1231 lines
41 KiB
C++
Executable File
1231 lines
41 KiB
C++
Executable File
/* -*- mode: C++; indent-tabs-mode: nil; -*-
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*
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* This file is a part of LEMON, a generic C++ optimization library.
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*
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* Copyright (C) 2003-2013
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* Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
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* (Egervary Research Group on Combinatorial Optimization, EGRES).
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*
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* Permission to use, modify and distribute this software is granted
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* provided that this copyright notice appears in all copies. For
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* precise terms see the accompanying LICENSE file.
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*
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* This software is provided "AS IS" with no warranty of any kind,
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* express or implied, and with no claim as to its suitability for any
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* purpose.
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*
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*/
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#ifndef LEMON_CYCLE_CANCELING_H
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#define LEMON_CYCLE_CANCELING_H
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/// \ingroup min_cost_flow_algs
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/// \file
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/// \brief Cycle-canceling algorithms for finding a minimum cost flow.
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#include <vector>
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#include <limits>
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#include <lemon/core.h>
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#include <lemon/maps.h>
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#include <lemon/path.h>
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#include <lemon/math.h>
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#include <lemon/static_graph.h>
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#include <lemon/adaptors.h>
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#include <lemon/circulation.h>
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#include <lemon/bellman_ford.h>
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#include <lemon/howard_mmc.h>
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#include <lemon/hartmann_orlin_mmc.h>
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namespace lemon {
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/// \addtogroup min_cost_flow_algs
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/// @{
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/// \brief Implementation of cycle-canceling algorithms for
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/// finding a \ref min_cost_flow "minimum cost flow".
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///
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/// \ref CycleCanceling implements three different cycle-canceling
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/// algorithms for finding a \ref min_cost_flow "minimum cost flow"
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/// \cite amo93networkflows, \cite klein67primal,
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/// \cite goldberg89cyclecanceling.
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/// The most efficent one is the \ref CANCEL_AND_TIGHTEN
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/// "Cancel-and-Tighten" algorithm, thus it is the default method.
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/// It runs in strongly polynomial time \f$O(n^2 m^2 \log n)\f$,
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/// but in practice, it is typically orders of magnitude slower than
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/// the scaling algorithms and \ref NetworkSimplex.
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/// (For more information, see \ref min_cost_flow_algs "the module page".)
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///
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/// Most of the parameters of the problem (except for the digraph)
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/// can be given using separate functions, and the algorithm can be
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/// executed using the \ref run() function. If some parameters are not
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/// specified, then default values will be used.
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///
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/// \tparam GR The digraph type the algorithm runs on.
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/// \tparam V The number type used for flow amounts, capacity bounds
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/// and supply values in the algorithm. By default, it is \c int.
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/// \tparam C The number type used for costs and potentials in the
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/// algorithm. By default, it is the same as \c V.
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///
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/// \warning Both \c V and \c C must be signed number types.
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/// \warning All input data (capacities, supply values, and costs) must
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/// be integer.
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/// \warning This algorithm does not support negative costs for
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/// arcs having infinite upper bound.
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///
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/// \note For more information about the three available methods,
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/// see \ref Method.
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#ifdef DOXYGEN
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template <typename GR, typename V, typename C>
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#else
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template <typename GR, typename V = int, typename C = V>
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#endif
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class CycleCanceling
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{
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public:
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/// The type of the digraph
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typedef GR Digraph;
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/// The type of the flow amounts, capacity bounds and supply values
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typedef V Value;
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/// The type of the arc costs
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typedef C Cost;
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public:
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/// \brief Problem type constants for the \c run() function.
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///
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/// Enum type containing the problem type constants that can be
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/// returned by the \ref run() function of the algorithm.
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enum ProblemType {
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/// The problem has no feasible solution (flow).
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INFEASIBLE,
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/// The problem has optimal solution (i.e. it is feasible and
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/// bounded), and the algorithm has found optimal flow and node
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/// potentials (primal and dual solutions).
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OPTIMAL,
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/// The digraph contains an arc of negative cost and infinite
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/// upper bound. It means that the objective function is unbounded
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/// on that arc, however, note that it could actually be bounded
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/// over the feasible flows, but this algroithm cannot handle
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/// these cases.
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UNBOUNDED
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};
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/// \brief Constants for selecting the used method.
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///
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/// Enum type containing constants for selecting the used method
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/// for the \ref run() function.
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///
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/// \ref CycleCanceling provides three different cycle-canceling
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/// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel-and-Tighten"
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/// is used, which is by far the most efficient and the most robust.
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/// However, the other methods can be selected using the \ref run()
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/// function with the proper parameter.
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enum Method {
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/// A simple cycle-canceling method, which uses the
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/// \ref BellmanFord "Bellman-Ford" algorithm for detecting negative
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/// cycles in the residual network.
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/// The number of Bellman-Ford iterations is bounded by a successively
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/// increased limit.
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SIMPLE_CYCLE_CANCELING,
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/// The "Minimum Mean Cycle-Canceling" algorithm, which is a
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/// well-known strongly polynomial method
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/// \cite goldberg89cyclecanceling. It improves along a
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/// \ref min_mean_cycle "minimum mean cycle" in each iteration.
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/// Its running time complexity is \f$O(n^2 m^3 \log n)\f$.
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MINIMUM_MEAN_CYCLE_CANCELING,
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/// The "Cancel-and-Tighten" algorithm, which can be viewed as an
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/// improved version of the previous method
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/// \cite goldberg89cyclecanceling.
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/// It is faster both in theory and in practice, its running time
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/// complexity is \f$O(n^2 m^2 \log n)\f$.
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CANCEL_AND_TIGHTEN
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};
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private:
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TEMPLATE_DIGRAPH_TYPEDEFS(GR);
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typedef std::vector<int> IntVector;
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typedef std::vector<double> DoubleVector;
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typedef std::vector<Value> ValueVector;
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typedef std::vector<Cost> CostVector;
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typedef std::vector<char> BoolVector;
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// Note: vector<char> is used instead of vector<bool> for efficiency reasons
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private:
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template <typename KT, typename VT>
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class StaticVectorMap {
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public:
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typedef KT Key;
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typedef VT Value;
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StaticVectorMap(std::vector<Value>& v) : _v(v) {}
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const Value& operator[](const Key& key) const {
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return _v[StaticDigraph::id(key)];
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}
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Value& operator[](const Key& key) {
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return _v[StaticDigraph::id(key)];
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}
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void set(const Key& key, const Value& val) {
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_v[StaticDigraph::id(key)] = val;
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}
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private:
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std::vector<Value>& _v;
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};
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typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
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typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
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private:
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// Data related to the underlying digraph
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const GR &_graph;
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int _node_num;
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int _arc_num;
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int _res_node_num;
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int _res_arc_num;
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int _root;
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// Parameters of the problem
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bool _has_lower;
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Value _sum_supply;
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// Data structures for storing the digraph
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IntNodeMap _node_id;
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IntArcMap _arc_idf;
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IntArcMap _arc_idb;
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IntVector _first_out;
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BoolVector _forward;
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IntVector _source;
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IntVector _target;
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IntVector _reverse;
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// Node and arc data
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ValueVector _lower;
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ValueVector _upper;
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CostVector _cost;
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ValueVector _supply;
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ValueVector _res_cap;
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CostVector _pi;
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// Data for a StaticDigraph structure
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typedef std::pair<int, int> IntPair;
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StaticDigraph _sgr;
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std::vector<IntPair> _arc_vec;
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std::vector<Cost> _cost_vec;
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IntVector _id_vec;
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CostArcMap _cost_map;
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CostNodeMap _pi_map;
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public:
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/// \brief Constant for infinite upper bounds (capacities).
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///
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/// Constant for infinite upper bounds (capacities).
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/// It is \c std::numeric_limits<Value>::infinity() if available,
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/// \c std::numeric_limits<Value>::max() otherwise.
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const Value INF;
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public:
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/// \brief Constructor.
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///
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/// The constructor of the class.
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///
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/// \param graph The digraph the algorithm runs on.
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CycleCanceling(const GR& graph) :
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_graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
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_cost_map(_cost_vec), _pi_map(_pi),
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INF(std::numeric_limits<Value>::has_infinity ?
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std::numeric_limits<Value>::infinity() :
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std::numeric_limits<Value>::max())
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{
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// Check the number types
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LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
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"The flow type of CycleCanceling must be signed");
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LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
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"The cost type of CycleCanceling must be signed");
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// Reset data structures
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reset();
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}
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/// \name Parameters
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/// The parameters of the algorithm can be specified using these
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/// functions.
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/// @{
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/// \brief Set the lower bounds on the arcs.
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///
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/// This function sets the lower bounds on the arcs.
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/// If it is not used before calling \ref run(), the lower bounds
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/// will be set to zero on all arcs.
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///
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/// \param map An arc map storing the lower bounds.
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/// Its \c Value type must be convertible to the \c Value type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template <typename LowerMap>
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CycleCanceling& lowerMap(const LowerMap& map) {
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_has_lower = true;
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for (ArcIt a(_graph); a != INVALID; ++a) {
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_lower[_arc_idf[a]] = map[a];
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}
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return *this;
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}
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/// \brief Set the upper bounds (capacities) on the arcs.
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///
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/// This function sets the upper bounds (capacities) on the arcs.
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/// If it is not used before calling \ref run(), the upper bounds
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/// will be set to \ref INF on all arcs (i.e. the flow value will be
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/// unbounded from above).
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///
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/// \param map An arc map storing the upper bounds.
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/// Its \c Value type must be convertible to the \c Value type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template<typename UpperMap>
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CycleCanceling& upperMap(const UpperMap& map) {
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for (ArcIt a(_graph); a != INVALID; ++a) {
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_upper[_arc_idf[a]] = map[a];
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}
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return *this;
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}
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/// \brief Set the costs of the arcs.
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///
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/// This function sets the costs of the arcs.
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/// If it is not used before calling \ref run(), the costs
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/// will be set to \c 1 on all arcs.
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///
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/// \param map An arc map storing the costs.
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/// Its \c Value type must be convertible to the \c Cost type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template<typename CostMap>
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CycleCanceling& costMap(const CostMap& map) {
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for (ArcIt a(_graph); a != INVALID; ++a) {
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_cost[_arc_idf[a]] = map[a];
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_cost[_arc_idb[a]] = -map[a];
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}
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return *this;
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}
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/// \brief Set the supply values of the nodes.
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///
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/// This function sets the supply values of the nodes.
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/// If neither this function nor \ref stSupply() is used before
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/// calling \ref run(), the supply of each node will be set to zero.
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///
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/// \param map A node map storing the supply values.
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/// Its \c Value type must be convertible to the \c Value type
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/// of the algorithm.
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///
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/// \return <tt>(*this)</tt>
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template<typename SupplyMap>
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CycleCanceling& supplyMap(const SupplyMap& map) {
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for (NodeIt n(_graph); n != INVALID; ++n) {
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_supply[_node_id[n]] = map[n];
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}
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return *this;
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}
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/// \brief Set single source and target nodes and a supply value.
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///
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/// This function sets a single source node and a single target node
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/// and the required flow value.
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/// If neither this function nor \ref supplyMap() is used before
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/// calling \ref run(), the supply of each node will be set to zero.
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///
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/// Using this function has the same effect as using \ref supplyMap()
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/// with a map in which \c k is assigned to \c s, \c -k is
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/// assigned to \c t and all other nodes have zero supply value.
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///
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/// \param s The source node.
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/// \param t The target node.
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/// \param k The required amount of flow from node \c s to node \c t
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/// (i.e. the supply of \c s and the demand of \c t).
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///
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/// \return <tt>(*this)</tt>
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CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
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for (int i = 0; i != _res_node_num; ++i) {
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_supply[i] = 0;
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}
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_supply[_node_id[s]] = k;
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_supply[_node_id[t]] = -k;
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return *this;
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}
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/// @}
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/// \name Execution control
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/// The algorithm can be executed using \ref run().
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/// @{
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/// \brief Run the algorithm.
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///
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/// This function runs the algorithm.
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/// The paramters can be specified using functions \ref lowerMap(),
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/// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
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/// For example,
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/// \code
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/// CycleCanceling<ListDigraph> cc(graph);
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/// cc.lowerMap(lower).upperMap(upper).costMap(cost)
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/// .supplyMap(sup).run();
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/// \endcode
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///
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/// This function can be called more than once. All the given parameters
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/// are kept for the next call, unless \ref resetParams() or \ref reset()
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/// is used, thus only the modified parameters have to be set again.
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/// If the underlying digraph was also modified after the construction
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/// of the class (or the last \ref reset() call), then the \ref reset()
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/// function must be called.
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///
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/// \param method The cycle-canceling method that will be used.
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/// For more information, see \ref Method.
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///
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/// \return \c INFEASIBLE if no feasible flow exists,
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/// \n \c OPTIMAL if the problem has optimal solution
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/// (i.e. it is feasible and bounded), and the algorithm has found
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/// optimal flow and node potentials (primal and dual solutions),
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/// \n \c UNBOUNDED if the digraph contains an arc of negative cost
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/// and infinite upper bound. It means that the objective function
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/// is unbounded on that arc, however, note that it could actually be
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/// bounded over the feasible flows, but this algroithm cannot handle
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/// these cases.
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///
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/// \see ProblemType, Method
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/// \see resetParams(), reset()
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ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
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ProblemType pt = init();
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if (pt != OPTIMAL) return pt;
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start(method);
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return OPTIMAL;
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}
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/// \brief Reset all the parameters that have been given before.
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///
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/// This function resets all the paramaters that have been given
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/// before using functions \ref lowerMap(), \ref upperMap(),
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/// \ref costMap(), \ref supplyMap(), \ref stSupply().
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///
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/// It is useful for multiple \ref run() calls. Basically, all the given
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/// parameters are kept for the next \ref run() call, unless
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/// \ref resetParams() or \ref reset() is used.
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/// If the underlying digraph was also modified after the construction
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/// of the class or the last \ref reset() call, then the \ref reset()
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/// function must be used, otherwise \ref resetParams() is sufficient.
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///
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/// For example,
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/// \code
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/// CycleCanceling<ListDigraph> cs(graph);
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///
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/// // First run
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/// cc.lowerMap(lower).upperMap(upper).costMap(cost)
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/// .supplyMap(sup).run();
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///
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/// // Run again with modified cost map (resetParams() is not called,
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/// // so only the cost map have to be set again)
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/// cost[e] += 100;
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/// cc.costMap(cost).run();
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///
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/// // Run again from scratch using resetParams()
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/// // (the lower bounds will be set to zero on all arcs)
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/// cc.resetParams();
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/// cc.upperMap(capacity).costMap(cost)
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/// .supplyMap(sup).run();
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/// \endcode
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///
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/// \return <tt>(*this)</tt>
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///
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/// \see reset(), run()
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CycleCanceling& resetParams() {
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for (int i = 0; i != _res_node_num; ++i) {
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_supply[i] = 0;
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}
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int limit = _first_out[_root];
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for (int j = 0; j != limit; ++j) {
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_lower[j] = 0;
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_upper[j] = INF;
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_cost[j] = _forward[j] ? 1 : -1;
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}
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for (int j = limit; j != _res_arc_num; ++j) {
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_lower[j] = 0;
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_upper[j] = INF;
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_cost[j] = 0;
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_cost[_reverse[j]] = 0;
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}
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_has_lower = false;
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return *this;
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}
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/// \brief Reset the internal data structures and all the parameters
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/// that have been given before.
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///
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/// This function resets the internal data structures and all the
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/// paramaters that have been given before using functions \ref lowerMap(),
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/// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
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///
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/// It is useful for multiple \ref run() calls. Basically, all the given
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/// parameters are kept for the next \ref run() call, unless
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/// \ref resetParams() or \ref reset() is used.
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/// If the underlying digraph was also modified after the construction
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/// of the class or the last \ref reset() call, then the \ref reset()
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/// function must be used, otherwise \ref resetParams() is sufficient.
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///
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/// See \ref resetParams() for examples.
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///
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/// \return <tt>(*this)</tt>
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///
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/// \see resetParams(), run()
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CycleCanceling& reset() {
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// Resize vectors
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_node_num = countNodes(_graph);
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_arc_num = countArcs(_graph);
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_res_node_num = _node_num + 1;
|
|
_res_arc_num = 2 * (_arc_num + _node_num);
|
|
_root = _node_num;
|
|
|
|
_first_out.resize(_res_node_num + 1);
|
|
_forward.resize(_res_arc_num);
|
|
_source.resize(_res_arc_num);
|
|
_target.resize(_res_arc_num);
|
|
_reverse.resize(_res_arc_num);
|
|
|
|
_lower.resize(_res_arc_num);
|
|
_upper.resize(_res_arc_num);
|
|
_cost.resize(_res_arc_num);
|
|
_supply.resize(_res_node_num);
|
|
|
|
_res_cap.resize(_res_arc_num);
|
|
_pi.resize(_res_node_num);
|
|
|
|
_arc_vec.reserve(_res_arc_num);
|
|
_cost_vec.reserve(_res_arc_num);
|
|
_id_vec.reserve(_res_arc_num);
|
|
|
|
// Copy the graph
|
|
int i = 0, j = 0, k = 2 * _arc_num + _node_num;
|
|
for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
|
|
_node_id[n] = i;
|
|
}
|
|
i = 0;
|
|
for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
|
|
_first_out[i] = j;
|
|
for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
|
|
_arc_idf[a] = j;
|
|
_forward[j] = true;
|
|
_source[j] = i;
|
|
_target[j] = _node_id[_graph.runningNode(a)];
|
|
}
|
|
for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
|
|
_arc_idb[a] = j;
|
|
_forward[j] = false;
|
|
_source[j] = i;
|
|
_target[j] = _node_id[_graph.runningNode(a)];
|
|
}
|
|
_forward[j] = false;
|
|
_source[j] = i;
|
|
_target[j] = _root;
|
|
_reverse[j] = k;
|
|
_forward[k] = true;
|
|
_source[k] = _root;
|
|
_target[k] = i;
|
|
_reverse[k] = j;
|
|
++j; ++k;
|
|
}
|
|
_first_out[i] = j;
|
|
_first_out[_res_node_num] = k;
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
int fi = _arc_idf[a];
|
|
int bi = _arc_idb[a];
|
|
_reverse[fi] = bi;
|
|
_reverse[bi] = fi;
|
|
}
|
|
|
|
// Reset parameters
|
|
resetParams();
|
|
return *this;
|
|
}
|
|
|
|
/// @}
|
|
|
|
/// \name Query Functions
|
|
/// The results of the algorithm can be obtained using these
|
|
/// functions.\n
|
|
/// The \ref run() function must be called before using them.
|
|
|
|
/// @{
|
|
|
|
/// \brief Return the total cost of the found flow.
|
|
///
|
|
/// This function returns the total cost of the found flow.
|
|
/// Its complexity is O(m).
|
|
///
|
|
/// \note The return type of the function can be specified as a
|
|
/// template parameter. For example,
|
|
/// \code
|
|
/// cc.totalCost<double>();
|
|
/// \endcode
|
|
/// It is useful if the total cost cannot be stored in the \c Cost
|
|
/// type of the algorithm, which is the default return type of the
|
|
/// function.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
template <typename Number>
|
|
Number totalCost() const {
|
|
Number c = 0;
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
int i = _arc_idb[a];
|
|
c += static_cast<Number>(_res_cap[i]) *
|
|
(-static_cast<Number>(_cost[i]));
|
|
}
|
|
return c;
|
|
}
|
|
|
|
#ifndef DOXYGEN
|
|
Cost totalCost() const {
|
|
return totalCost<Cost>();
|
|
}
|
|
#endif
|
|
|
|
/// \brief Return the flow on the given arc.
|
|
///
|
|
/// This function returns the flow on the given arc.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
Value flow(const Arc& a) const {
|
|
return _res_cap[_arc_idb[a]];
|
|
}
|
|
|
|
/// \brief Copy the flow values (the primal solution) into the
|
|
/// given map.
|
|
///
|
|
/// This function copies the flow value on each arc into the given
|
|
/// map. The \c Value type of the algorithm must be convertible to
|
|
/// the \c Value type of the map.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
template <typename FlowMap>
|
|
void flowMap(FlowMap &map) const {
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
map.set(a, _res_cap[_arc_idb[a]]);
|
|
}
|
|
}
|
|
|
|
/// \brief Return the potential (dual value) of the given node.
|
|
///
|
|
/// This function returns the potential (dual value) of the
|
|
/// given node.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
Cost potential(const Node& n) const {
|
|
return static_cast<Cost>(_pi[_node_id[n]]);
|
|
}
|
|
|
|
/// \brief Copy the potential values (the dual solution) into the
|
|
/// given map.
|
|
///
|
|
/// This function copies the potential (dual value) of each node
|
|
/// into the given map.
|
|
/// The \c Cost type of the algorithm must be convertible to the
|
|
/// \c Value type of the map.
|
|
///
|
|
/// \pre \ref run() must be called before using this function.
|
|
template <typename PotentialMap>
|
|
void potentialMap(PotentialMap &map) const {
|
|
for (NodeIt n(_graph); n != INVALID; ++n) {
|
|
map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
|
|
}
|
|
}
|
|
|
|
/// @}
|
|
|
|
private:
|
|
|
|
// Initialize the algorithm
|
|
ProblemType init() {
|
|
if (_res_node_num <= 1) return INFEASIBLE;
|
|
|
|
// Check the sum of supply values
|
|
_sum_supply = 0;
|
|
for (int i = 0; i != _root; ++i) {
|
|
_sum_supply += _supply[i];
|
|
}
|
|
if (_sum_supply > 0) return INFEASIBLE;
|
|
|
|
// Check lower and upper bounds
|
|
LEMON_DEBUG(checkBoundMaps(),
|
|
"Upper bounds must be greater or equal to the lower bounds");
|
|
|
|
|
|
// Initialize vectors
|
|
for (int i = 0; i != _res_node_num; ++i) {
|
|
_pi[i] = 0;
|
|
}
|
|
ValueVector excess(_supply);
|
|
|
|
// Remove infinite upper bounds and check negative arcs
|
|
const Value MAX = std::numeric_limits<Value>::max();
|
|
int last_out;
|
|
if (_has_lower) {
|
|
for (int i = 0; i != _root; ++i) {
|
|
last_out = _first_out[i+1];
|
|
for (int j = _first_out[i]; j != last_out; ++j) {
|
|
if (_forward[j]) {
|
|
Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
|
|
if (c >= MAX) return UNBOUNDED;
|
|
excess[i] -= c;
|
|
excess[_target[j]] += c;
|
|
}
|
|
}
|
|
}
|
|
} else {
|
|
for (int i = 0; i != _root; ++i) {
|
|
last_out = _first_out[i+1];
|
|
for (int j = _first_out[i]; j != last_out; ++j) {
|
|
if (_forward[j] && _cost[j] < 0) {
|
|
Value c = _upper[j];
|
|
if (c >= MAX) return UNBOUNDED;
|
|
excess[i] -= c;
|
|
excess[_target[j]] += c;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
Value ex, max_cap = 0;
|
|
for (int i = 0; i != _res_node_num; ++i) {
|
|
ex = excess[i];
|
|
if (ex < 0) max_cap -= ex;
|
|
}
|
|
for (int j = 0; j != _res_arc_num; ++j) {
|
|
if (_upper[j] >= MAX) _upper[j] = max_cap;
|
|
}
|
|
|
|
// Initialize maps for Circulation and remove non-zero lower bounds
|
|
ConstMap<Arc, Value> low(0);
|
|
typedef typename Digraph::template ArcMap<Value> ValueArcMap;
|
|
typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
|
|
ValueArcMap cap(_graph), flow(_graph);
|
|
ValueNodeMap sup(_graph);
|
|
for (NodeIt n(_graph); n != INVALID; ++n) {
|
|
sup[n] = _supply[_node_id[n]];
|
|
}
|
|
if (_has_lower) {
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
int j = _arc_idf[a];
|
|
Value c = _lower[j];
|
|
cap[a] = _upper[j] - c;
|
|
sup[_graph.source(a)] -= c;
|
|
sup[_graph.target(a)] += c;
|
|
}
|
|
} else {
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
cap[a] = _upper[_arc_idf[a]];
|
|
}
|
|
}
|
|
|
|
// Find a feasible flow using Circulation
|
|
Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
|
|
circ(_graph, low, cap, sup);
|
|
if (!circ.flowMap(flow).run()) return INFEASIBLE;
|
|
|
|
// Set residual capacities and handle GEQ supply type
|
|
if (_sum_supply < 0) {
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
Value fa = flow[a];
|
|
_res_cap[_arc_idf[a]] = cap[a] - fa;
|
|
_res_cap[_arc_idb[a]] = fa;
|
|
sup[_graph.source(a)] -= fa;
|
|
sup[_graph.target(a)] += fa;
|
|
}
|
|
for (NodeIt n(_graph); n != INVALID; ++n) {
|
|
excess[_node_id[n]] = sup[n];
|
|
}
|
|
for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
|
|
int u = _target[a];
|
|
int ra = _reverse[a];
|
|
_res_cap[a] = -_sum_supply + 1;
|
|
_res_cap[ra] = -excess[u];
|
|
_cost[a] = 0;
|
|
_cost[ra] = 0;
|
|
}
|
|
} else {
|
|
for (ArcIt a(_graph); a != INVALID; ++a) {
|
|
Value fa = flow[a];
|
|
_res_cap[_arc_idf[a]] = cap[a] - fa;
|
|
_res_cap[_arc_idb[a]] = fa;
|
|
}
|
|
for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
|
|
int ra = _reverse[a];
|
|
_res_cap[a] = 1;
|
|
_res_cap[ra] = 0;
|
|
_cost[a] = 0;
|
|
_cost[ra] = 0;
|
|
}
|
|
}
|
|
|
|
return OPTIMAL;
|
|
}
|
|
|
|
// Check if the upper bound is greater than or equal to the lower bound
|
|
// on each forward arc.
|
|
bool checkBoundMaps() {
|
|
for (int j = 0; j != _res_arc_num; ++j) {
|
|
if (_forward[j] && _upper[j] < _lower[j]) return false;
|
|
}
|
|
return true;
|
|
}
|
|
|
|
// Build a StaticDigraph structure containing the current
|
|
// residual network
|
|
void buildResidualNetwork() {
|
|
_arc_vec.clear();
|
|
_cost_vec.clear();
|
|
_id_vec.clear();
|
|
for (int j = 0; j != _res_arc_num; ++j) {
|
|
if (_res_cap[j] > 0) {
|
|
_arc_vec.push_back(IntPair(_source[j], _target[j]));
|
|
_cost_vec.push_back(_cost[j]);
|
|
_id_vec.push_back(j);
|
|
}
|
|
}
|
|
_sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
|
|
}
|
|
|
|
// Execute the algorithm and transform the results
|
|
void start(Method method) {
|
|
// Execute the algorithm
|
|
switch (method) {
|
|
case SIMPLE_CYCLE_CANCELING:
|
|
startSimpleCycleCanceling();
|
|
break;
|
|
case MINIMUM_MEAN_CYCLE_CANCELING:
|
|
startMinMeanCycleCanceling();
|
|
break;
|
|
case CANCEL_AND_TIGHTEN:
|
|
startCancelAndTighten();
|
|
break;
|
|
}
|
|
|
|
// Compute node potentials
|
|
if (method != SIMPLE_CYCLE_CANCELING) {
|
|
buildResidualNetwork();
|
|
typename BellmanFord<StaticDigraph, CostArcMap>
|
|
::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
|
|
bf.distMap(_pi_map);
|
|
bf.init(0);
|
|
bf.start();
|
|
}
|
|
|
|
// Handle non-zero lower bounds
|
|
if (_has_lower) {
|
|
int limit = _first_out[_root];
|
|
for (int j = 0; j != limit; ++j) {
|
|
if (_forward[j]) _res_cap[_reverse[j]] += _lower[j];
|
|
}
|
|
}
|
|
}
|
|
|
|
// Execute the "Simple Cycle Canceling" method
|
|
void startSimpleCycleCanceling() {
|
|
// Constants for computing the iteration limits
|
|
const int BF_FIRST_LIMIT = 2;
|
|
const double BF_LIMIT_FACTOR = 1.5;
|
|
|
|
typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
|
|
typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
|
|
typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
|
|
typedef typename BellmanFord<ResDigraph, CostArcMap>
|
|
::template SetDistMap<CostNodeMap>
|
|
::template SetPredMap<PredMap>::Create BF;
|
|
|
|
// Build the residual network
|
|
_arc_vec.clear();
|
|
_cost_vec.clear();
|
|
for (int j = 0; j != _res_arc_num; ++j) {
|
|
_arc_vec.push_back(IntPair(_source[j], _target[j]));
|
|
_cost_vec.push_back(_cost[j]);
|
|
}
|
|
_sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
|
|
|
|
FilterMap filter_map(_res_cap);
|
|
ResDigraph rgr(_sgr, filter_map);
|
|
std::vector<int> cycle;
|
|
std::vector<StaticDigraph::Arc> pred(_res_arc_num);
|
|
PredMap pred_map(pred);
|
|
BF bf(rgr, _cost_map);
|
|
bf.distMap(_pi_map).predMap(pred_map);
|
|
|
|
int length_bound = BF_FIRST_LIMIT;
|
|
bool optimal = false;
|
|
while (!optimal) {
|
|
bf.init(0);
|
|
int iter_num = 0;
|
|
bool cycle_found = false;
|
|
while (!cycle_found) {
|
|
// Perform some iterations of the Bellman-Ford algorithm
|
|
int curr_iter_num = iter_num + length_bound <= _node_num ?
|
|
length_bound : _node_num - iter_num;
|
|
iter_num += curr_iter_num;
|
|
int real_iter_num = curr_iter_num;
|
|
for (int i = 0; i < curr_iter_num; ++i) {
|
|
if (bf.processNextWeakRound()) {
|
|
real_iter_num = i;
|
|
break;
|
|
}
|
|
}
|
|
if (real_iter_num < curr_iter_num) {
|
|
// Optimal flow is found
|
|
optimal = true;
|
|
break;
|
|
} else {
|
|
// Search for node disjoint negative cycles
|
|
std::vector<int> state(_res_node_num, 0);
|
|
int id = 0;
|
|
for (int u = 0; u != _res_node_num; ++u) {
|
|
if (state[u] != 0) continue;
|
|
++id;
|
|
int v = u;
|
|
for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
|
|
-1 : rgr.id(rgr.source(pred[v]))) {
|
|
state[v] = id;
|
|
}
|
|
if (v != -1 && state[v] == id) {
|
|
// A negative cycle is found
|
|
cycle_found = true;
|
|
cycle.clear();
|
|
StaticDigraph::Arc a = pred[v];
|
|
Value d, delta = _res_cap[rgr.id(a)];
|
|
cycle.push_back(rgr.id(a));
|
|
while (rgr.id(rgr.source(a)) != v) {
|
|
a = pred_map[rgr.source(a)];
|
|
d = _res_cap[rgr.id(a)];
|
|
if (d < delta) delta = d;
|
|
cycle.push_back(rgr.id(a));
|
|
}
|
|
|
|
// Augment along the cycle
|
|
for (int i = 0; i < int(cycle.size()); ++i) {
|
|
int j = cycle[i];
|
|
_res_cap[j] -= delta;
|
|
_res_cap[_reverse[j]] += delta;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Increase iteration limit if no cycle is found
|
|
if (!cycle_found) {
|
|
length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Execute the "Minimum Mean Cycle Canceling" method
|
|
void startMinMeanCycleCanceling() {
|
|
typedef Path<StaticDigraph> SPath;
|
|
typedef typename SPath::ArcIt SPathArcIt;
|
|
typedef typename HowardMmc<StaticDigraph, CostArcMap>
|
|
::template SetPath<SPath>::Create HwMmc;
|
|
typedef typename HartmannOrlinMmc<StaticDigraph, CostArcMap>
|
|
::template SetPath<SPath>::Create HoMmc;
|
|
|
|
const double HW_ITER_LIMIT_FACTOR = 1.0;
|
|
const int HW_ITER_LIMIT_MIN_VALUE = 5;
|
|
|
|
const int hw_iter_limit =
|
|
std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
|
|
HW_ITER_LIMIT_MIN_VALUE);
|
|
|
|
SPath cycle;
|
|
HwMmc hw_mmc(_sgr, _cost_map);
|
|
hw_mmc.cycle(cycle);
|
|
buildResidualNetwork();
|
|
while (true) {
|
|
|
|
typename HwMmc::TerminationCause hw_tc =
|
|
hw_mmc.findCycleMean(hw_iter_limit);
|
|
if (hw_tc == HwMmc::ITERATION_LIMIT) {
|
|
// Howard's algorithm reached the iteration limit, start a
|
|
// strongly polynomial algorithm instead
|
|
HoMmc ho_mmc(_sgr, _cost_map);
|
|
ho_mmc.cycle(cycle);
|
|
// Find a minimum mean cycle (Hartmann-Orlin algorithm)
|
|
if (!(ho_mmc.findCycleMean() && ho_mmc.cycleCost() < 0)) break;
|
|
ho_mmc.findCycle();
|
|
} else {
|
|
// Find a minimum mean cycle (Howard algorithm)
|
|
if (!(hw_tc == HwMmc::OPTIMAL && hw_mmc.cycleCost() < 0)) break;
|
|
hw_mmc.findCycle();
|
|
}
|
|
|
|
// Compute delta value
|
|
Value delta = INF;
|
|
for (SPathArcIt a(cycle); a != INVALID; ++a) {
|
|
Value d = _res_cap[_id_vec[_sgr.id(a)]];
|
|
if (d < delta) delta = d;
|
|
}
|
|
|
|
// Augment along the cycle
|
|
for (SPathArcIt a(cycle); a != INVALID; ++a) {
|
|
int j = _id_vec[_sgr.id(a)];
|
|
_res_cap[j] -= delta;
|
|
_res_cap[_reverse[j]] += delta;
|
|
}
|
|
|
|
// Rebuild the residual network
|
|
buildResidualNetwork();
|
|
}
|
|
}
|
|
|
|
// Execute the "Cancel-and-Tighten" method
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void startCancelAndTighten() {
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// Constants for the min mean cycle computations
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const double LIMIT_FACTOR = 1.0;
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const int MIN_LIMIT = 5;
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const double HW_ITER_LIMIT_FACTOR = 1.0;
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const int HW_ITER_LIMIT_MIN_VALUE = 5;
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const int hw_iter_limit =
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std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
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HW_ITER_LIMIT_MIN_VALUE);
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// Contruct auxiliary data vectors
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DoubleVector pi(_res_node_num, 0.0);
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IntVector level(_res_node_num);
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BoolVector reached(_res_node_num);
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BoolVector processed(_res_node_num);
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IntVector pred_node(_res_node_num);
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IntVector pred_arc(_res_node_num);
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std::vector<int> stack(_res_node_num);
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std::vector<int> proc_vector(_res_node_num);
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// Initialize epsilon
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double epsilon = 0;
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for (int a = 0; a != _res_arc_num; ++a) {
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if (_res_cap[a] > 0 && -_cost[a] > epsilon)
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epsilon = -_cost[a];
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}
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// Start phases
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Tolerance<double> tol;
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tol.epsilon(1e-6);
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int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
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if (limit < MIN_LIMIT) limit = MIN_LIMIT;
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int iter = limit;
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while (epsilon * _res_node_num >= 1) {
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// Find and cancel cycles in the admissible network using DFS
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for (int u = 0; u != _res_node_num; ++u) {
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reached[u] = false;
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processed[u] = false;
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}
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int stack_head = -1;
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int proc_head = -1;
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for (int start = 0; start != _res_node_num; ++start) {
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if (reached[start]) continue;
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// New start node
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reached[start] = true;
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pred_arc[start] = -1;
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pred_node[start] = -1;
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// Find the first admissible outgoing arc
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double p = pi[start];
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int a = _first_out[start];
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int last_out = _first_out[start+1];
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for (; a != last_out && (_res_cap[a] == 0 ||
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!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
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if (a == last_out) {
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processed[start] = true;
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proc_vector[++proc_head] = start;
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continue;
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}
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stack[++stack_head] = a;
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while (stack_head >= 0) {
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int sa = stack[stack_head];
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int u = _source[sa];
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int v = _target[sa];
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if (!reached[v]) {
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// A new node is reached
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reached[v] = true;
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pred_node[v] = u;
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pred_arc[v] = sa;
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p = pi[v];
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a = _first_out[v];
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last_out = _first_out[v+1];
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for (; a != last_out && (_res_cap[a] == 0 ||
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!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
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stack[++stack_head] = a == last_out ? -1 : a;
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} else {
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if (!processed[v]) {
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// A cycle is found
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int n, w = u;
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Value d, delta = _res_cap[sa];
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for (n = u; n != v; n = pred_node[n]) {
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d = _res_cap[pred_arc[n]];
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if (d <= delta) {
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delta = d;
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w = pred_node[n];
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}
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}
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// Augment along the cycle
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_res_cap[sa] -= delta;
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_res_cap[_reverse[sa]] += delta;
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for (n = u; n != v; n = pred_node[n]) {
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int pa = pred_arc[n];
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_res_cap[pa] -= delta;
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_res_cap[_reverse[pa]] += delta;
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}
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for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
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--stack_head;
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reached[n] = false;
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}
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u = w;
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}
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v = u;
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// Find the next admissible outgoing arc
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p = pi[v];
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a = stack[stack_head] + 1;
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last_out = _first_out[v+1];
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for (; a != last_out && (_res_cap[a] == 0 ||
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!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
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stack[stack_head] = a == last_out ? -1 : a;
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}
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while (stack_head >= 0 && stack[stack_head] == -1) {
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processed[v] = true;
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proc_vector[++proc_head] = v;
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if (--stack_head >= 0) {
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// Find the next admissible outgoing arc
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v = _source[stack[stack_head]];
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p = pi[v];
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a = stack[stack_head] + 1;
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last_out = _first_out[v+1];
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for (; a != last_out && (_res_cap[a] == 0 ||
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!tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
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stack[stack_head] = a == last_out ? -1 : a;
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}
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}
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}
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}
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// Tighten potentials and epsilon
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if (--iter > 0) {
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for (int u = 0; u != _res_node_num; ++u) {
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level[u] = 0;
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}
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for (int i = proc_head; i > 0; --i) {
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int u = proc_vector[i];
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double p = pi[u];
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int l = level[u] + 1;
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int last_out = _first_out[u+1];
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for (int a = _first_out[u]; a != last_out; ++a) {
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int v = _target[a];
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if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
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l > level[v]) level[v] = l;
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}
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}
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// Modify potentials
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double q = std::numeric_limits<double>::max();
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for (int u = 0; u != _res_node_num; ++u) {
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int lu = level[u];
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double p, pu = pi[u];
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int last_out = _first_out[u+1];
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for (int a = _first_out[u]; a != last_out; ++a) {
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if (_res_cap[a] == 0) continue;
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int v = _target[a];
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int ld = lu - level[v];
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if (ld > 0) {
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p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
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if (p < q) q = p;
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}
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}
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}
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for (int u = 0; u != _res_node_num; ++u) {
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pi[u] -= q * level[u];
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}
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// Modify epsilon
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epsilon = 0;
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for (int u = 0; u != _res_node_num; ++u) {
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double curr, pu = pi[u];
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int last_out = _first_out[u+1];
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for (int a = _first_out[u]; a != last_out; ++a) {
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if (_res_cap[a] == 0) continue;
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curr = _cost[a] + pu - pi[_target[a]];
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if (-curr > epsilon) epsilon = -curr;
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}
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}
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} else {
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typedef HowardMmc<StaticDigraph, CostArcMap> HwMmc;
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typedef HartmannOrlinMmc<StaticDigraph, CostArcMap> HoMmc;
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typedef typename BellmanFord<StaticDigraph, CostArcMap>
|
|
::template SetDistMap<CostNodeMap>::Create BF;
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// Set epsilon to the minimum cycle mean
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Cost cycle_cost = 0;
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int cycle_size = 1;
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buildResidualNetwork();
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HwMmc hw_mmc(_sgr, _cost_map);
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if (hw_mmc.findCycleMean(hw_iter_limit) == HwMmc::ITERATION_LIMIT) {
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// Howard's algorithm reached the iteration limit, start a
|
|
// strongly polynomial algorithm instead
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HoMmc ho_mmc(_sgr, _cost_map);
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ho_mmc.findCycleMean();
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|
epsilon = -ho_mmc.cycleMean();
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cycle_cost = ho_mmc.cycleCost();
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cycle_size = ho_mmc.cycleSize();
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} else {
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// Set epsilon
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epsilon = -hw_mmc.cycleMean();
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cycle_cost = hw_mmc.cycleCost();
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cycle_size = hw_mmc.cycleSize();
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}
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|
|
// Compute feasible potentials for the current epsilon
|
|
for (int i = 0; i != int(_cost_vec.size()); ++i) {
|
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_cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
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}
|
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BF bf(_sgr, _cost_map);
|
|
bf.distMap(_pi_map);
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|
bf.init(0);
|
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bf.start();
|
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for (int u = 0; u != _res_node_num; ++u) {
|
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pi[u] = static_cast<double>(_pi[u]) / cycle_size;
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}
|
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|
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iter = limit;
|
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}
|
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}
|
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}
|
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|
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}; //class CycleCanceling
|
|
|
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///@}
|
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|
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} //namespace lemon
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|
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#endif //LEMON_CYCLE_CANCELING_H
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