a point.
|
||
Dimensionality is inferred from the column number.
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
</p>
|
||
<p><b>See also: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
<a class="link" href="igraph-Spatial.html#igraph_lune_beta_skeleton" title="2.5. igraph_lune_beta_skeleton — The lune based β-skeleton of a spatial point set."><code class="function">igraph_lune_beta_skeleton()</code></a> to compute the lune based β-skeleton
|
||
for <code class="literal">β = 2</code> or other β values.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time Complexity: Around O(n^floor(d/2) log n), where n is the number of points
|
||
and d is the dimensionality of the point set.
|
||
|
||
|
||
</p>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h3 class="title">
|
||
<a name="igraph_lune_beta_skeleton"></a>2.5. <code class="function">igraph_lune_beta_skeleton</code> — The lune based β-skeleton of a spatial point set.</h3></div></div></div>
|
||
<a class="indexterm" name="id-1.15.3.6.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_lune_beta_skeleton(igraph_t *graph, const igraph_matrix_t *points, igraph_real_t beta);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
</p>
|
||
<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
|
||
<h3 class="title">Warning</h3>
|
||
<p>This function is experimental and its signature is not considered final yet.
|
||
We reserve the right to change the function signature without changing the
|
||
major version of igraph. Use it at your own risk.</p>
|
||
</div>
|
||
<p>This function constructs the lune-based β-skeleton of an n-dimensional
|
||
spatial point set.
|
||
|
||
</p>
|
||
<p>
|
||
A larger β results in a larger region, and a sparser graph.
|
||
Values of β < 1 are only supported in 2D, and are considerably slower.
|
||
|
||
</p>
|
||
<p>
|
||
The Gabriel graph is a special case of beta skeleton where <code class="literal">β = 1</code>.
|
||
|
||
</p>
|
||
<p>
|
||
The Relative Neighborhood graph is a special case of beta skeleton where
|
||
β approaches
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
A pointer to the graph that will be created.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
|
||
<td><p>
|
||
A matrix containing the points that will be used to create the
|
||
graph. Each row is a point, dimensionality is inferred from the column count.</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time Complexity: Around O(n^floor(d/2) log n), where n is the number of points
|
||
and d is the dimensionality of the point set.
|
||
|
||
|
||
</p>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h3 class="title">
|
||
<a name="igraph_circle_beta_skeleton"></a>2.6. <code class="function">igraph_circle_beta_skeleton</code> — The circle based β-skeleton of a 2D spatial point set.</h3></div></div></div>
|
||
<a class="indexterm" name="id-1.15.3.7.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_circle_beta_skeleton(igraph_t *graph, const igraph_matrix_t *points, igraph_real_t beta);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
</p>
|
||
<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
|
||
<h3 class="title">Warning</h3>
|
||
<p>This function is experimental and its signature is not considered final yet.
|
||
We reserve the right to change the function signature without changing the
|
||
major version of igraph. Use it at your own risk.</p>
|
||
</div>
|
||
<p>This function constructs the circle based β-skeleton of a 2D spatial point set.
|
||
|
||
</p>
|
||
<p>
|
||
A larger <em class="parameter"><code>beta</code></em> value results in a larger region, and a sparser graph.
|
||
Values of beta < 1 are considerably slower
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
A pointer to the graph that will be created.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
|
||
<td><p>
|
||
An n-by-2 matrix containing the points that will be used to
|
||
create the graph. Each row is a point.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>beta</code></em>:</span></p></td>
|
||
<td><p>
|
||
A positive real value used to parameterize the graph.
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time Complexity: Around O(n^floor(d/2) log n), where n is the number of points
|
||
and d is the dimensionality of the point set.
|
||
|
||
|
||
</p>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h3 class="title">
|
||
<a name="igraph_beta_weighted_gabriel_graph"></a>2.7. <code class="function">igraph_beta_weighted_gabriel_graph</code> — A Gabriel graph, with edges weighted by the β value at which it disappears.</h3></div></div></div>
|
||
<a class="indexterm" name="id-1.15.3.8.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_beta_weighted_gabriel_graph(
|
||
igraph_t *graph,
|
||
igraph_vector_t *weights,
|
||
const igraph_matrix_t *points,
|
||
igraph_real_t max_beta);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
</p>
|
||
<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
|
||
<h3 class="title">Warning</h3>
|
||
<p>This function is experimental and its signature is not considered final yet.
|
||
We reserve the right to change the function signature without changing the
|
||
major version of igraph. Use it at your own risk.</p>
|
||
</div>
|
||
<p>This function generates a Gabriel graph, and for each edge of this graph it
|
||
computes the threshold β value at which the edge ceases to be part of the
|
||
lune-based β-skeleton. For edges that continue to be part of β-skeletons
|
||
for arbitrarily large β, <code class="constant">IGRAPH_INFINITΥ</code> is returned.
|
||
|
||
</p>
|
||
<p>
|
||
The <em class="parameter"><code>max_beta</code></em> cutoff parameter controls the largest β value to consider
|
||
For edges that persist above this β value, <code class="constant">IGRAPH_INFINITΥ</code> is returned.
|
||
This parameter serves to improve performance: the smaller this cutoff,
|
||
the faster the computation. Pass <code class="constant">IGRAPH_INFINITY</code> to use no cutoff.
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
A pointer to the graph that will be created.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>weights</code></em>:</span></p></td>
|
||
<td><p>
|
||
Will contain the edge weights corresponding to the edge
|
||
indices from the graph.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
|
||
<td><p>
|
||
A matrix containing the points that will be used.
|
||
Each row is a point, dimensionality is inferred from the column count.
|
||
There must be no duplicate points.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>max_beta</code></em>:</span></p></td>
|
||
<td><p>
|
||
Maximum value of beta to search to, higher values will be
|
||
represented as <code class="constant">IGRAPH_INFINITY</code>.
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
</p>
|
||
<p><b>See also: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
<a class="link" href="igraph-Spatial.html#igraph_lune_beta_skeleton" title="2.5. igraph_lune_beta_skeleton — The lune based β-skeleton of a spatial point set."><code class="function">igraph_lune_beta_skeleton()</code></a> or <a class="link" href="igraph-Spatial.html#igraph_circle_beta_skeleton" title="2.6. igraph_circle_beta_skeleton — The circle based β-skeleton of a 2D spatial point set."><code class="function">igraph_circle_beta_skeleton()</code></a>
|
||
to generate a graph with a given value of beta; <a class="link" href="igraph-Spatial.html#igraph_gabriel_graph" title="2.3. igraph_gabriel_graph — The Gabriel graph of a point set."><code class="function">igraph_gabriel_graph()</code></a>
|
||
to only generate a Gabriel graph, without edge weights.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
|
||
Time Complexity: Around O(n^floor(d/2) log n), where n is the number of points
|
||
and d is the dimensionality of the point set. Though large values of max_beta
|
||
can cause long run times if there are edges that disappear only at large betas.
|
||
|
||
|
||
</p>
|
||
</div>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
<a name="spatial-properties"></a>3. Properties of spatial graphs</h2></div></div></div>
|
||
<div class="toc"><dl class="toc"><dt><span class="section"><a href="igraph-Spatial.html#igraph_spatial_edge_lengths">3.1. <code class="function">igraph_spatial_edge_lengths</code> — Edge lengths based on spatial vertex coordinates.</a></span></dt></dl></div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h3 class="title">
|
||
<a name="igraph_spatial_edge_lengths"></a>3.1. <code class="function">igraph_spatial_edge_lengths</code> — Edge lengths based on spatial vertex coordinates.</h3></div></div></div>
|
||
<a class="indexterm" name="id-1.15.4.2.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_spatial_edge_lengths(
|
||
const igraph_t *graph,
|
||
igraph_vector_t *lengths,
|
||
const igraph_matrix_t *points,
|
||
igraph_metric_t metric);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
</p>
|
||
<div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;">
|
||
<h3 class="title">Warning</h3>
|
||
<p>This function is experimental and its signature is not considered final yet.
|
||
We reserve the right to change the function signature without changing the
|
||
major version of igraph. Use it at your own risk.</p>
|
||
</div>
|
||
<p>The length of each edge is computed based on spatial coordinates. The length
|
||
can be employed by several igraph functions, such as <a class="link" href="igraph-Structural.html#igraph_voronoi" title="3.28. igraph_voronoi — Voronoi partitioning of a graph."><code class="function">igraph_voronoi()</code></a>,
|
||
<a class="link" href="igraph-Structural.html#igraph_betweenness" title="11.3. igraph_betweenness — Betweenness centrality of some vertices."><code class="function">igraph_betweenness()</code></a>, <a class="link" href="igraph-Structural.html#igraph_closeness" title="11.1. igraph_closeness — Closeness centrality calculations for some vertices."><code class="function">igraph_closeness()</code></a> and others.
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>graph</code></em>:</span></p></td>
|
||
<td><p>
|
||
The graph whose edge lengths are to be computed.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>lengths</code></em>:</span></p></td>
|
||
<td><p>
|
||
An initialized vector. Length will be stored here, in the
|
||
order of edge IDs. It will be resized as needed.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>points</code></em>:</span></p></td>
|
||
<td><p>
|
||
A matrix of vertex coordinates. Each row contains the
|
||
coordinates of the corresponding vertex, in the order of vertex IDs.
|
||
Arbitrary dimensional point sets are supported.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>metric</code></em>:</span></p></td>
|
||
<td><p>
|
||
The distance metric to use. See <a class="link" href="igraph-Spatial.html#igraph_metric_t" title="1.1. igraph_metric_t — Metric functions for use with spatial computation."><code class="function">igraph_metric_t</code></a> for
|
||
valid values.
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
</p>
|
||
<p><b>See also: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
<a class="link" href="igraph-Spatial.html#igraph_nearest_neighbor_graph" title="2.2. igraph_nearest_neighbor_graph — Computes the nearest neighbor graph for a spatial point set."><code class="function">igraph_nearest_neighbor_graph()</code></a> computes a k nearest neighbor graph
|
||
and <a class="link" href="igraph-Spatial.html#igraph_delaunay_graph" title="2.1. igraph_delaunay_graph — Computes the Delaunay graph of a spatial point set."><code class="function">igraph_delaunay_graph()</code></a> computes a Delaunay graph based on a set of
|
||
spatial points.
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time complexity: O(|E| d) where |E| is the number of edges and d is the
|
||
dimensionality of the point set.
|
||
|
||
</p>
|
||
</div>
|
||
</div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||
<a name="nongraph-spatial"></a>4. Non-graph related spatial processing</h2></div></div></div>
|
||
<div class="toc"><dl class="toc"><dt><span class="section"><a href="igraph-Spatial.html#igraph_convex_hull_2d">4.1. <code class="function">igraph_convex_hull_2d</code> — Determines the convex hull of a given set of points in the 2D plane.</a></span></dt></dl></div>
|
||
<div class="section">
|
||
<div class="titlepage"><div><div><h3 class="title">
|
||
<a name="igraph_convex_hull_2d"></a>4.1. <code class="function">igraph_convex_hull_2d</code> — Determines the convex hull of a given set of points in the 2D plane.</h3></div></div></div>
|
||
<a class="indexterm" name="id-1.15.5.2.2"></a><p>
|
||
</p>
|
||
<div class="informalexample"><pre class="programlisting">
|
||
igraph_error_t igraph_convex_hull_2d(
|
||
const igraph_matrix_t *data,
|
||
igraph_vector_int_t *resverts,
|
||
igraph_matrix_t *rescoords);
|
||
</pre></div>
|
||
<p>
|
||
</p>
|
||
<p>
|
||
|
||
|
||
|
||
</p>
|
||
<p>
|
||
The convex hull is determined by the Graham scan algorithm.
|
||
See the following reference for details:
|
||
|
||
</p>
|
||
<p>
|
||
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford
|
||
Stein. Introduction to Algorithms, Second Edition. MIT Press and
|
||
McGraw-Hill, 2001. ISBN 0262032937. Pages 949-955 of section 33.3:
|
||
Finding the convex hull.
|
||
|
||
</p>
|
||
<p><b>Arguments: </b>
|
||
</p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>data</code></em>:</span></p></td>
|
||
<td><p>
|
||
vector containing the coordinates. The length of the
|
||
vector must be even, since it contains X-Y coordinate pairs.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>resverts</code></em>:</span></p></td>
|
||
<td><p>
|
||
the vector containing the result, e.g. the vector of
|
||
vertex indices used as the corners of the convex hull. Supply
|
||
<code class="constant">NULL</code> here if you are only interested in the coordinates of
|
||
the convex hull corners.
|
||
</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td><p><span class="term"><em class="parameter"><code>rescoords</code></em>:</span></p></td>
|
||
<td><p>
|
||
the matrix containing the coordinates of the selected
|
||
corner vertices. Supply <code class="constant">NULL</code> here if you are only interested in
|
||
the vertex indices.
|
||
</p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<p>
|
||
</p>
|
||
<p><b>Returns: </b></p>
|
||
<div class="variablelist"><table border="0" class="variablelist">
|
||
<colgroup>
|
||
<col align="left" valign="top">
|
||
<col>
|
||
</colgroup>
|
||
<tbody><tr>
|
||
<td><p><span class="term"><em class="parameter"><code></code></em></span></p></td>
|
||
<td><p>
|
||
Error code:
|
||
<code class="constant">IGRAPH_ENOMEM</code>: not enough memory
|
||
</p></td>
|
||
</tr></tbody>
|
||
</table></div>
|
||
<p>
|
||
|
||
Time complexity: O(n log(n)) where n is the number of vertices.
|
||
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<table class="navigation-footer" width="100%" summary="Navigation footer" cellpadding="2" cellspacing="0"><tr valign="middle">
|
||
<td align="left"><a accesskey="p" href="igraph-Bipartite.html"><b>← Chapter 13. Bipartite, i.e. two-mode graphs</b></a></td>
|
||
<td align="right"><a accesskey="n" href="igraph-Operators.html"><b>Chapter 15. Graph operators →</b></a></td>
|
||
</tr></table>
|
||
</body>
|
||
</html>
|