/* -*- mode: C++; indent-tabs-mode: nil; -*- * * This file is a part of LEMON, a generic C++ optimization library. * * Copyright (C) 2003-2010 * 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 #include #include #include #include #include #include #include #include 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 #else template < typename GR, typename V = int, typename C = V, bool integer = std::numeric_limits::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 struct CostScalingDefaultTraits { 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" \ref amo93networkflows, \ref goldberg90approximation, /// \ref goldberg97efficient, \ref 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. /// /// In general, \ref NetworkSimplex and \ref CostScaling are the fastest /// implementations available in LEMON for this problem. /// /// 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". /// 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 #else template < typename GR, typename V = int, typename C = V, typename TR = CostScalingDefaultTraits > #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; /// The \ref 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 IntVector; typedef std::vector ValueVector; typedef std::vector CostVector; typedef std::vector LargeCostVector; typedef std::vector BoolVector; // Note: vector is used instead of vector for efficiency reasons private: template class StaticVectorMap { public: typedef KT Key; typedef VT Value; StaticVectorMap(std::vector& 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& _v; }; typedef StaticVectorMap 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 _have_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 _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::infinity() if available, /// \c std::numeric_limits::max() otherwise. const Value INF; public: /// \name Named Template Parameters /// @{ template 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 struct SetLargeCost : public CostScaling > { typedef CostScaling > 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::has_infinity ? std::numeric_limits::infinity() : std::numeric_limits::max()) { // Check the number types LEMON_ASSERT(std::numeric_limits::is_signed, "The flow type of CostScaling must be signed"); LEMON_ASSERT(std::numeric_limits::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 (*this) template CostScaling& lowerMap(const LowerMap& map) { _have_lower = true; for (ArcIt a(_graph); a != INVALID; ++a) { _lower[_arc_idf[a]] = map[a]; _lower[_arc_idb[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 (*this) template 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 (*this) template 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 (*this) template 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 (*this) 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 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 larger than one. /// /// \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 = 8) { _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 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 (*this) /// /// \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; } _have_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 (*this) /// /// \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(e). /// /// \note The return type of the function can be specified as a /// template parameter. For example, /// \code /// cs.totalCost(); /// \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 Number totalCost() const { Number c = 0; for (ArcIt a(_graph); a != INVALID; ++a) { int i = _arc_idb[a]; c += static_cast(_res_cap[i]) * (-static_cast(_scost[i])); } return c; } #ifndef DOXYGEN Cost totalCost() const { return totalCost(); } #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 Return the flow map (the primal solution). /// /// 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 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(_pi[_node_id[n]]); } /// \brief Return the potential map (the dual solution). /// /// 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 void potentialMap(PotentialMap &map) const { for (NodeIt n(_graph); n != INVALID; ++n) { map.set(n, static_cast(_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; // 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::max(); int last_out; if (_have_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(_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 low(0); typedef typename Digraph::template ArcMap ValueArcMap; typedef typename Digraph::template NodeMap ValueNodeMap; ValueArcMap cap(_graph), flow(_graph); ValueNodeMap sup(_graph); for (NodeIt n(_graph); n != INVALID; ++n) { sup[n] = _supply[_node_id[n]]; } if (_have_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, 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; } // 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(_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 IntPair; StaticDigraph sgr; std::vector arc_vec; std::vector 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::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 (_have_lower) { int limit = _first_out[_root]; for (int j = 0; j != limit; ++j) { if (!_forward[j]) _res_cap[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((-rc - 0.5) / _epsilon); if (nrc < LargeCost(_max_rank)) { int new_rank_v = rank_u + static_cast(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(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 incomming 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(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(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::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(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::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