diff --git a/lemon/capacity_scaling.h b/lemon/capacity_scaling.h --- a/lemon/capacity_scaling.h +++ b/lemon/capacity_scaling.h @@ -19,396 +19,417 @@ #ifndef LEMON_CAPACITY_SCALING_H #define LEMON_CAPACITY_SCALING_H -/// \ingroup min_cost_flow +/// \ingroup min_cost_flow_algs /// /// \file -/// \brief Capacity scaling algorithm for finding a minimum cost flow. +/// \brief Capacity Scaling algorithm for finding a minimum cost flow. #include +#include +#include #include namespace lemon { - /// \addtogroup min_cost_flow + /// \addtogroup min_cost_flow_algs /// @{ - /// \brief Implementation of the capacity scaling algorithm for - /// finding a minimum cost flow. + /// \brief Implementation of the Capacity Scaling algorithm for + /// finding a \ref min_cost_flow "minimum cost flow". /// /// \ref CapacityScaling implements the capacity scaling version - /// of the successive shortest path algorithm for finding a minimum - /// cost flow. + /// of the successive shortest path algorithm for finding a + /// \ref min_cost_flow "minimum cost flow". It is an efficient dual + /// solution method. /// - /// \tparam Digraph The digraph type the algorithm runs on. - /// \tparam LowerMap The type of the lower bound map. - /// \tparam CapacityMap The type of the capacity (upper bound) map. - /// \tparam CostMap The type of the cost (length) map. - /// \tparam SupplyMap The type of the supply map. + /// 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. /// - /// \warning - /// - Arc capacities and costs should be \e non-negative \e integers. - /// - Supply values should be \e signed \e integers. - /// - The value types of the maps should be convertible to each other. - /// - \c CostMap::Value must be signed type. + /// \tparam GR The digraph type the algorithm runs on. + /// \tparam V The value type used for flow amounts, capacity bounds + /// and supply values in the algorithm. By default it is \c int. + /// \tparam C The value type used for costs and potentials in the + /// algorithm. By default it is the same as \c V. /// - /// \author Peter Kovacs - template < typename Digraph, - typename LowerMap = typename Digraph::template ArcMap, - typename CapacityMap = typename Digraph::template ArcMap, - typename CostMap = typename Digraph::template ArcMap, - typename SupplyMap = typename Digraph::template NodeMap > + /// \warning Both value types must be signed and all input data must + /// be integer. + /// \warning This algorithm does not support negative costs for such + /// arcs that have infinite upper bound. + template class CapacityScaling { - TEMPLATE_DIGRAPH_TYPEDEFS(Digraph); + public: - typedef typename CapacityMap::Value Capacity; - typedef typename CostMap::Value Cost; - typedef typename SupplyMap::Value Supply; - typedef typename Digraph::template ArcMap CapacityArcMap; - typedef typename Digraph::template NodeMap SupplyNodeMap; - typedef typename Digraph::template NodeMap PredMap; + /// The type of the flow amounts, capacity bounds and supply values + typedef V Value; + /// The type of the arc costs + typedef C Cost; public: - /// The type of the flow map. - typedef typename Digraph::template ArcMap FlowMap; - /// The type of the potential map. - typedef typename Digraph::template NodeMap PotentialMap; + /// \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 + }; + + private: + + TEMPLATE_DIGRAPH_TYPEDEFS(GR); + + typedef std::vector ArcVector; + typedef std::vector NodeVector; + typedef std::vector IntVector; + typedef std::vector BoolVector; + typedef std::vector ValueVector; + typedef std::vector CostVector; private: - /// \brief Special implementation of the \ref Dijkstra algorithm - /// for finding shortest paths in the residual network. + // Data related to the underlying digraph + const GR &_graph; + int _node_num; + int _arc_num; + int _res_arc_num; + int _root; + + // Parameters of the problem + bool _have_lower; + Value _sum_supply; + + // 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 _cost; + ValueVector _supply; + + ValueVector _res_cap; + CostVector _pi; + ValueVector _excess; + IntVector _excess_nodes; + IntVector _deficit_nodes; + + Value _delta; + int _phase_num; + IntVector _pred; + + public: + + /// \brief Constant for infinite upper bounds (capacities). /// - /// \ref ResidualDijkstra is a special implementation of the - /// \ref Dijkstra algorithm for finding shortest paths in the - /// residual network of the digraph with respect to the reduced arc - /// costs and modifying the node potentials according to the - /// distance of the nodes. + /// Constant for infinite upper bounds (capacities). + /// It is \c std::numeric_limits::infinity() if available, + /// \c std::numeric_limits::max() otherwise. + const Value INF; + + private: + + // Special implementation of the Dijkstra algorithm for finding + // shortest paths in the residual network of the digraph with + // respect to the reduced arc costs and modifying the node + // potentials according to the found distance labels. class ResidualDijkstra { - typedef typename Digraph::template NodeMap HeapCrossRef; + typedef RangeMap HeapCrossRef; typedef BinHeap Heap; private: - // The digraph the algorithm runs on - const Digraph &_graph; - - // The main maps - const FlowMap &_flow; - const CapacityArcMap &_res_cap; - const CostMap &_cost; - const SupplyNodeMap &_excess; - PotentialMap &_potential; - - // The distance map - PotentialMap _dist; - // The pred arc map - PredMap &_pred; - // The processed (i.e. permanently labeled) nodes - std::vector _proc_nodes; - + int _node_num; + const IntVector &_first_out; + const IntVector &_target; + const CostVector &_cost; + const ValueVector &_res_cap; + const ValueVector &_excess; + CostVector &_pi; + IntVector &_pred; + + IntVector _proc_nodes; + CostVector _dist; + public: - /// Constructor. - ResidualDijkstra( const Digraph &digraph, - const FlowMap &flow, - const CapacityArcMap &res_cap, - const CostMap &cost, - const SupplyMap &excess, - PotentialMap &potential, - PredMap &pred ) : - _graph(digraph), _flow(flow), _res_cap(res_cap), _cost(cost), - _excess(excess), _potential(potential), _dist(digraph), - _pred(pred) + ResidualDijkstra(CapacityScaling& cs) : + _node_num(cs._node_num), _first_out(cs._first_out), + _target(cs._target), _cost(cs._cost), _res_cap(cs._res_cap), + _excess(cs._excess), _pi(cs._pi), _pred(cs._pred), + _dist(cs._node_num) {} - /// Run the algorithm from the given source node. - Node run(Node s, Capacity delta = 1) { - HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP); + int run(int s, Value delta = 1) { + HeapCrossRef heap_cross_ref(_node_num, Heap::PRE_HEAP); Heap heap(heap_cross_ref); heap.push(s, 0); - _pred[s] = INVALID; + _pred[s] = -1; _proc_nodes.clear(); - // Processing nodes + // Process nodes while (!heap.empty() && _excess[heap.top()] > -delta) { - Node u = heap.top(), v; - Cost d = heap.prio() + _potential[u], nd; + int u = heap.top(), v; + Cost d = heap.prio() + _pi[u], dn; _dist[u] = heap.prio(); + _proc_nodes.push_back(u); heap.pop(); - _proc_nodes.push_back(u); - // Traversing outgoing arcs - for (OutArcIt e(_graph, u); e != INVALID; ++e) { - if (_res_cap[e] >= delta) { - v = _graph.target(e); - switch(heap.state(v)) { + // Traverse outgoing residual arcs + for (int a = _first_out[u]; a != _first_out[u+1]; ++a) { + if (_res_cap[a] < delta) continue; + v = _target[a]; + switch (heap.state(v)) { case Heap::PRE_HEAP: - heap.push(v, d + _cost[e] - _potential[v]); - _pred[v] = e; + heap.push(v, d + _cost[a] - _pi[v]); + _pred[v] = a; break; case Heap::IN_HEAP: - nd = d + _cost[e] - _potential[v]; - if (nd < heap[v]) { - heap.decrease(v, nd); - _pred[v] = e; + dn = d + _cost[a] - _pi[v]; + if (dn < heap[v]) { + heap.decrease(v, dn); + _pred[v] = a; } break; case Heap::POST_HEAP: break; - } - } - } - - // Traversing incoming arcs - for (InArcIt e(_graph, u); e != INVALID; ++e) { - if (_flow[e] >= delta) { - v = _graph.source(e); - switch(heap.state(v)) { - case Heap::PRE_HEAP: - heap.push(v, d - _cost[e] - _potential[v]); - _pred[v] = e; - break; - case Heap::IN_HEAP: - nd = d - _cost[e] - _potential[v]; - if (nd < heap[v]) { - heap.decrease(v, nd); - _pred[v] = e; - } - break; - case Heap::POST_HEAP: - break; - } } } } - if (heap.empty()) return INVALID; + if (heap.empty()) return -1; - // Updating potentials of processed nodes - Node t = heap.top(); - Cost t_dist = heap.prio(); - for (int i = 0; i < int(_proc_nodes.size()); ++i) - _potential[_proc_nodes[i]] += _dist[_proc_nodes[i]] - t_dist; + // Update potentials of processed nodes + int t = heap.top(); + Cost dt = heap.prio(); + for (int i = 0; i < int(_proc_nodes.size()); ++i) { + _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt; + } return t; } }; //class ResidualDijkstra - private: - - // The digraph the algorithm runs on - const Digraph &_graph; - // The original lower bound map - const LowerMap *_lower; - // The modified capacity map - CapacityArcMap _capacity; - // The original cost map - const CostMap &_cost; - // The modified supply map - SupplyNodeMap _supply; - bool _valid_supply; - - // Arc map of the current flow - FlowMap *_flow; - bool _local_flow; - // Node map of the current potentials - PotentialMap *_potential; - bool _local_potential; - - // The residual capacity map - CapacityArcMap _res_cap; - // The excess map - SupplyNodeMap _excess; - // The excess nodes (i.e. nodes with positive excess) - std::vector _excess_nodes; - // The deficit nodes (i.e. nodes with negative excess) - std::vector _deficit_nodes; - - // The delta parameter used for capacity scaling - Capacity _delta; - // The maximum number of phases - int _phase_num; - - // The pred arc map - PredMap _pred; - // Implementation of the Dijkstra algorithm for finding augmenting - // shortest paths in the residual network - ResidualDijkstra *_dijkstra; - public: - /// \brief General constructor (with lower bounds). + /// \brief Constructor. /// - /// General constructor (with lower bounds). + /// The constructor of the class. /// - /// \param digraph The digraph the algorithm runs on. - /// \param lower The lower bounds of the arcs. - /// \param capacity The capacities (upper bounds) of the arcs. - /// \param cost The cost (length) values of the arcs. - /// \param supply The supply values of the nodes (signed). - CapacityScaling( const Digraph &digraph, - const LowerMap &lower, - const CapacityMap &capacity, - const CostMap &cost, - const SupplyMap &supply ) : - _graph(digraph), _lower(&lower), _capacity(digraph), _cost(cost), - _supply(digraph), _flow(NULL), _local_flow(false), - _potential(NULL), _local_potential(false), - _res_cap(digraph), _excess(digraph), _pred(digraph), _dijkstra(NULL) + /// \param graph The digraph the algorithm runs on. + CapacityScaling(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()) { - Supply sum = 0; - for (NodeIt n(_graph); n != INVALID; ++n) { - _supply[n] = supply[n]; - _excess[n] = supply[n]; - sum += supply[n]; + // Check the value types + LEMON_ASSERT(std::numeric_limits::is_signed, + "The flow type of CapacityScaling must be signed"); + LEMON_ASSERT(std::numeric_limits::is_signed, + "The cost type of CapacityScaling must be signed"); + + // Resize vectors + _node_num = countNodes(_graph); + _arc_num = countArcs(_graph); + _res_arc_num = 2 * (_arc_num + _node_num); + _root = _node_num; + ++_node_num; + + _first_out.resize(_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(_node_num); + + _res_cap.resize(_res_arc_num); + _pi.resize(_node_num); + _excess.resize(_node_num); + _pred.resize(_node_num); + + // Copy the graph + int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1; + for (NodeIt n(_graph); n != INVALID; ++n, ++i) { + _node_id[n] = i; } - _valid_supply = sum == 0; + 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[_node_num] = k; for (ArcIt a(_graph); a != INVALID; ++a) { - _capacity[a] = capacity[a]; - _res_cap[a] = capacity[a]; + int fi = _arc_idf[a]; + int bi = _arc_idb[a]; + _reverse[fi] = bi; + _reverse[bi] = fi; } - - // Remove non-zero lower bounds - typename LowerMap::Value lcap; - for (ArcIt e(_graph); e != INVALID; ++e) { - if ((lcap = lower[e]) != 0) { - _capacity[e] -= lcap; - _res_cap[e] -= lcap; - _supply[_graph.source(e)] -= lcap; - _supply[_graph.target(e)] += lcap; - _excess[_graph.source(e)] -= lcap; - _excess[_graph.target(e)] += lcap; - } - } - } -/* - /// \brief General constructor (without lower bounds). - /// - /// General constructor (without lower bounds). - /// - /// \param digraph The digraph the algorithm runs on. - /// \param capacity The capacities (upper bounds) of the arcs. - /// \param cost The cost (length) values of the arcs. - /// \param supply The supply values of the nodes (signed). - CapacityScaling( const Digraph &digraph, - const CapacityMap &capacity, - const CostMap &cost, - const SupplyMap &supply ) : - _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost), - _supply(supply), _flow(NULL), _local_flow(false), - _potential(NULL), _local_potential(false), - _res_cap(capacity), _excess(supply), _pred(digraph), _dijkstra(NULL) - { - // Check the sum of supply values - Supply sum = 0; - for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n]; - _valid_supply = sum == 0; + + // Reset parameters + reset(); } - /// \brief Simple constructor (with lower bounds). + /// \name Parameters + /// The parameters of the algorithm can be specified using these + /// functions. + + /// @{ + + /// \brief Set the lower bounds on the arcs. /// - /// Simple constructor (with lower bounds). + /// 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 digraph The digraph the algorithm runs on. - /// \param lower The lower bounds of the arcs. - /// \param capacity The capacities (upper bounds) of the arcs. - /// \param cost The cost (length) values of the arcs. - /// \param s The source node. - /// \param t The target node. - /// \param flow_value 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). - CapacityScaling( const Digraph &digraph, - const LowerMap &lower, - const CapacityMap &capacity, - const CostMap &cost, - Node s, Node t, - Supply flow_value ) : - _graph(digraph), _lower(&lower), _capacity(capacity), _cost(cost), - _supply(digraph, 0), _flow(NULL), _local_flow(false), - _potential(NULL), _local_potential(false), - _res_cap(capacity), _excess(digraph, 0), _pred(digraph), _dijkstra(NULL) - { - // Remove non-zero lower bounds - _supply[s] = _excess[s] = flow_value; - _supply[t] = _excess[t] = -flow_value; - typename LowerMap::Value lcap; - for (ArcIt e(_graph); e != INVALID; ++e) { - if ((lcap = lower[e]) != 0) { - _capacity[e] -= lcap; - _res_cap[e] -= lcap; - _supply[_graph.source(e)] -= lcap; - _supply[_graph.target(e)] += lcap; - _excess[_graph.source(e)] -= lcap; - _excess[_graph.target(e)] += lcap; - } + /// \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 + CapacityScaling& 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]; } - _valid_supply = true; - } - - /// \brief Simple constructor (without lower bounds). - /// - /// Simple constructor (without lower bounds). - /// - /// \param digraph The digraph the algorithm runs on. - /// \param capacity The capacities (upper bounds) of the arcs. - /// \param cost The cost (length) values of the arcs. - /// \param s The source node. - /// \param t The target node. - /// \param flow_value 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). - CapacityScaling( const Digraph &digraph, - const CapacityMap &capacity, - const CostMap &cost, - Node s, Node t, - Supply flow_value ) : - _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost), - _supply(digraph, 0), _flow(NULL), _local_flow(false), - _potential(NULL), _local_potential(false), - _res_cap(capacity), _excess(digraph, 0), _pred(digraph), _dijkstra(NULL) - { - _supply[s] = _excess[s] = flow_value; - _supply[t] = _excess[t] = -flow_value; - _valid_supply = true; - } -*/ - /// Destructor. - ~CapacityScaling() { - if (_local_flow) delete _flow; - if (_local_potential) delete _potential; - delete _dijkstra; - } - - /// \brief Set the flow map. - /// - /// Set the flow map. - /// - /// \return \c (*this) - CapacityScaling& flowMap(FlowMap &map) { - if (_local_flow) { - delete _flow; - _local_flow = false; - } - _flow = ↦ return *this; } - /// \brief Set the potential map. + /// \brief Set the upper bounds (capacities) on the arcs. /// - /// Set the potential map. + /// 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 on each arc). /// - /// \return \c (*this) - CapacityScaling& potentialMap(PotentialMap &map) { - if (_local_potential) { - delete _potential; - _local_potential = false; + /// \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 + CapacityScaling& upperMap(const UpperMap& map) { + for (ArcIt a(_graph); a != INVALID; ++a) { + _upper[_arc_idf[a]] = map[a]; } - _potential = ↦ 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 + CapacityScaling& costMap(const CostMap& map) { + for (ArcIt a(_graph); a != INVALID; ++a) { + _cost[_arc_idf[a]] = map[a]; + _cost[_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 + CapacityScaling& 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 such 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) + CapacityScaling& stSupply(const Node& s, const Node& t, Value k) { + for (int i = 0; i != _node_num; ++i) { + _supply[i] = 0; + } + _supply[_node_id[s]] = k; + _supply[_node_id[t]] = -k; + return *this; + } + + /// @} + /// \name Execution control /// @{ @@ -416,15 +437,88 @@ /// \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 + /// CapacityScaling cs(graph); + /// cs.lowerMap(lower).upperMap(upper).costMap(cost) + /// .supplyMap(sup).run(); + /// \endcode + /// + /// This function can be called more than once. All the parameters + /// that have been given are kept for the next call, unless + /// \ref reset() is called, thus only the modified parameters + /// have to be set again. See \ref reset() for examples. + /// However the underlying digraph must not be modified after this + /// class have been constructed, since it copies the digraph. /// /// \param scaling Enable or disable capacity scaling. - /// If the maximum arc capacity and/or the amount of total supply + /// If the maximum upper bound and/or the amount of total supply /// is rather small, the algorithm could be slightly faster without /// scaling. /// - /// \return \c true if a feasible flow can be found. - bool run(bool scaling = true) { - return init(scaling) && start(); + /// \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 + ProblemType run(bool scaling = true) { + ProblemType pt = init(scaling); + if (pt != OPTIMAL) return pt; + return start(); + } + + /// \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 run() calls. If this function is not + /// used, all the parameters given before are kept for the next + /// \ref run() call. + /// However the underlying digraph must not be modified after this + /// class have been constructed, since it copies and extends the graph. + /// + /// For example, + /// \code + /// CapacityScaling cs(graph); + /// + /// // First run + /// cs.lowerMap(lower).upperMap(upper).costMap(cost) + /// .supplyMap(sup).run(); + /// + /// // Run again with modified cost map (reset() 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 reset() + /// // (the lower bounds will be set to zero on all arcs) + /// cs.reset(); + /// cs.upperMap(capacity).costMap(cost) + /// .supplyMap(sup).run(); + /// \endcode + /// + /// \return (*this) + CapacityScaling& reset() { + for (int i = 0; i != _node_num; ++i) { + _supply[i] = 0; + } + for (int j = 0; j != _res_arc_num; ++j) { + _lower[j] = 0; + _upper[j] = INF; + _cost[j] = _forward[j] ? 1 : -1; + } + _have_lower = false; + return *this; } /// @} @@ -432,97 +526,186 @@ /// \name Query Functions /// The results of the algorithm can be obtained using these /// functions.\n - /// \ref lemon::CapacityScaling::run() "run()" must be called before - /// using them. + /// The \ref run() function must be called before using them. /// @{ - /// \brief Return a const reference to the arc map storing the - /// found flow. + /// \brief Return the total cost of the found flow. /// - /// Return a const reference to the arc map storing 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. - const FlowMap& flowMap() const { - return *_flow; + 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(_cost[i])); + } + return c; } - /// \brief Return a const reference to the node map storing the - /// found potentials (the dual solution). - /// - /// Return a const reference to the node map storing the found - /// potentials (the dual solution). - /// - /// \pre \ref run() must be called before using this function. - const PotentialMap& potentialMap() const { - return *_potential; +#ifndef DOXYGEN + Cost totalCost() const { + return totalCost(); } +#endif /// \brief Return the flow on the given arc. /// - /// 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. - Capacity flow(const Arc& arc) const { - return (*_flow)[arc]; + Value flow(const Arc& a) const { + return _res_cap[_arc_idb[a]]; } - /// \brief Return the potential of the given node. + /// \brief Return the flow map (the primal solution). /// - /// Return the potential of the given node. + /// 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. - Cost potential(const Node& node) const { - return (*_potential)[node]; + template + void flowMap(FlowMap &map) const { + for (ArcIt a(_graph); a != INVALID; ++a) { + map.set(a, _res_cap[_arc_idb[a]]); + } } - /// \brief Return the total cost of the found flow. + /// \brief Return the potential (dual value) of the given node. /// - /// Return the total cost of the found flow. The complexity of the - /// function is \f$ O(e) \f$. + /// This function returns the potential (dual value) of the + /// given node. /// /// \pre \ref run() must be called before using this function. - Cost totalCost() const { - Cost c = 0; - for (ArcIt e(_graph); e != INVALID; ++e) - c += (*_flow)[e] * _cost[e]; - return c; + Cost potential(const Node& n) const { + return _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, _pi[_node_id[n]]); + } } /// @} private: - /// Initialize the algorithm. - bool init(bool scaling) { - if (!_valid_supply) return false; + // Initialize the algorithm + ProblemType init(bool scaling) { + if (_node_num == 0) return INFEASIBLE; - // Initializing maps - if (!_flow) { - _flow = new FlowMap(_graph); - _local_flow = true; + // Check the sum of supply values + _sum_supply = 0; + for (int i = 0; i != _root; ++i) { + _sum_supply += _supply[i]; } - if (!_potential) { - _potential = new PotentialMap(_graph); - _local_potential = true; + if (_sum_supply > 0) return INFEASIBLE; + + // Initialize maps + for (int i = 0; i != _root; ++i) { + _pi[i] = 0; + _excess[i] = _supply[i]; } - for (ArcIt e(_graph); e != INVALID; ++e) (*_flow)[e] = 0; - for (NodeIt n(_graph); n != INVALID; ++n) (*_potential)[n] = 0; - _dijkstra = new ResidualDijkstra( _graph, *_flow, _res_cap, _cost, - _excess, *_potential, _pred ); + // Remove non-zero lower bounds + if (_have_lower) { + for (int i = 0; i != _root; ++i) { + for (int j = _first_out[i]; j != _first_out[i+1]; ++j) { + if (_forward[j]) { + Value c = _lower[j]; + if (c >= 0) { + _res_cap[j] = _upper[j] < INF ? _upper[j] - c : INF; + } else { + _res_cap[j] = _upper[j] < INF + c ? _upper[j] - c : INF; + } + _excess[i] -= c; + _excess[_target[j]] += c; + } else { + _res_cap[j] = 0; + } + } + } + } else { + for (int j = 0; j != _res_arc_num; ++j) { + _res_cap[j] = _forward[j] ? _upper[j] : 0; + } + } - // Initializing delta value + // Handle negative costs + for (int u = 0; u != _root; ++u) { + for (int a = _first_out[u]; a != _first_out[u+1]; ++a) { + Value rc = _res_cap[a]; + if (_cost[a] < 0 && rc > 0) { + if (rc == INF) return UNBOUNDED; + _excess[u] -= rc; + _excess[_target[a]] += rc; + _res_cap[a] = 0; + _res_cap[_reverse[a]] += rc; + } + } + } + + // Handle GEQ supply type + if (_sum_supply < 0) { + _pi[_root] = 0; + _excess[_root] = -_sum_supply; + for (int a = _first_out[_root]; a != _res_arc_num; ++a) { + int u = _target[a]; + if (_excess[u] < 0) { + _res_cap[a] = -_excess[u] + 1; + } else { + _res_cap[a] = 1; + } + _res_cap[_reverse[a]] = 0; + _cost[a] = 0; + _cost[_reverse[a]] = 0; + } + } else { + _pi[_root] = 0; + _excess[_root] = 0; + for (int a = _first_out[_root]; a != _res_arc_num; ++a) { + _res_cap[a] = 1; + _res_cap[_reverse[a]] = 0; + _cost[a] = 0; + _cost[_reverse[a]] = 0; + } + } + + // Initialize delta value if (scaling) { // With scaling - Supply max_sup = 0, max_dem = 0; - for (NodeIt n(_graph); n != INVALID; ++n) { - if ( _supply[n] > max_sup) max_sup = _supply[n]; - if (-_supply[n] > max_dem) max_dem = -_supply[n]; + Value max_sup = 0, max_dem = 0; + for (int i = 0; i != _node_num; ++i) { + if ( _excess[i] > max_sup) max_sup = _excess[i]; + if (-_excess[i] > max_dem) max_dem = -_excess[i]; } - Capacity max_cap = 0; - for (ArcIt e(_graph); e != INVALID; ++e) { - if (_capacity[e] > max_cap) max_cap = _capacity[e]; + Value max_cap = 0; + for (int j = 0; j != _res_arc_num; ++j) { + if (_res_cap[j] > max_cap) max_cap = _res_cap[j]; } max_sup = std::min(std::min(max_sup, max_dem), max_cap); _phase_num = 0; @@ -533,53 +716,70 @@ _delta = 1; } - return true; + return OPTIMAL; } - bool start() { + ProblemType start() { + // Execute the algorithm + ProblemType pt; if (_delta > 1) - return startWithScaling(); + pt = startWithScaling(); else - return startWithoutScaling(); + pt = startWithoutScaling(); + + // Handle non-zero lower bounds + if (_have_lower) { + for (int j = 0; j != _res_arc_num - _node_num + 1; ++j) { + if (!_forward[j]) _res_cap[j] += _lower[j]; + } + } + + // Shift potentials if necessary + Cost pr = _pi[_root]; + if (_sum_supply < 0 || pr > 0) { + for (int i = 0; i != _node_num; ++i) { + _pi[i] -= pr; + } + } + + return pt; } - /// Execute the capacity scaling algorithm. - bool startWithScaling() { - // Processing capacity scaling phases - Node s, t; + // Execute the capacity scaling algorithm + ProblemType startWithScaling() { + // Process capacity scaling phases + int s, t; int phase_cnt = 0; int factor = 4; + ResidualDijkstra _dijkstra(*this); while (true) { - // Saturating all arcs not satisfying the optimality condition - for (ArcIt e(_graph); e != INVALID; ++e) { - Node u = _graph.source(e), v = _graph.target(e); - Cost c = _cost[e] + (*_potential)[u] - (*_potential)[v]; - if (c < 0 && _res_cap[e] >= _delta) { - _excess[u] -= _res_cap[e]; - _excess[v] += _res_cap[e]; - (*_flow)[e] = _capacity[e]; - _res_cap[e] = 0; - } - else if (c > 0 && (*_flow)[e] >= _delta) { - _excess[u] += (*_flow)[e]; - _excess[v] -= (*_flow)[e]; - (*_flow)[e] = 0; - _res_cap[e] = _capacity[e]; + // Saturate all arcs not satisfying the optimality condition + for (int u = 0; u != _node_num; ++u) { + for (int a = _first_out[u]; a != _first_out[u+1]; ++a) { + int v = _target[a]; + Cost c = _cost[a] + _pi[u] - _pi[v]; + Value rc = _res_cap[a]; + if (c < 0 && rc >= _delta) { + _excess[u] -= rc; + _excess[v] += rc; + _res_cap[a] = 0; + _res_cap[_reverse[a]] += rc; + } } } - // Finding excess nodes and deficit nodes + // Find excess nodes and deficit nodes _excess_nodes.clear(); _deficit_nodes.clear(); - for (NodeIt n(_graph); n != INVALID; ++n) { - if (_excess[n] >= _delta) _excess_nodes.push_back(n); - if (_excess[n] <= -_delta) _deficit_nodes.push_back(n); + for (int u = 0; u != _node_num; ++u) { + if (_excess[u] >= _delta) _excess_nodes.push_back(u); + if (_excess[u] <= -_delta) _deficit_nodes.push_back(u); } int next_node = 0, next_def_node = 0; - // Finding augmenting shortest paths + // Find augmenting shortest paths while (next_node < int(_excess_nodes.size())) { - // Checking deficit nodes + // Check deficit nodes if (_delta > 1) { bool delta_deficit = false; for ( ; next_def_node < int(_deficit_nodes.size()); @@ -592,44 +792,31 @@ if (!delta_deficit) break; } - // Running Dijkstra + // Run Dijkstra in the residual network s = _excess_nodes[next_node]; - if ((t = _dijkstra->run(s, _delta)) == INVALID) { + if ((t = _dijkstra.run(s, _delta)) == -1) { if (_delta > 1) { ++next_node; continue; } - return false; + return INFEASIBLE; } - // Augmenting along a shortest path from s to t. - Capacity d = std::min(_excess[s], -_excess[t]); - Node u = t; - Arc e; + // Augment along a shortest path from s to t + Value d = std::min(_excess[s], -_excess[t]); + int u = t; + int a; if (d > _delta) { - while ((e = _pred[u]) != INVALID) { - Capacity rc; - if (u == _graph.target(e)) { - rc = _res_cap[e]; - u = _graph.source(e); - } else { - rc = (*_flow)[e]; - u = _graph.target(e); - } - if (rc < d) d = rc; + while ((a = _pred[u]) != -1) { + if (_res_cap[a] < d) d = _res_cap[a]; + u = _source[a]; } } u = t; - while ((e = _pred[u]) != INVALID) { - if (u == _graph.target(e)) { - (*_flow)[e] += d; - _res_cap[e] -= d; - u = _graph.source(e); - } else { - (*_flow)[e] -= d; - _res_cap[e] += d; - u = _graph.target(e); - } + while ((a = _pred[u]) != -1) { + _res_cap[a] -= d; + _res_cap[_reverse[a]] += d; + u = _source[a]; } _excess[s] -= d; _excess[t] += d; @@ -638,74 +825,54 @@ } if (_delta == 1) break; - if (++phase_cnt > _phase_num / 4) factor = 2; + if (++phase_cnt == _phase_num / 4) factor = 2; _delta = _delta <= factor ? 1 : _delta / factor; } - // Handling non-zero lower bounds - if (_lower) { - for (ArcIt e(_graph); e != INVALID; ++e) - (*_flow)[e] += (*_lower)[e]; - } - return true; + return OPTIMAL; } - /// Execute the successive shortest path algorithm. - bool startWithoutScaling() { - // Finding excess nodes - for (NodeIt n(_graph); n != INVALID; ++n) - if (_excess[n] > 0) _excess_nodes.push_back(n); - if (_excess_nodes.size() == 0) return true; + // Execute the successive shortest path algorithm + ProblemType startWithoutScaling() { + // Find excess nodes + _excess_nodes.clear(); + for (int i = 0; i != _node_num; ++i) { + if (_excess[i] > 0) _excess_nodes.push_back(i); + } + if (_excess_nodes.size() == 0) return OPTIMAL; int next_node = 0; - // Finding shortest paths - Node s, t; + // Find shortest paths + int s, t; + ResidualDijkstra _dijkstra(*this); while ( _excess[_excess_nodes[next_node]] > 0 || ++next_node < int(_excess_nodes.size()) ) { - // Running Dijkstra + // Run Dijkstra in the residual network s = _excess_nodes[next_node]; - if ((t = _dijkstra->run(s)) == INVALID) return false; + if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE; - // Augmenting along a shortest path from s to t - Capacity d = std::min(_excess[s], -_excess[t]); - Node u = t; - Arc e; + // Augment along a shortest path from s to t + Value d = std::min(_excess[s], -_excess[t]); + int u = t; + int a; if (d > 1) { - while ((e = _pred[u]) != INVALID) { - Capacity rc; - if (u == _graph.target(e)) { - rc = _res_cap[e]; - u = _graph.source(e); - } else { - rc = (*_flow)[e]; - u = _graph.target(e); - } - if (rc < d) d = rc; + while ((a = _pred[u]) != -1) { + if (_res_cap[a] < d) d = _res_cap[a]; + u = _source[a]; } } u = t; - while ((e = _pred[u]) != INVALID) { - if (u == _graph.target(e)) { - (*_flow)[e] += d; - _res_cap[e] -= d; - u = _graph.source(e); - } else { - (*_flow)[e] -= d; - _res_cap[e] += d; - u = _graph.target(e); - } + while ((a = _pred[u]) != -1) { + _res_cap[a] -= d; + _res_cap[_reverse[a]] += d; + u = _source[a]; } _excess[s] -= d; _excess[t] += d; } - // Handling non-zero lower bounds - if (_lower) { - for (ArcIt e(_graph); e != INVALID; ++e) - (*_flow)[e] += (*_lower)[e]; - } - return true; + return OPTIMAL; } }; //class CapacityScaling