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diff --git a/extern/quadriflow/3rd/lemon-1.3.1/lemon/cost_scaling.h b/extern/quadriflow/3rd/lemon-1.3.1/lemon/cost_scaling.h new file mode 100644 index 00000000000..efecdfe77c7 --- /dev/null +++ b/extern/quadriflow/3rd/lemon-1.3.1/lemon/cost_scaling.h @@ -0,0 +1,1607 @@ +/* -*- mode: C++; indent-tabs-mode: nil; -*- + * + * This file is a part of LEMON, a generic C++ optimization library. + * + * Copyright (C) 2003-2013 + * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport + * (Egervary Research Group on Combinatorial Optimization, EGRES). + * + * Permission to use, modify and distribute this software is granted + * provided that this copyright notice appears in all copies. For + * precise terms see the accompanying LICENSE file. + * + * This software is provided "AS IS" with no warranty of any kind, + * express or implied, and with no claim as to its suitability for any + * purpose. + * + */ + +#ifndef LEMON_COST_SCALING_H +#define LEMON_COST_SCALING_H + +/// \ingroup min_cost_flow_algs +/// \file +/// \brief Cost scaling algorithm for finding a minimum cost flow. + +#include <vector> +#include <deque> +#include <limits> + +#include <lemon/core.h> +#include <lemon/maps.h> +#include <lemon/math.h> +#include <lemon/static_graph.h> +#include <lemon/circulation.h> +#include <lemon/bellman_ford.h> + +namespace lemon { + + /// \brief Default traits class of CostScaling algorithm. + /// + /// Default traits class of CostScaling algorithm. + /// \tparam GR Digraph type. + /// \tparam V The number type used for flow amounts, capacity bounds + /// and supply values. By default it is \c int. + /// \tparam C The number type used for costs and potentials. + /// By default it is the same as \c V. +#ifdef DOXYGEN + template <typename GR, typename V = int, typename C = V> +#else + template < typename GR, typename V = int, typename C = V, + bool integer = std::numeric_limits<C>::is_integer > +#endif + struct CostScalingDefaultTraits + { + /// The type of the digraph + typedef GR Digraph; + /// The type of the flow amounts, capacity bounds and supply values + typedef V Value; + /// The type of the arc costs + typedef C Cost; + + /// \brief The large cost type used for internal computations + /// + /// The large cost type used for internal computations. + /// It is \c long \c long if the \c Cost type is integer, + /// otherwise it is \c double. + /// \c Cost must be convertible to \c LargeCost. + typedef double LargeCost; + }; + + // Default traits class for integer cost types + template <typename GR, typename V, typename C> + struct CostScalingDefaultTraits<GR, V, C, true> + { + typedef GR Digraph; + typedef V Value; + typedef C Cost; +#ifdef LEMON_HAVE_LONG_LONG + typedef long long LargeCost; +#else + typedef long LargeCost; +#endif + }; + + + /// \addtogroup min_cost_flow_algs + /// @{ + + /// \brief Implementation of the Cost Scaling algorithm for + /// finding a \ref min_cost_flow "minimum cost flow". + /// + /// \ref CostScaling implements a cost scaling algorithm that performs + /// push/augment and relabel operations for finding a \ref min_cost_flow + /// "minimum cost flow" \cite amo93networkflows, + /// \cite goldberg90approximation, + /// \cite goldberg97efficient, \cite bunnagel98efficient. + /// It is a highly efficient primal-dual solution method, which + /// can be viewed as the generalization of the \ref Preflow + /// "preflow push-relabel" algorithm for the maximum flow problem. + /// It is a polynomial algorithm, its running time complexity is + /// \f$O(n^2m\log(nK))\f$, where <i>K</i> denotes the maximum arc cost. + /// + /// In general, \ref NetworkSimplex and \ref CostScaling are the fastest + /// implementations available in LEMON for solving this problem. + /// (For more information, see \ref min_cost_flow_algs "the module page".) + /// + /// Most of the parameters of the problem (except for the digraph) + /// can be given using separate functions, and the algorithm can be + /// executed using the \ref run() function. If some parameters are not + /// specified, then default values will be used. + /// + /// \tparam GR The digraph type the algorithm runs on. + /// \tparam V The number type used for flow amounts, capacity bounds + /// and supply values in the algorithm. By default, it is \c int. + /// \tparam C The number type used for costs and potentials in the + /// algorithm. By default, it is the same as \c V. + /// \tparam TR The traits class that defines various types used by the + /// algorithm. By default, it is \ref CostScalingDefaultTraits + /// "CostScalingDefaultTraits<GR, V, C>". + /// In most cases, this parameter should not be set directly, + /// consider to use the named template parameters instead. + /// + /// \warning Both \c V and \c C must be signed number types. + /// \warning All input data (capacities, supply values, and costs) must + /// be integer. + /// \warning This algorithm does not support negative costs for + /// arcs having infinite upper bound. + /// + /// \note %CostScaling provides three different internal methods, + /// from which the most efficient one is used by default. + /// For more information, see \ref Method. +#ifdef DOXYGEN + template <typename GR, typename V, typename C, typename TR> +#else + template < typename GR, typename V = int, typename C = V, + typename TR = CostScalingDefaultTraits<GR, V, C> > +#endif + class CostScaling + { + public: + + /// The type of the digraph + typedef typename TR::Digraph Digraph; + /// The type of the flow amounts, capacity bounds and supply values + typedef typename TR::Value Value; + /// The type of the arc costs + typedef typename TR::Cost Cost; + + /// \brief The large cost type + /// + /// The large cost type used for internal computations. + /// By default, it is \c long \c long if the \c Cost type is integer, + /// otherwise it is \c double. + typedef typename TR::LargeCost LargeCost; + + /// \brief The \ref lemon::CostScalingDefaultTraits "traits class" + /// of the algorithm + typedef TR Traits; + + public: + + /// \brief Problem type constants for the \c run() function. + /// + /// Enum type containing the problem type constants that can be + /// returned by the \ref run() function of the algorithm. + enum ProblemType { + /// The problem has no feasible solution (flow). + INFEASIBLE, + /// The problem has optimal solution (i.e. it is feasible and + /// bounded), and the algorithm has found optimal flow and node + /// potentials (primal and dual solutions). + OPTIMAL, + /// The digraph contains an arc of negative cost and infinite + /// upper bound. It means that the objective function is unbounded + /// on that arc, however, note that it could actually be bounded + /// over the feasible flows, but this algroithm cannot handle + /// these cases. + UNBOUNDED + }; + + /// \brief Constants for selecting the internal method. + /// + /// Enum type containing constants for selecting the internal method + /// for the \ref run() function. + /// + /// \ref CostScaling provides three internal methods that differ mainly + /// in their base operations, which are used in conjunction with the + /// relabel operation. + /// By default, the so called \ref PARTIAL_AUGMENT + /// "Partial Augment-Relabel" method is used, which turned out to be + /// the most efficient and the most robust on various test inputs. + /// However, the other methods can be selected using the \ref run() + /// function with the proper parameter. + enum Method { + /// Local push operations are used, i.e. flow is moved only on one + /// admissible arc at once. + PUSH, + /// Augment operations are used, i.e. flow is moved on admissible + /// paths from a node with excess to a node with deficit. + AUGMENT, + /// Partial augment operations are used, i.e. flow is moved on + /// admissible paths started from a node with excess, but the + /// lengths of these paths are limited. This method can be viewed + /// as a combined version of the previous two operations. + PARTIAL_AUGMENT + }; + + private: + + TEMPLATE_DIGRAPH_TYPEDEFS(GR); + + typedef std::vector<int> IntVector; + typedef std::vector<Value> ValueVector; + typedef std::vector<Cost> CostVector; + typedef std::vector<LargeCost> LargeCostVector; + typedef std::vector<char> BoolVector; + // Note: vector<char> is used instead of vector<bool> + // for efficiency reasons + + private: + + template <typename KT, typename VT> + class StaticVectorMap { + public: + typedef KT Key; + typedef VT Value; + + StaticVectorMap(std::vector<Value>& v) : _v(v) {} + + const Value& operator[](const Key& key) const { + return _v[StaticDigraph::id(key)]; + } + + Value& operator[](const Key& key) { + return _v[StaticDigraph::id(key)]; + } + + void set(const Key& key, const Value& val) { + _v[StaticDigraph::id(key)] = val; + } + + private: + std::vector<Value>& _v; + }; + + typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap; + + private: + + // Data related to the underlying digraph + const GR &_graph; + int _node_num; + int _arc_num; + int _res_node_num; + int _res_arc_num; + int _root; + + // Parameters of the problem + bool _has_lower; + Value _sum_supply; + int _sup_node_num; + + // Data structures for storing the digraph + IntNodeMap _node_id; + IntArcMap _arc_idf; + IntArcMap _arc_idb; + IntVector _first_out; + BoolVector _forward; + IntVector _source; + IntVector _target; + IntVector _reverse; + + // Node and arc data + ValueVector _lower; + ValueVector _upper; + CostVector _scost; + ValueVector _supply; + + ValueVector _res_cap; + LargeCostVector _cost; + LargeCostVector _pi; + ValueVector _excess; + IntVector _next_out; + std::deque<int> _active_nodes; + + // Data for scaling + LargeCost _epsilon; + int _alpha; + + IntVector _buckets; + IntVector _bucket_next; + IntVector _bucket_prev; + IntVector _rank; + int _max_rank; + + public: + + /// \brief Constant for infinite upper bounds (capacities). + /// + /// Constant for infinite upper bounds (capacities). + /// It is \c std::numeric_limits<Value>::infinity() if available, + /// \c std::numeric_limits<Value>::max() otherwise. + const Value INF; + + public: + + /// \name Named Template Parameters + /// @{ + + template <typename T> + struct SetLargeCostTraits : public Traits { + typedef T LargeCost; + }; + + /// \brief \ref named-templ-param "Named parameter" for setting + /// \c LargeCost type. + /// + /// \ref named-templ-param "Named parameter" for setting \c LargeCost + /// type, which is used for internal computations in the algorithm. + /// \c Cost must be convertible to \c LargeCost. + template <typename T> + struct SetLargeCost + : public CostScaling<GR, V, C, SetLargeCostTraits<T> > { + typedef CostScaling<GR, V, C, SetLargeCostTraits<T> > Create; + }; + + /// @} + + protected: + + CostScaling() {} + + public: + + /// \brief Constructor. + /// + /// The constructor of the class. + /// + /// \param graph The digraph the algorithm runs on. + CostScaling(const GR& graph) : + _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph), + INF(std::numeric_limits<Value>::has_infinity ? + std::numeric_limits<Value>::infinity() : + std::numeric_limits<Value>::max()) + { + // Check the number types + LEMON_ASSERT(std::numeric_limits<Value>::is_signed, + "The flow type of CostScaling must be signed"); + LEMON_ASSERT(std::numeric_limits<Cost>::is_signed, + "The cost type of CostScaling must be signed"); + + // Reset data structures + reset(); + } + + /// \name Parameters + /// The parameters of the algorithm can be specified using these + /// functions. + + /// @{ + + /// \brief Set the lower bounds on the arcs. + /// + /// This function sets the lower bounds on the arcs. + /// If it is not used before calling \ref run(), the lower bounds + /// will be set to zero on all arcs. + /// + /// \param map An arc map storing the lower bounds. + /// Its \c Value type must be convertible to the \c Value type + /// of the algorithm. + /// + /// \return <tt>(*this)</tt> + template <typename LowerMap> + CostScaling& lowerMap(const LowerMap& map) { + _has_lower = true; + for (ArcIt a(_graph); a != INVALID; ++a) { + _lower[_arc_idf[a]] = map[a]; + } + return *this; + } + + /// \brief Set the upper bounds (capacities) on the arcs. + /// + /// This function sets the upper bounds (capacities) on the arcs. + /// If it is not used before calling \ref run(), the upper bounds + /// will be set to \ref INF on all arcs (i.e. the flow value will be + /// unbounded from above). + /// + /// \param map An arc map storing the upper bounds. + /// Its \c Value type must be convertible to the \c Value type + /// of the algorithm. + /// + /// \return <tt>(*this)</tt> + template<typename UpperMap> + CostScaling& upperMap(const UpperMap& map) { + for (ArcIt a(_graph); a != INVALID; ++a) { + _upper[_arc_idf[a]] = map[a]; + } + return *this; + } + + /// \brief Set the costs of the arcs. + /// + /// This function sets the costs of the arcs. + /// If it is not used before calling \ref run(), the costs + /// will be set to \c 1 on all arcs. + /// + /// \param map An arc map storing the costs. + /// Its \c Value type must be convertible to the \c Cost type + /// of the algorithm. + /// + /// \return <tt>(*this)</tt> + template<typename CostMap> + CostScaling& costMap(const CostMap& map) { + for (ArcIt a(_graph); a != INVALID; ++a) { + _scost[_arc_idf[a]] = map[a]; + _scost[_arc_idb[a]] = -map[a]; + } + return *this; + } + + /// \brief Set the supply values of the nodes. + /// + /// This function sets the supply values of the nodes. + /// If neither this function nor \ref stSupply() is used before + /// calling \ref run(), the supply of each node will be set to zero. + /// + /// \param map A node map storing the supply values. + /// Its \c Value type must be convertible to the \c Value type + /// of the algorithm. + /// + /// \return <tt>(*this)</tt> + template<typename SupplyMap> + CostScaling& supplyMap(const SupplyMap& map) { + for (NodeIt n(_graph); n != INVALID; ++n) { + _supply[_node_id[n]] = map[n]; + } + return *this; + } + + /// \brief Set single source and target nodes and a supply value. + /// + /// This function sets a single source node and a single target node + /// and the required flow value. + /// If neither this function nor \ref supplyMap() is used before + /// calling \ref run(), the supply of each node will be set to zero. + /// + /// Using this function has the same effect as using \ref supplyMap() + /// with a map in which \c k is assigned to \c s, \c -k is + /// assigned to \c t and all other nodes have zero supply value. + /// + /// \param s The source node. + /// \param t The target node. + /// \param k The required amount of flow from node \c s to node \c t + /// (i.e. the supply of \c s and the demand of \c t). + /// + /// \return <tt>(*this)</tt> + CostScaling& stSupply(const Node& s, const Node& t, Value k) { + for (int i = 0; i != _res_node_num; ++i) { + _supply[i] = 0; + } + _supply[_node_id[s]] = k; + _supply[_node_id[t]] = -k; + return *this; + } + + /// @} + + /// \name Execution control + /// The algorithm can be executed using \ref run(). + + /// @{ + + /// \brief Run the algorithm. + /// + /// This function runs the algorithm. + /// The paramters can be specified using functions \ref lowerMap(), + /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(). + /// For example, + /// \code + /// CostScaling<ListDigraph> cs(graph); + /// cs.lowerMap(lower).upperMap(upper).costMap(cost) + /// .supplyMap(sup).run(); + /// \endcode + /// + /// This function can be called more than once. All the given parameters + /// are kept for the next call, unless \ref resetParams() or \ref reset() + /// is used, thus only the modified parameters have to be set again. + /// If the underlying digraph was also modified after the construction + /// of the class (or the last \ref reset() call), then the \ref reset() + /// function must be called. + /// + /// \param method The internal method that will be used in the + /// algorithm. For more information, see \ref Method. + /// \param factor The cost scaling factor. It must be at least two. + /// + /// \return \c INFEASIBLE if no feasible flow exists, + /// \n \c OPTIMAL if the problem has optimal solution + /// (i.e. it is feasible and bounded), and the algorithm has found + /// optimal flow and node potentials (primal and dual solutions), + /// \n \c UNBOUNDED if the digraph contains an arc of negative cost + /// and infinite upper bound. It means that the objective function + /// is unbounded on that arc, however, note that it could actually be + /// bounded over the feasible flows, but this algroithm cannot handle + /// these cases. + /// + /// \see ProblemType, Method + /// \see resetParams(), reset() + ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 16) { + LEMON_ASSERT(factor >= 2, "The scaling factor must be at least 2"); + _alpha = factor; + ProblemType pt = init(); + if (pt != OPTIMAL) return pt; + start(method); + return OPTIMAL; + } + + /// \brief Reset all the parameters that have been given before. + /// + /// This function resets all the paramaters that have been given + /// before using functions \ref lowerMap(), \ref upperMap(), + /// \ref costMap(), \ref supplyMap(), \ref stSupply(). + /// + /// It is useful for multiple \ref run() calls. Basically, all the given + /// parameters are kept for the next \ref run() call, unless + /// \ref resetParams() or \ref reset() is used. + /// If the underlying digraph was also modified after the construction + /// of the class or the last \ref reset() call, then the \ref reset() + /// function must be used, otherwise \ref resetParams() is sufficient. + /// + /// For example, + /// \code + /// CostScaling<ListDigraph> cs(graph); + /// + /// // First run + /// cs.lowerMap(lower).upperMap(upper).costMap(cost) + /// .supplyMap(sup).run(); + /// + /// // Run again with modified cost map (resetParams() is not called, + /// // so only the cost map have to be set again) + /// cost[e] += 100; + /// cs.costMap(cost).run(); + /// + /// // Run again from scratch using resetParams() + /// // (the lower bounds will be set to zero on all arcs) + /// cs.resetParams(); + /// cs.upperMap(capacity).costMap(cost) + /// .supplyMap(sup).run(); + /// \endcode + /// + /// \return <tt>(*this)</tt> + /// + /// \see reset(), run() + CostScaling& resetParams() { + for (int i = 0; i != _res_node_num; ++i) { + _supply[i] = 0; + } + int limit = _first_out[_root]; + for (int j = 0; j != limit; ++j) { + _lower[j] = 0; + _upper[j] = INF; + _scost[j] = _forward[j] ? 1 : -1; + } + for (int j = limit; j != _res_arc_num; ++j) { + _lower[j] = 0; + _upper[j] = INF; + _scost[j] = 0; + _scost[_reverse[j]] = 0; + } + _has_lower = false; + return *this; + } + + /// \brief Reset the internal data structures and all the parameters + /// that have been given before. + /// + /// This function resets the internal data structures and all the + /// paramaters that have been given before using functions \ref lowerMap(), + /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(). + /// + /// It is useful for multiple \ref run() calls. By default, all the given + /// parameters are kept for the next \ref run() call, unless + /// \ref resetParams() or \ref reset() is used. + /// If the underlying digraph was also modified after the construction + /// of the class or the last \ref reset() call, then the \ref reset() + /// function must be used, otherwise \ref resetParams() is sufficient. + /// + /// See \ref resetParams() for examples. + /// + /// \return <tt>(*this)</tt> + /// + /// \see resetParams(), run() + CostScaling& reset() { + // Resize vectors + _node_num = countNodes(_graph); + _arc_num = countArcs(_graph); + _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); + _scost.resize(_res_arc_num); + _supply.resize(_res_node_num); + + _res_cap.resize(_res_arc_num); + _cost.resize(_res_arc_num); + _pi.resize(_res_node_num); + _excess.resize(_res_node_num); + _next_out.resize(_res_node_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 + /// cs.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>(_scost[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; + _excess[i] = _supply[i]; + } + + // 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 = _scost[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] && _scost[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]; + _excess[i] = 0; + if (ex < 0) max_cap -= ex; + } + for (int j = 0; j != _res_arc_num; ++j) { + if (_upper[j] >= MAX) _upper[j] = max_cap; + } + + // Initialize the large cost vector and the epsilon parameter + _epsilon = 0; + LargeCost lc; + for (int i = 0; i != _root; ++i) { + last_out = _first_out[i+1]; + for (int j = _first_out[i]; j != last_out; ++j) { + lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha; + _cost[j] = lc; + if (lc > _epsilon) _epsilon = lc; + } + } + _epsilon /= _alpha; + + // 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]]; + } + } + + _sup_node_num = 0; + for (NodeIt n(_graph); n != INVALID; ++n) { + if (sup[n] > 0) ++_sup_node_num; + } + + // 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; + _excess[u] = 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] = 0; + _res_cap[ra] = 0; + _cost[a] = 0; + _cost[ra] = 0; + } + } + + // Initialize data structures for buckets + _max_rank = _alpha * _res_node_num; + _buckets.resize(_max_rank); + _bucket_next.resize(_res_node_num + 1); + _bucket_prev.resize(_res_node_num + 1); + _rank.resize(_res_node_num + 1); + + 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; + } + + // Execute the algorithm and transform the results + void start(Method method) { + const int MAX_PARTIAL_PATH_LENGTH = 4; + + switch (method) { + case PUSH: + startPush(); + break; + case AUGMENT: + startAugment(_res_node_num - 1); + break; + case PARTIAL_AUGMENT: + startAugment(MAX_PARTIAL_PATH_LENGTH); + break; + } + + // Compute node potentials (dual solution) + for (int i = 0; i != _res_node_num; ++i) { + _pi[i] = static_cast<Cost>(_pi[i] / (_res_node_num * _alpha)); + } + bool optimal = true; + for (int i = 0; optimal && i != _res_node_num; ++i) { + LargeCost pi_i = _pi[i]; + int last_out = _first_out[i+1]; + for (int j = _first_out[i]; j != last_out; ++j) { + if (_res_cap[j] > 0 && _scost[j] + pi_i - _pi[_target[j]] < 0) { + optimal = false; + break; + } + } + } + + if (!optimal) { + // Compute node potentials for the original costs with BellmanFord + // (if it is necessary) + typedef std::pair<int, int> IntPair; + StaticDigraph sgr; + std::vector<IntPair> arc_vec; + std::vector<LargeCost> cost_vec; + LargeCostArcMap cost_map(cost_vec); + + arc_vec.clear(); + cost_vec.clear(); + for (int j = 0; j != _res_arc_num; ++j) { + if (_res_cap[j] > 0) { + int u = _source[j], v = _target[j]; + arc_vec.push_back(IntPair(u, v)); + cost_vec.push_back(_scost[j] + _pi[u] - _pi[v]); + } + } + sgr.build(_res_node_num, arc_vec.begin(), arc_vec.end()); + + typename BellmanFord<StaticDigraph, LargeCostArcMap>::Create + bf(sgr, cost_map); + bf.init(0); + bf.start(); + + for (int i = 0; i != _res_node_num; ++i) { + _pi[i] += bf.dist(sgr.node(i)); + } + } + + // Shift potentials to meet the requirements of the GEQ type + // optimality conditions + LargeCost max_pot = _pi[_root]; + for (int i = 0; i != _res_node_num; ++i) { + if (_pi[i] > max_pot) max_pot = _pi[i]; + } + if (max_pot != 0) { + for (int i = 0; i != _res_node_num; ++i) { + _pi[i] -= max_pot; + } + } + + // 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]; + } + } + } + + // Initialize a cost scaling phase + void initPhase() { + // Saturate arcs not satisfying the optimality condition + for (int u = 0; u != _res_node_num; ++u) { + int last_out = _first_out[u+1]; + LargeCost pi_u = _pi[u]; + for (int a = _first_out[u]; a != last_out; ++a) { + Value delta = _res_cap[a]; + if (delta > 0) { + int v = _target[a]; + if (_cost[a] + pi_u - _pi[v] < 0) { + _excess[u] -= delta; + _excess[v] += delta; + _res_cap[a] = 0; + _res_cap[_reverse[a]] += delta; + } + } + } + } + + // Find active nodes (i.e. nodes with positive excess) + for (int u = 0; u != _res_node_num; ++u) { + if (_excess[u] > 0) _active_nodes.push_back(u); + } + + // Initialize the next arcs + for (int u = 0; u != _res_node_num; ++u) { + _next_out[u] = _first_out[u]; + } + } + + // Price (potential) refinement heuristic + bool priceRefinement() { + + // Stack for stroing the topological order + IntVector stack(_res_node_num); + int stack_top; + + // Perform phases + while (topologicalSort(stack, stack_top)) { + + // Compute node ranks in the acyclic admissible network and + // store the nodes in buckets + for (int i = 0; i != _res_node_num; ++i) { + _rank[i] = 0; + } + const int bucket_end = _root + 1; + for (int r = 0; r != _max_rank; ++r) { + _buckets[r] = bucket_end; + } + int top_rank = 0; + for ( ; stack_top >= 0; --stack_top) { + int u = stack[stack_top], v; + int rank_u = _rank[u]; + + LargeCost rc, pi_u = _pi[u]; + int last_out = _first_out[u+1]; + for (int a = _first_out[u]; a != last_out; ++a) { + if (_res_cap[a] > 0) { + v = _target[a]; + rc = _cost[a] + pi_u - _pi[v]; + if (rc < 0) { + LargeCost nrc = static_cast<LargeCost>((-rc - 0.5) / _epsilon); + if (nrc < LargeCost(_max_rank)) { + int new_rank_v = rank_u + static_cast<int>(nrc); + if (new_rank_v > _rank[v]) { + _rank[v] = new_rank_v; + } + } + } + } + } + + if (rank_u > 0) { + top_rank = std::max(top_rank, rank_u); + int bfirst = _buckets[rank_u]; + _bucket_next[u] = bfirst; + _bucket_prev[bfirst] = u; + _buckets[rank_u] = u; + } + } + + // Check if the current flow is epsilon-optimal + if (top_rank == 0) { + return true; + } + + // Process buckets in top-down order + for (int rank = top_rank; rank > 0; --rank) { + while (_buckets[rank] != bucket_end) { + // Remove the first node from the current bucket + int u = _buckets[rank]; + _buckets[rank] = _bucket_next[u]; + + // Search the outgoing arcs of u + LargeCost rc, pi_u = _pi[u]; + int last_out = _first_out[u+1]; + int v, old_rank_v, new_rank_v; + for (int a = _first_out[u]; a != last_out; ++a) { + if (_res_cap[a] > 0) { + v = _target[a]; + old_rank_v = _rank[v]; + + if (old_rank_v < rank) { + + // Compute the new rank of node v + rc = _cost[a] + pi_u - _pi[v]; + if (rc < 0) { + new_rank_v = rank; + } else { + LargeCost nrc = rc / _epsilon; + new_rank_v = 0; + if (nrc < LargeCost(_max_rank)) { + new_rank_v = rank - 1 - static_cast<int>(nrc); + } + } + + // Change the rank of node v + if (new_rank_v > old_rank_v) { + _rank[v] = new_rank_v; + + // Remove v from its old bucket + if (old_rank_v > 0) { + if (_buckets[old_rank_v] == v) { + _buckets[old_rank_v] = _bucket_next[v]; + } else { + int pv = _bucket_prev[v], nv = _bucket_next[v]; + _bucket_next[pv] = nv; + _bucket_prev[nv] = pv; + } + } + + // Insert v into its new bucket + int nv = _buckets[new_rank_v]; + _bucket_next[v] = nv; + _bucket_prev[nv] = v; + _buckets[new_rank_v] = v; + } + } + } + } + + // Refine potential of node u + _pi[u] -= rank * _epsilon; + } + } + + } + + return false; + } + + // Find and cancel cycles in the admissible network and + // determine topological order using DFS + bool topologicalSort(IntVector &stack, int &stack_top) { + const int MAX_CYCLE_CANCEL = 1; + + BoolVector reached(_res_node_num, false); + BoolVector processed(_res_node_num, false); + IntVector pred(_res_node_num); + for (int i = 0; i != _res_node_num; ++i) { + _next_out[i] = _first_out[i]; + } + stack_top = -1; + + int cycle_cnt = 0; + for (int start = 0; start != _res_node_num; ++start) { + if (reached[start]) continue; + + // Start DFS search from this start node + pred[start] = -1; + int tip = start, v; + while (true) { + // Check the outgoing arcs of the current tip node + reached[tip] = true; + LargeCost pi_tip = _pi[tip]; + int a, last_out = _first_out[tip+1]; + for (a = _next_out[tip]; a != last_out; ++a) { + if (_res_cap[a] > 0) { + v = _target[a]; + if (_cost[a] + pi_tip - _pi[v] < 0) { + if (!reached[v]) { + // A new node is reached + reached[v] = true; + pred[v] = tip; + _next_out[tip] = a; + tip = v; + a = _next_out[tip]; + last_out = _first_out[tip+1]; + break; + } + else if (!processed[v]) { + // A cycle is found + ++cycle_cnt; + _next_out[tip] = a; + + // Find the minimum residual capacity along the cycle + Value d, delta = _res_cap[a]; + int u, delta_node = tip; + for (u = tip; u != v; ) { + u = pred[u]; + d = _res_cap[_next_out[u]]; + if (d <= delta) { + delta = d; + delta_node = u; + } + } + + // Augment along the cycle + _res_cap[a] -= delta; + _res_cap[_reverse[a]] += delta; + for (u = tip; u != v; ) { + u = pred[u]; + int ca = _next_out[u]; + _res_cap[ca] -= delta; + _res_cap[_reverse[ca]] += delta; + } + + // Check the maximum number of cycle canceling + if (cycle_cnt >= MAX_CYCLE_CANCEL) { + return false; + } + + // Roll back search to delta_node + if (delta_node != tip) { + for (u = tip; u != delta_node; u = pred[u]) { + reached[u] = false; + } + tip = delta_node; + a = _next_out[tip] + 1; + last_out = _first_out[tip+1]; + break; + } + } + } + } + } + + // Step back to the previous node + if (a == last_out) { + processed[tip] = true; + stack[++stack_top] = tip; + tip = pred[tip]; + if (tip < 0) { + // Finish DFS from the current start node + break; + } + ++_next_out[tip]; + } + } + + } + + return (cycle_cnt == 0); + } + + // Global potential update heuristic + void globalUpdate() { + const int bucket_end = _root + 1; + + // Initialize buckets + for (int r = 0; r != _max_rank; ++r) { + _buckets[r] = bucket_end; + } + Value total_excess = 0; + int b0 = bucket_end; + for (int i = 0; i != _res_node_num; ++i) { + if (_excess[i] < 0) { + _rank[i] = 0; + _bucket_next[i] = b0; + _bucket_prev[b0] = i; + b0 = i; + } else { + total_excess += _excess[i]; + _rank[i] = _max_rank; + } + } + if (total_excess == 0) return; + _buckets[0] = b0; + + // Search the buckets + int r = 0; + for ( ; r != _max_rank; ++r) { + while (_buckets[r] != bucket_end) { + // Remove the first node from the current bucket + int u = _buckets[r]; + _buckets[r] = _bucket_next[u]; + + // Search the incoming arcs of u + LargeCost pi_u = _pi[u]; + int last_out = _first_out[u+1]; + for (int a = _first_out[u]; a != last_out; ++a) { + int ra = _reverse[a]; + if (_res_cap[ra] > 0) { + int v = _source[ra]; + int old_rank_v = _rank[v]; + if (r < old_rank_v) { + // Compute the new rank of v + LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon; + int new_rank_v = old_rank_v; + if (nrc < LargeCost(_max_rank)) { + new_rank_v = r + 1 + static_cast<int>(nrc); + } + + // Change the rank of v + if (new_rank_v < old_rank_v) { + _rank[v] = new_rank_v; + _next_out[v] = _first_out[v]; + + // Remove v from its old bucket + if (old_rank_v < _max_rank) { + if (_buckets[old_rank_v] == v) { + _buckets[old_rank_v] = _bucket_next[v]; + } else { + int pv = _bucket_prev[v], nv = _bucket_next[v]; + _bucket_next[pv] = nv; + _bucket_prev[nv] = pv; + } + } + + // Insert v into its new bucket + int nv = _buckets[new_rank_v]; + _bucket_next[v] = nv; + _bucket_prev[nv] = v; + _buckets[new_rank_v] = v; + } + } + } + } + + // Finish search if there are no more active nodes + if (_excess[u] > 0) { + total_excess -= _excess[u]; + if (total_excess <= 0) break; + } + } + if (total_excess <= 0) break; + } + + // Relabel nodes + for (int u = 0; u != _res_node_num; ++u) { + int k = std::min(_rank[u], r); + if (k > 0) { + _pi[u] -= _epsilon * k; + _next_out[u] = _first_out[u]; + } + } + } + + /// Execute the algorithm performing augment and relabel operations + void startAugment(int max_length) { + // Paramters for heuristics + const int PRICE_REFINEMENT_LIMIT = 2; + const double GLOBAL_UPDATE_FACTOR = 1.0; + const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR * + (_res_node_num + _sup_node_num * _sup_node_num)); + int next_global_update_limit = global_update_skip; + + // Perform cost scaling phases + IntVector path; + BoolVector path_arc(_res_arc_num, false); + int relabel_cnt = 0; + int eps_phase_cnt = 0; + for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ? + 1 : _epsilon / _alpha ) + { + ++eps_phase_cnt; + + // Price refinement heuristic + if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) { + if (priceRefinement()) continue; + } + + // Initialize current phase + initPhase(); + + // Perform partial augment and relabel operations + while (true) { + // Select an active node (FIFO selection) + while (_active_nodes.size() > 0 && + _excess[_active_nodes.front()] <= 0) { + _active_nodes.pop_front(); + } + if (_active_nodes.size() == 0) break; + int start = _active_nodes.front(); + + // Find an augmenting path from the start node + int tip = start; + while (int(path.size()) < max_length && _excess[tip] >= 0) { + int u; + LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max(); + LargeCost pi_tip = _pi[tip]; + int last_out = _first_out[tip+1]; + for (int a = _next_out[tip]; a != last_out; ++a) { + if (_res_cap[a] > 0) { + u = _target[a]; + rc = _cost[a] + pi_tip - _pi[u]; + if (rc < 0) { + path.push_back(a); + _next_out[tip] = a; + if (path_arc[a]) { + goto augment; // a cycle is found, stop path search + } + tip = u; + path_arc[a] = true; + goto next_step; + } + else if (rc < min_red_cost) { + min_red_cost = rc; + } + } + } + + // Relabel tip node + if (tip != start) { + int ra = _reverse[path.back()]; + min_red_cost = + std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]); + } + last_out = _next_out[tip]; + for (int a = _first_out[tip]; a != last_out; ++a) { + if (_res_cap[a] > 0) { + rc = _cost[a] + pi_tip - _pi[_target[a]]; + if (rc < min_red_cost) { + min_red_cost = rc; + } + } + } + _pi[tip] -= min_red_cost + _epsilon; + _next_out[tip] = _first_out[tip]; + ++relabel_cnt; + + // Step back + if (tip != start) { + int pa = path.back(); + path_arc[pa] = false; + tip = _source[pa]; + path.pop_back(); + } + + next_step: ; + } + + // Augment along the found path (as much flow as possible) + augment: + Value delta; + int pa, u, v = start; + for (int i = 0; i != int(path.size()); ++i) { + pa = path[i]; + u = v; + v = _target[pa]; + path_arc[pa] = false; + delta = std::min(_res_cap[pa], _excess[u]); + _res_cap[pa] -= delta; + _res_cap[_reverse[pa]] += delta; + _excess[u] -= delta; + _excess[v] += delta; + if (_excess[v] > 0 && _excess[v] <= delta) { + _active_nodes.push_back(v); + } + } + path.clear(); + + // Global update heuristic + if (relabel_cnt >= next_global_update_limit) { + globalUpdate(); + next_global_update_limit += global_update_skip; + } + } + + } + + } + + /// Execute the algorithm performing push and relabel operations + void startPush() { + // Paramters for heuristics + const int PRICE_REFINEMENT_LIMIT = 2; + const double GLOBAL_UPDATE_FACTOR = 2.0; + + const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR * + (_res_node_num + _sup_node_num * _sup_node_num)); + int next_global_update_limit = global_update_skip; + + // Perform cost scaling phases + BoolVector hyper(_res_node_num, false); + LargeCostVector hyper_cost(_res_node_num); + int relabel_cnt = 0; + int eps_phase_cnt = 0; + for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ? + 1 : _epsilon / _alpha ) + { + ++eps_phase_cnt; + + // Price refinement heuristic + if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) { + if (priceRefinement()) continue; + } + + // Initialize current phase + initPhase(); + + // Perform push and relabel operations + while (_active_nodes.size() > 0) { + LargeCost min_red_cost, rc, pi_n; + Value delta; + int n, t, a, last_out = _res_arc_num; + + next_node: + // Select an active node (FIFO selection) + n = _active_nodes.front(); + last_out = _first_out[n+1]; + pi_n = _pi[n]; + + // Perform push operations if there are admissible arcs + if (_excess[n] > 0) { + for (a = _next_out[n]; a != last_out; ++a) { + if (_res_cap[a] > 0 && + _cost[a] + pi_n - _pi[_target[a]] < 0) { + delta = std::min(_res_cap[a], _excess[n]); + t = _target[a]; + + // Push-look-ahead heuristic + Value ahead = -_excess[t]; + int last_out_t = _first_out[t+1]; + LargeCost pi_t = _pi[t]; + for (int ta = _next_out[t]; ta != last_out_t; ++ta) { + if (_res_cap[ta] > 0 && + _cost[ta] + pi_t - _pi[_target[ta]] < 0) + ahead += _res_cap[ta]; + if (ahead >= delta) break; + } + if (ahead < 0) ahead = 0; + + // Push flow along the arc + if (ahead < delta && !hyper[t]) { + _res_cap[a] -= ahead; + _res_cap[_reverse[a]] += ahead; + _excess[n] -= ahead; + _excess[t] += ahead; + _active_nodes.push_front(t); + hyper[t] = true; + hyper_cost[t] = _cost[a] + pi_n - pi_t; + _next_out[n] = a; + goto next_node; + } else { + _res_cap[a] -= delta; + _res_cap[_reverse[a]] += delta; + _excess[n] -= delta; + _excess[t] += delta; + if (_excess[t] > 0 && _excess[t] <= delta) + _active_nodes.push_back(t); + } + + if (_excess[n] == 0) { + _next_out[n] = a; + goto remove_nodes; + } + } + } + _next_out[n] = a; + } + + // Relabel the node if it is still active (or hyper) + if (_excess[n] > 0 || hyper[n]) { + min_red_cost = hyper[n] ? -hyper_cost[n] : + std::numeric_limits<LargeCost>::max(); + for (int a = _first_out[n]; a != last_out; ++a) { + if (_res_cap[a] > 0) { + rc = _cost[a] + pi_n - _pi[_target[a]]; + if (rc < min_red_cost) { + min_red_cost = rc; + } + } + } + _pi[n] -= min_red_cost + _epsilon; + _next_out[n] = _first_out[n]; + hyper[n] = false; + ++relabel_cnt; + } + + // Remove nodes that are not active nor hyper + remove_nodes: + while ( _active_nodes.size() > 0 && + _excess[_active_nodes.front()] <= 0 && + !hyper[_active_nodes.front()] ) { + _active_nodes.pop_front(); + } + + // Global update heuristic + if (relabel_cnt >= next_global_update_limit) { + globalUpdate(); + for (int u = 0; u != _res_node_num; ++u) + hyper[u] = false; + next_global_update_limit += global_update_skip; + } + } + } + } + + }; //class CostScaling + + ///@} + +} //namespace lemon + +#endif //LEMON_COST_SCALING_H |