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templateThe maximum_cycle_ratio() function calculates the maximum cycle ratio of a weighted directed multigraph G=(V,E,W1,W2), where V is a vertex set, E is an edge set, W1 and W2 are edge weight functions, W2 is nonnegative. As a multigraph, G can have multiple edges connecting a pair of vertices.dobule maximum_cycle_ratio(const Graph &g, VertexIndexMap vim, EdgeWeight1 ewm, EdgeWeight2 ew2m, std::vector ::edge_descriptor> *pcc = 0); template typename FloatTraits::float_type maximum_cycle_ratio(const Graph &g, VertexIndexMap vim, EdgeWeight1 ewm, EdgeWeight2 ew2m, std::vector ::edge_descriptor> *pcc = 0, FloatTraits = FloatTraits()); template dobule minimum_cycle_ratio(const Graph &g, VertexIndexMap vim, EdgeWeight1 ewm, EdgeWeight2 ew2m, std::vector ::edge_descriptor> *pcc = 0); template Graph, typename VertexIndexMap, typename EdgeWeight1, typename EdgeWeight2> typename FloatTraits::float_type minimum_cycle_ratio(const Graph &g, VertexIndexMap vim, EdgeWeight1 ewm, EdgeWeight2 ew2m, std::vector ::edge_descriptor> *pcc = 0, FloatTraits = FloatTraits());
Let every edge e has two weights W1(e) and W2(e). Let c be a cycle of the graphg. Then, the cycle ratio, cr(c), is defined as:
The algorithm is due to Howard's iteration policy algorithm, descibed in [1]. Ali Dasdan, Sandy S. Irani and Rajesh K.Gupta in their paper Efficient Algorithms for Optimum Cycle Mean and Optimum Cost to Time Ratio Problemsstate that this is the most efficient algorithm to the time being (1999).
For the convenience, functions maximum_cycle_mean() and minimum_cycle_mean() are also provided. They have the following signatures:
templatedouble maximum_cycle_mean(const Graph &g, VertexIndexMap vim, EdgeWeightMap ewm, EdgeIndexMap eim, std::vector ::edge_descriptor> *pcc = 0); template typename FloatTraits::float_type maximum_cycle_mean(const Graph &g, VertexIndexMap vim, EdgeWeightMap ewm, EdgeIndexMap eim, std::vector ::edge_descriptor> *pcc = 0, FloatTraits = FloatTraits()); template double minimum_cycle_mean(const Graph &g, VertexIndexMap vim, EdgeWeightMap ewm, EdgeIndexMap eim, std::vector ::edge_descriptor> *pcc = 0); template typename FloatTraits::float_type minimum_cycle_mean(const Graph &g, VertexIndexMap vim, EdgeWeightMap ewm, EdgeIndexMap eim, std::vector ::edge_descriptor> *pcc = 0, FloatTraits = FloatTraits());
boost/graph/howard_cycle_ratio.hpp
IN: FloatTraits
The FloatTrats encapsulates customizable limits-like information for floating point types. This type must provide an associated type, value_type for the floating point type. The default value is boost::mcr_float<>which has the following definition:
templateThe value FloatTraits::epsilon() controls the "tolerance" of the algorithm. By increasing the absolute value of epsilon you may slightly decrease the execution time but there is a risk to skip a global optima. By decreasing the absolute value you may fall to the infinite loop. The best option is to leave this parameter unchanged.struct mcr_float { typedef Float value_type; static Float infinity() { return (std::numeric_limits ::max)(); } static Float epsilon() { return Float(-0.005); } };
IN: const Graph& g
A weighted directed multigraph. The graph's type must be a model of VertexListGraph and IncidenceGraph
IN: VertexIndexMap vim
Maps each vertex of the graph to a unique integer in the range [0, num_vertices(g)).
IN: EdgeWeight1 ew1m
The W1 edge weight function.
IN: EdgeWeight2 ew2m
The W2 edge weight function. Should be nonnegative. The actual limitation of the algorithm is the positivity of the total weight of each directed cycle of the graph.
OUT: std::vector
If non zero then one critical cycle will be stored in the std::vector. Default value is 0.
IN (only for maximum/minimal_cycle_mean()): EdgeIndexMap eim
Maps each edge of the graph to a unique integer in the range [0, num_edges(g)).
All property maps must be models of Readable Property Map
There is no known precise upper bound for the time complexity of the algorithm. Imperical time complexity is O(|E|), where the constant tends to be pretty small (about 20-30). Space complexity is equal to 7*|V| plus the space required to store a graph.
The program in libs/graph/example/cycle_ratio_example.cpp generates a random graph and computes its maximum cycle ratio.
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