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337 lines
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Plaintext
337 lines
12 KiB
Plaintext
[/
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/ Copyright (c) 2007 Andrew Sutton
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/
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/ Distributed under the Boost Software License, Version 1.0. (See accompanying
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/ file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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/]
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[section Eccentricity]
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[template ex_eccentricity[] [link
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boost_graph.reference.algorithms.measures.eccentricity.examples.eccentricity__radius__and_diameter
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Eccentricity Example]]
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[heading Overview]
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The /eccentricity/ of a vertex is the maximum geodesic distance (shortest path) from
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that vertex to any other vertex in the graph. Intuitively, vertices with low
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eccentricity can reach all other vertices in the graph more "quickly" than vertices
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with high eccentricity. Eccentricity values are used to help compute the /radius/ and
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/diameter/.
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Note that if vertex /u/ is disconnected from any other vertex /v/ the distance from
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/u/ to /v/ is infinite. This implies the eccentricity of /v/ is infinite as well.
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If any vertex in a graph has infinite eccentricity, the graph also has an infinite
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diameter.
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Consider the following social network represented by an undirected, unweighted
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graph.
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[figure
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images/reference/social_network.png
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*Figure 1.* A network of friends
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]
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Because the graph is small and relatively well, connected the eccentricies are
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exaclty 2 or 3 for all vertices. The radius (the smaller of these numbers)
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is 2, and the diameter is 3. See the [ex_eccentricity] for details on computing
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eccentricity and related measures.
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[section [^eccentricity()]]
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#include <boost/graph/eccentricity.hpp>
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template <typename Graph, typename DistanceMap>
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typename property_traits<DistanceMap>::value_type
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eccentricity(const Graph& g, DistanceMap dm)
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The `eccentricity()` function computes the eccentricty of a single vertex
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given the shortest paths to every other vertex in the graph. The distance map can
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be computed using a shortest paths algorithms (e.g., [dijkstra_shortest_paths] or a
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[breadth_first_search]. The output of this function is returned to the caller.
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[heading Parameters]
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[table
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[[Type] [Parameter] [Description]]
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[
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[required, in] [`const Graph& g`]
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[
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The graph for which the eccentricity measures are being comptued.
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*Requirements:* The `Graph` type must be a model of the
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[VertexListGraph] concepts.
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]
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]
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[
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[required, in] [`DistanceMap dm`]
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[
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Given a vertex `v`, the `dm` parameter provides the length of the
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shortest path between a vertex `u` and `v`. The vertex `u` is the
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vertex for which the distance map was initially computed.
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*Requirements:* `DistanceMap` must be a model of [ReadablePropertyMap].
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The `key_type` of the distance map must be the same as the `vertex_descriptor`
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of the `Graph` parameter. The `value_type` is required to be a model of
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[NumericValue].
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]
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]
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]
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[heading Return Value]
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The `eccentricity()` function returns the eccentricity of the vertex. If the graph
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is unconnected, this returns the infinite value of the distance map's `value_type`.
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[heading Complexity]
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The `eccentricity()` returns in linear with respect to the number of vertices
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in the graph.
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[endsect]
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[section [^eccentricities()]]
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#include <boost/graph/eccentricity.hpp>
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template <typename Graph, typename DistanceMatrixMap, typename EccentricityMap>
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std::pair<typename property_traits<EccentricityMap>::value_type,
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typename property_traits<EccentricityMap>::value_type>
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eccentricity(const Graph& g, DistanceMatrixMap dm, EccentricityMap em)
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The `eccentricities()` function computes the eccentricity of each vertix in the graph,
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given a matrix that provides the shortest paths between all pairs of vertices. This
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matrix can be computed as the output of an all-pairs shortest paths algorithm (e.g., [floyd_warshall_all_pairs_shortest_paths]) or repeated application of a
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[breadth_first_search] (if the graph is unweighted). Theoutput of this function is
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stored in the eccentricity property map.
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This function also computes the radius and diameter of the graph, returning those
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values in a pair.
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[heading Parameters]
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[table
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[[Type] [Parameter] [Description]]
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[
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[required, in] [`const Graph& g`]
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[
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The graph for which the eccentricity measures are being comptued.
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*Requirements:* The `Graph` type must be a model of the
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[VertexListGraph] concepts.
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]
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]
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[
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[required, in] [`DistanceMatrixMap dm`]
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[
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Given vertices /u/ and /v/, the `dm` parameter provides the length
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of the shortest path between the two vertices. Note that the
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distance between a vertex and itself should be 0 (i.e., `dm[u][u] == 0`).
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*Requirements:* `DistanceMatrixMap` must be a model of [ReadablePropertyMap].
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The `key_type` of the distance matrixc must be the same as the `vertex_descriptor`
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of the `Graph` parameter. The `value_type` must be a [ReadWritePropertyMap]
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whose `key_type` is also the `vertex_descriptor` of the `Graph` and
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whose `value_type` is a model of [NumericValue].
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]
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]
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[
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[required, out] [`EccentricityMap em`]
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[
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The eccentricity map `em` stores the computed eccentricity values for
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each vertex in the graph.
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*Requirements:* The `EccentricityMap` type must model the [WritablePropertyMap]
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concept. The `key_type` of the property map must be the `vertex_descriptor`
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of the `Graph` parameter. The `value_type` must be the same type as the
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`value_type` of the `DistanceMatrixMap` parameter.
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]
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]
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]
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[heading Return]
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The `eccentricities()` function returns a pair containing the radius and diameter
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as the `first` and `second` elements respectively.
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[heading Complexity]
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The `eccentricities()` function returns in ['O(n[sup 2])] where /n/ is the number
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of vertices in the graph.
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[endsect]
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[section [^radius()]]
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#include <boost/graph/eccentricity.hpp>
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template <typename Graph, typename EccentricityMap>
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inline typename property_traits<EccentricityMap>::value_type
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radius(const Graph& g, EccentricityMap ecc)
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The `radius()` function returns the radius of the graph, which is defined as the
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minimum eccentricity of any vertex in the graph. The radius is computed from the
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eccentricities that have been previously computed using the [eccentricity] function.
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[heading Parameters]
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[table
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[[Type] [Parameter] [Description]]
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[
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[required, in] [`const Graph& g`]
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[
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The graph for which the eccentricity measures are being comptued.
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*Requirements:* The `Graph` type must be a model of the
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[VertexListGraph] concepts.
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]
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]
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[
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[required, out] [`EccentricityMap em`]
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[
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The eccentricity map `em` provides access to the previously computed
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eccentricities of all vertices in the graph.
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*Requirements:* The `EccentricityMap` type must model the [ReadablePropertyMap]
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concept. The `key_type` of the property map must be the `vertex_descriptor`
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of the `Graph` parameter. The `value_type` must model the [NumericValue]
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concept.
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]
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]
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]
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[heading Return Value]
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The `radius()` function return the radius of the graph. The return type of
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this function is the same as the `value_type` of the `EccentricityMap` parameter.
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[heading Complexity]
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The `radius()` function returns in linear time with respect to the number
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of vertices in the graph.
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[endsect]
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[section [^diameter()]]
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#include <boost/graph/eccentricity.hpp>
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template <typename Graph, typename EccentricityMap>
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inline typename property_traits<EccentricityMap>::value_type
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diameter(const Graph& g, EccentricityMap ecc)
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The `diameter()` function returns the diameter of the graph, which is defined as the
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maximum eccentricity of any vertex in the graph. The diameter is computed from the
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eccentricities that have been previously computed using the [eccentricity] function.
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[heading Parameters]
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[table
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[[Type] [Parameter] [Description]]
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[
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[required, in] [`const Graph& g`]
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[
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The graph for which the eccentricity measures are being comptued.
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*Requirements:* The `Graph` type must be a model of the
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[VertexListGraph] concepts.
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]
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]
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[
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[required, out] [`EccentricityMap em`]
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[
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The eccentricity map `em` provides access to the previously computed
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eccentricities of all vertices in the graph.
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*Requirements:* The `EccentricityMap` type must model the [ReadablePropertyMap]
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concept. The `key_type` of the property map must be the `vertex_descriptor`
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of the `Graph` parameter. The `value_type` must model the [NumericValue]
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concept.
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]
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]
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]
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[heading Return Value]
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The `diameter()` function return the diameter of the graph. If any vertices
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are disconnected from the graph, the diameter will be infinite. The return type of
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this function is the same as the `value_type` of the `EccentricityMap` parameter.
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[heading Complexity]
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The `diameter()` function returns in linear time with respect to the number
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of vertices in the graph.
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[endsect]
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[section [^radius_and_diameter()]]
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#include <boost/graph/eccentricity.hpp>
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template <typename Graph, typename EccentricityMap>
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std::pair<typename property_traits<EccentricityMap>::value_type,
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typename property_traits<EccentricityMap>::value_type>
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radius_and_diameter(const Graph& g, EccentricityMap ecc)
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The `radius_and_diameter()` function returns both the radius and diameter of the
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graph as a pair such that the `first` element of the pair is the radius and the
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`second` element is the diameter. These values are computed from the eccentricities
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that have been previously computed using the [eccentricity] function.
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[heading Parameters]
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[table
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[[Type] [Parameter] [Description]]
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[
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[required, in] [`const Graph& g`]
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[
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The graph for which the eccentricity measures are being comptued.
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*Requirements:* The `Graph` type must be a model of the
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[VertexListGraph] concepts.
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]
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]
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[
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[required, out] [`EccentricityMap em`]
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[
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The eccentricity map `em` provides access to the previously computed
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eccentricities of all vertices in the graph.
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*Requirements:* The `EccentricityMap` type must model the [ReadablePropertyMap]
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concept. The `key_type` of the property map must be the `vertex_descriptor`
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of the `Graph` parameter. The `value_type` must model the [NumericValue]
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concept.
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]
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]
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]
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[heading Return Value]
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The `radius_and_diameter()` funtion returns a pair containing both the radius (`first`)
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and diameter (`second`) of the graph. The types of these values are the same as the
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`value_type` of the `EccentricityMap`.
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[heading Complexity]
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The `radius_radius_diameter()` returns in linear time with respect to the number of vertices
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in the graph.
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[endsect]
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[section Examples]
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[heading Eccentricity, Radius, and Diameter]
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This example computes the eccentricities for all vertices in a graph, and the radius
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and diameter of the graph itself. This example includes the following files:
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* [^examples/eccentricity.cpp]
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* [^examples/helper.hpp]
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[eccentricity_example]
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Note that we can also compute the radius and diameter separatel. For exmaple:
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// Compute the radius and diameter after computing eccentricites,
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// but separately.
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int r = radius(g, em);
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int d = diameter(g, em);
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// Compute the radius and diameter after computing eccentricities,
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// but in one step.
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int r, d;
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tie(r, d) = radius_diameter(g, em);
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For small graphs, choosing one method over the other makes little difference. For
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large graphs, however, multiple iterations over the vertex set may result in lower
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performance.
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If given the `social_network.graph` file as input, the output will be:
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[pre
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Scott 2
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Jill 3
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Mary 3
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Bill 3
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Josh 3
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Frank 2
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Laurie 3
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Anne 3
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Howard 3
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radius: 2
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diamter: 3
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]
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[endsect]
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[endsect] |