3 * This file is a part of LEMON, a generic C++ optimization library
5 * Copyright (C) 2003-2008
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
19 #ifndef LEMON_COST_SCALING_H
20 #define LEMON_COST_SCALING_H
22 /// \ingroup min_cost_flow_algs
24 /// \brief Cost scaling algorithm for finding a minimum cost flow.
30 #include <lemon/core.h>
31 #include <lemon/maps.h>
32 #include <lemon/math.h>
33 #include <lemon/static_graph.h>
34 #include <lemon/circulation.h>
35 #include <lemon/bellman_ford.h>
39 /// \brief Default traits class of CostScaling algorithm.
41 /// Default traits class of CostScaling algorithm.
42 /// \tparam GR Digraph type.
43 /// \tparam V The number type used for flow amounts, capacity bounds
44 /// and supply values. By default it is \c int.
45 /// \tparam C The number type used for costs and potentials.
46 /// By default it is the same as \c V.
48 template <typename GR, typename V = int, typename C = V>
50 template < typename GR, typename V = int, typename C = V,
51 bool integer = std::numeric_limits<C>::is_integer >
53 struct CostScalingDefaultTraits
55 /// The type of the digraph
57 /// The type of the flow amounts, capacity bounds and supply values
59 /// The type of the arc costs
62 /// \brief The large cost type used for internal computations
64 /// The large cost type used for internal computations.
65 /// It is \c long \c long if the \c Cost type is integer,
66 /// otherwise it is \c double.
67 /// \c Cost must be convertible to \c LargeCost.
68 typedef double LargeCost;
71 // Default traits class for integer cost types
72 template <typename GR, typename V, typename C>
73 struct CostScalingDefaultTraits<GR, V, C, true>
78 #ifdef LEMON_HAVE_LONG_LONG
79 typedef long long LargeCost;
81 typedef long LargeCost;
86 /// \addtogroup min_cost_flow_algs
89 /// \brief Implementation of the Cost Scaling algorithm for
90 /// finding a \ref min_cost_flow "minimum cost flow".
92 /// \ref CostScaling implements a cost scaling algorithm that performs
93 /// push/augment and relabel operations for finding a \ref min_cost_flow
94 /// "minimum cost flow" \ref amo93networkflows, \ref goldberg90approximation,
95 /// \ref goldberg97efficient, \ref bunnagel98efficient.
96 /// It is a highly efficient primal-dual solution method, which
97 /// can be viewed as the generalization of the \ref Preflow
98 /// "preflow push-relabel" algorithm for the maximum flow problem.
100 /// Most of the parameters of the problem (except for the digraph)
101 /// can be given using separate functions, and the algorithm can be
102 /// executed using the \ref run() function. If some parameters are not
103 /// specified, then default values will be used.
105 /// \tparam GR The digraph type the algorithm runs on.
106 /// \tparam V The number type used for flow amounts, capacity bounds
107 /// and supply values in the algorithm. By default it is \c int.
108 /// \tparam C The number type used for costs and potentials in the
109 /// algorithm. By default it is the same as \c V.
111 /// \warning Both number types must be signed and all input data must
113 /// \warning This algorithm does not support negative costs for such
114 /// arcs that have infinite upper bound.
116 /// \note %CostScaling provides three different internal methods,
117 /// from which the most efficient one is used by default.
118 /// For more information, see \ref Method.
120 template <typename GR, typename V, typename C, typename TR>
122 template < typename GR, typename V = int, typename C = V,
123 typename TR = CostScalingDefaultTraits<GR, V, C> >
129 /// The type of the digraph
130 typedef typename TR::Digraph Digraph;
131 /// The type of the flow amounts, capacity bounds and supply values
132 typedef typename TR::Value Value;
133 /// The type of the arc costs
134 typedef typename TR::Cost Cost;
136 /// \brief The large cost type
138 /// The large cost type used for internal computations.
139 /// Using the \ref CostScalingDefaultTraits "default traits class",
140 /// it is \c long \c long if the \c Cost type is integer,
141 /// otherwise it is \c double.
142 typedef typename TR::LargeCost LargeCost;
144 /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
149 /// \brief Problem type constants for the \c run() function.
151 /// Enum type containing the problem type constants that can be
152 /// returned by the \ref run() function of the algorithm.
154 /// The problem has no feasible solution (flow).
156 /// The problem has optimal solution (i.e. it is feasible and
157 /// bounded), and the algorithm has found optimal flow and node
158 /// potentials (primal and dual solutions).
160 /// The digraph contains an arc of negative cost and infinite
161 /// upper bound. It means that the objective function is unbounded
162 /// on that arc, however, note that it could actually be bounded
163 /// over the feasible flows, but this algroithm cannot handle
168 /// \brief Constants for selecting the internal method.
170 /// Enum type containing constants for selecting the internal method
171 /// for the \ref run() function.
173 /// \ref CostScaling provides three internal methods that differ mainly
174 /// in their base operations, which are used in conjunction with the
175 /// relabel operation.
176 /// By default, the so called \ref PARTIAL_AUGMENT
177 /// "Partial Augment-Relabel" method is used, which proved to be
178 /// the most efficient and the most robust on various test inputs.
179 /// However, the other methods can be selected using the \ref run()
180 /// function with the proper parameter.
182 /// Local push operations are used, i.e. flow is moved only on one
183 /// admissible arc at once.
185 /// Augment operations are used, i.e. flow is moved on admissible
186 /// paths from a node with excess to a node with deficit.
188 /// Partial augment operations are used, i.e. flow is moved on
189 /// admissible paths started from a node with excess, but the
190 /// lengths of these paths are limited. This method can be viewed
191 /// as a combined version of the previous two operations.
197 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
199 typedef std::vector<int> IntVector;
200 typedef std::vector<char> BoolVector;
201 typedef std::vector<Value> ValueVector;
202 typedef std::vector<Cost> CostVector;
203 typedef std::vector<LargeCost> LargeCostVector;
207 template <typename KT, typename VT>
208 class StaticVectorMap {
213 StaticVectorMap(std::vector<Value>& v) : _v(v) {}
215 const Value& operator[](const Key& key) const {
216 return _v[StaticDigraph::id(key)];
219 Value& operator[](const Key& key) {
220 return _v[StaticDigraph::id(key)];
223 void set(const Key& key, const Value& val) {
224 _v[StaticDigraph::id(key)] = val;
228 std::vector<Value>& _v;
231 typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
232 typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
236 // Data related to the underlying digraph
244 // Parameters of the problem
248 // Data structures for storing the digraph
252 IntVector _first_out;
264 ValueVector _res_cap;
265 LargeCostVector _cost;
269 std::deque<int> _active_nodes;
275 // Data for a StaticDigraph structure
276 typedef std::pair<int, int> IntPair;
278 std::vector<IntPair> _arc_vec;
279 std::vector<LargeCost> _cost_vec;
280 LargeCostArcMap _cost_map;
281 LargeCostNodeMap _pi_map;
285 /// \brief Constant for infinite upper bounds (capacities).
287 /// Constant for infinite upper bounds (capacities).
288 /// It is \c std::numeric_limits<Value>::infinity() if available,
289 /// \c std::numeric_limits<Value>::max() otherwise.
294 /// \name Named Template Parameters
297 template <typename T>
298 struct SetLargeCostTraits : public Traits {
302 /// \brief \ref named-templ-param "Named parameter" for setting
303 /// \c LargeCost type.
305 /// \ref named-templ-param "Named parameter" for setting \c LargeCost
306 /// type, which is used for internal computations in the algorithm.
307 /// \c Cost must be convertible to \c LargeCost.
308 template <typename T>
310 : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
311 typedef CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
318 /// \brief Constructor.
320 /// The constructor of the class.
322 /// \param graph The digraph the algorithm runs on.
323 CostScaling(const GR& graph) :
324 _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
325 _cost_map(_cost_vec), _pi_map(_pi),
326 INF(std::numeric_limits<Value>::has_infinity ?
327 std::numeric_limits<Value>::infinity() :
328 std::numeric_limits<Value>::max())
330 // Check the number types
331 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
332 "The flow type of CostScaling must be signed");
333 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
334 "The cost type of CostScaling must be signed");
336 // Reset data structures
341 /// The parameters of the algorithm can be specified using these
346 /// \brief Set the lower bounds on the arcs.
348 /// This function sets the lower bounds on the arcs.
349 /// If it is not used before calling \ref run(), the lower bounds
350 /// will be set to zero on all arcs.
352 /// \param map An arc map storing the lower bounds.
353 /// Its \c Value type must be convertible to the \c Value type
354 /// of the algorithm.
356 /// \return <tt>(*this)</tt>
357 template <typename LowerMap>
358 CostScaling& lowerMap(const LowerMap& map) {
360 for (ArcIt a(_graph); a != INVALID; ++a) {
361 _lower[_arc_idf[a]] = map[a];
362 _lower[_arc_idb[a]] = map[a];
367 /// \brief Set the upper bounds (capacities) on the arcs.
369 /// This function sets the upper bounds (capacities) on the arcs.
370 /// If it is not used before calling \ref run(), the upper bounds
371 /// will be set to \ref INF on all arcs (i.e. the flow value will be
372 /// unbounded from above).
374 /// \param map An arc map storing the upper bounds.
375 /// Its \c Value type must be convertible to the \c Value type
376 /// of the algorithm.
378 /// \return <tt>(*this)</tt>
379 template<typename UpperMap>
380 CostScaling& upperMap(const UpperMap& map) {
381 for (ArcIt a(_graph); a != INVALID; ++a) {
382 _upper[_arc_idf[a]] = map[a];
387 /// \brief Set the costs of the arcs.
389 /// This function sets the costs of the arcs.
390 /// If it is not used before calling \ref run(), the costs
391 /// will be set to \c 1 on all arcs.
393 /// \param map An arc map storing the costs.
394 /// Its \c Value type must be convertible to the \c Cost type
395 /// of the algorithm.
397 /// \return <tt>(*this)</tt>
398 template<typename CostMap>
399 CostScaling& costMap(const CostMap& map) {
400 for (ArcIt a(_graph); a != INVALID; ++a) {
401 _scost[_arc_idf[a]] = map[a];
402 _scost[_arc_idb[a]] = -map[a];
407 /// \brief Set the supply values of the nodes.
409 /// This function sets the supply values of the nodes.
410 /// If neither this function nor \ref stSupply() is used before
411 /// calling \ref run(), the supply of each node will be set to zero.
413 /// \param map A node map storing the supply values.
414 /// Its \c Value type must be convertible to the \c Value type
415 /// of the algorithm.
417 /// \return <tt>(*this)</tt>
418 template<typename SupplyMap>
419 CostScaling& supplyMap(const SupplyMap& map) {
420 for (NodeIt n(_graph); n != INVALID; ++n) {
421 _supply[_node_id[n]] = map[n];
426 /// \brief Set single source and target nodes and a supply value.
428 /// This function sets a single source node and a single target node
429 /// and the required flow value.
430 /// If neither this function nor \ref supplyMap() is used before
431 /// calling \ref run(), the supply of each node will be set to zero.
433 /// Using this function has the same effect as using \ref supplyMap()
434 /// with such a map in which \c k is assigned to \c s, \c -k is
435 /// assigned to \c t and all other nodes have zero supply value.
437 /// \param s The source node.
438 /// \param t The target node.
439 /// \param k The required amount of flow from node \c s to node \c t
440 /// (i.e. the supply of \c s and the demand of \c t).
442 /// \return <tt>(*this)</tt>
443 CostScaling& stSupply(const Node& s, const Node& t, Value k) {
444 for (int i = 0; i != _res_node_num; ++i) {
447 _supply[_node_id[s]] = k;
448 _supply[_node_id[t]] = -k;
454 /// \name Execution control
455 /// The algorithm can be executed using \ref run().
459 /// \brief Run the algorithm.
461 /// This function runs the algorithm.
462 /// The paramters can be specified using functions \ref lowerMap(),
463 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
466 /// CostScaling<ListDigraph> cs(graph);
467 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
468 /// .supplyMap(sup).run();
471 /// This function can be called more than once. All the given parameters
472 /// are kept for the next call, unless \ref resetParams() or \ref reset()
473 /// is used, thus only the modified parameters have to be set again.
474 /// If the underlying digraph was also modified after the construction
475 /// of the class (or the last \ref reset() call), then the \ref reset()
476 /// function must be called.
478 /// \param method The internal method that will be used in the
479 /// algorithm. For more information, see \ref Method.
480 /// \param factor The cost scaling factor. It must be larger than one.
482 /// \return \c INFEASIBLE if no feasible flow exists,
483 /// \n \c OPTIMAL if the problem has optimal solution
484 /// (i.e. it is feasible and bounded), and the algorithm has found
485 /// optimal flow and node potentials (primal and dual solutions),
486 /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
487 /// and infinite upper bound. It means that the objective function
488 /// is unbounded on that arc, however, note that it could actually be
489 /// bounded over the feasible flows, but this algroithm cannot handle
492 /// \see ProblemType, Method
493 /// \see resetParams(), reset()
494 ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
496 ProblemType pt = init();
497 if (pt != OPTIMAL) return pt;
502 /// \brief Reset all the parameters that have been given before.
504 /// This function resets all the paramaters that have been given
505 /// before using functions \ref lowerMap(), \ref upperMap(),
506 /// \ref costMap(), \ref supplyMap(), \ref stSupply().
508 /// It is useful for multiple \ref run() calls. Basically, all the given
509 /// parameters are kept for the next \ref run() call, unless
510 /// \ref resetParams() or \ref reset() is used.
511 /// If the underlying digraph was also modified after the construction
512 /// of the class or the last \ref reset() call, then the \ref reset()
513 /// function must be used, otherwise \ref resetParams() is sufficient.
517 /// CostScaling<ListDigraph> cs(graph);
520 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
521 /// .supplyMap(sup).run();
523 /// // Run again with modified cost map (resetParams() is not called,
524 /// // so only the cost map have to be set again)
526 /// cs.costMap(cost).run();
528 /// // Run again from scratch using resetParams()
529 /// // (the lower bounds will be set to zero on all arcs)
530 /// cs.resetParams();
531 /// cs.upperMap(capacity).costMap(cost)
532 /// .supplyMap(sup).run();
535 /// \return <tt>(*this)</tt>
537 /// \see reset(), run()
538 CostScaling& resetParams() {
539 for (int i = 0; i != _res_node_num; ++i) {
542 int limit = _first_out[_root];
543 for (int j = 0; j != limit; ++j) {
546 _scost[j] = _forward[j] ? 1 : -1;
548 for (int j = limit; j != _res_arc_num; ++j) {
552 _scost[_reverse[j]] = 0;
558 /// \brief Reset all the parameters that have been given before.
560 /// This function resets all the paramaters that have been given
561 /// before using functions \ref lowerMap(), \ref upperMap(),
562 /// \ref costMap(), \ref supplyMap(), \ref stSupply().
564 /// It is useful for multiple run() calls. If this function is not
565 /// used, all the parameters given before are kept for the next
567 /// However, the underlying digraph must not be modified after this
568 /// class have been constructed, since it copies and extends the graph.
569 /// \return <tt>(*this)</tt>
570 CostScaling& reset() {
572 _node_num = countNodes(_graph);
573 _arc_num = countArcs(_graph);
574 _res_node_num = _node_num + 1;
575 _res_arc_num = 2 * (_arc_num + _node_num);
578 _first_out.resize(_res_node_num + 1);
579 _forward.resize(_res_arc_num);
580 _source.resize(_res_arc_num);
581 _target.resize(_res_arc_num);
582 _reverse.resize(_res_arc_num);
584 _lower.resize(_res_arc_num);
585 _upper.resize(_res_arc_num);
586 _scost.resize(_res_arc_num);
587 _supply.resize(_res_node_num);
589 _res_cap.resize(_res_arc_num);
590 _cost.resize(_res_arc_num);
591 _pi.resize(_res_node_num);
592 _excess.resize(_res_node_num);
593 _next_out.resize(_res_node_num);
595 _arc_vec.reserve(_res_arc_num);
596 _cost_vec.reserve(_res_arc_num);
599 int i = 0, j = 0, k = 2 * _arc_num + _node_num;
600 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
604 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
606 for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
610 _target[j] = _node_id[_graph.runningNode(a)];
612 for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
616 _target[j] = _node_id[_graph.runningNode(a)];
629 _first_out[_res_node_num] = k;
630 for (ArcIt a(_graph); a != INVALID; ++a) {
631 int fi = _arc_idf[a];
632 int bi = _arc_idb[a];
644 /// \name Query Functions
645 /// The results of the algorithm can be obtained using these
647 /// The \ref run() function must be called before using them.
651 /// \brief Return the total cost of the found flow.
653 /// This function returns the total cost of the found flow.
654 /// Its complexity is O(e).
656 /// \note The return type of the function can be specified as a
657 /// template parameter. For example,
659 /// cs.totalCost<double>();
661 /// It is useful if the total cost cannot be stored in the \c Cost
662 /// type of the algorithm, which is the default return type of the
665 /// \pre \ref run() must be called before using this function.
666 template <typename Number>
667 Number totalCost() const {
669 for (ArcIt a(_graph); a != INVALID; ++a) {
671 c += static_cast<Number>(_res_cap[i]) *
672 (-static_cast<Number>(_scost[i]));
678 Cost totalCost() const {
679 return totalCost<Cost>();
683 /// \brief Return the flow on the given arc.
685 /// This function returns the flow on the given arc.
687 /// \pre \ref run() must be called before using this function.
688 Value flow(const Arc& a) const {
689 return _res_cap[_arc_idb[a]];
692 /// \brief Return the flow map (the primal solution).
694 /// This function copies the flow value on each arc into the given
695 /// map. The \c Value type of the algorithm must be convertible to
696 /// the \c Value type of the map.
698 /// \pre \ref run() must be called before using this function.
699 template <typename FlowMap>
700 void flowMap(FlowMap &map) const {
701 for (ArcIt a(_graph); a != INVALID; ++a) {
702 map.set(a, _res_cap[_arc_idb[a]]);
706 /// \brief Return the potential (dual value) of the given node.
708 /// This function returns the potential (dual value) of the
711 /// \pre \ref run() must be called before using this function.
712 Cost potential(const Node& n) const {
713 return static_cast<Cost>(_pi[_node_id[n]]);
716 /// \brief Return the potential map (the dual solution).
718 /// This function copies the potential (dual value) of each node
719 /// into the given map.
720 /// The \c Cost type of the algorithm must be convertible to the
721 /// \c Value type of the map.
723 /// \pre \ref run() must be called before using this function.
724 template <typename PotentialMap>
725 void potentialMap(PotentialMap &map) const {
726 for (NodeIt n(_graph); n != INVALID; ++n) {
727 map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
735 // Initialize the algorithm
737 if (_res_node_num <= 1) return INFEASIBLE;
739 // Check the sum of supply values
741 for (int i = 0; i != _root; ++i) {
742 _sum_supply += _supply[i];
744 if (_sum_supply > 0) return INFEASIBLE;
747 // Initialize vectors
748 for (int i = 0; i != _res_node_num; ++i) {
750 _excess[i] = _supply[i];
753 // Remove infinite upper bounds and check negative arcs
754 const Value MAX = std::numeric_limits<Value>::max();
757 for (int i = 0; i != _root; ++i) {
758 last_out = _first_out[i+1];
759 for (int j = _first_out[i]; j != last_out; ++j) {
761 Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
762 if (c >= MAX) return UNBOUNDED;
764 _excess[_target[j]] += c;
769 for (int i = 0; i != _root; ++i) {
770 last_out = _first_out[i+1];
771 for (int j = _first_out[i]; j != last_out; ++j) {
772 if (_forward[j] && _scost[j] < 0) {
774 if (c >= MAX) return UNBOUNDED;
776 _excess[_target[j]] += c;
781 Value ex, max_cap = 0;
782 for (int i = 0; i != _res_node_num; ++i) {
785 if (ex < 0) max_cap -= ex;
787 for (int j = 0; j != _res_arc_num; ++j) {
788 if (_upper[j] >= MAX) _upper[j] = max_cap;
791 // Initialize the large cost vector and the epsilon parameter
794 for (int i = 0; i != _root; ++i) {
795 last_out = _first_out[i+1];
796 for (int j = _first_out[i]; j != last_out; ++j) {
797 lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
799 if (lc > _epsilon) _epsilon = lc;
804 // Initialize maps for Circulation and remove non-zero lower bounds
805 ConstMap<Arc, Value> low(0);
806 typedef typename Digraph::template ArcMap<Value> ValueArcMap;
807 typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
808 ValueArcMap cap(_graph), flow(_graph);
809 ValueNodeMap sup(_graph);
810 for (NodeIt n(_graph); n != INVALID; ++n) {
811 sup[n] = _supply[_node_id[n]];
814 for (ArcIt a(_graph); a != INVALID; ++a) {
817 cap[a] = _upper[j] - c;
818 sup[_graph.source(a)] -= c;
819 sup[_graph.target(a)] += c;
822 for (ArcIt a(_graph); a != INVALID; ++a) {
823 cap[a] = _upper[_arc_idf[a]];
827 // Find a feasible flow using Circulation
828 Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
829 circ(_graph, low, cap, sup);
830 if (!circ.flowMap(flow).run()) return INFEASIBLE;
832 // Set residual capacities and handle GEQ supply type
833 if (_sum_supply < 0) {
834 for (ArcIt a(_graph); a != INVALID; ++a) {
836 _res_cap[_arc_idf[a]] = cap[a] - fa;
837 _res_cap[_arc_idb[a]] = fa;
838 sup[_graph.source(a)] -= fa;
839 sup[_graph.target(a)] += fa;
841 for (NodeIt n(_graph); n != INVALID; ++n) {
842 _excess[_node_id[n]] = sup[n];
844 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
846 int ra = _reverse[a];
847 _res_cap[a] = -_sum_supply + 1;
848 _res_cap[ra] = -_excess[u];
854 for (ArcIt a(_graph); a != INVALID; ++a) {
856 _res_cap[_arc_idf[a]] = cap[a] - fa;
857 _res_cap[_arc_idb[a]] = fa;
859 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
860 int ra = _reverse[a];
871 // Execute the algorithm and transform the results
872 void start(Method method) {
873 // Maximum path length for partial augment
874 const int MAX_PATH_LENGTH = 4;
876 // Execute the algorithm
884 case PARTIAL_AUGMENT:
885 startAugment(MAX_PATH_LENGTH);
889 // Compute node potentials for the original costs
892 for (int j = 0; j != _res_arc_num; ++j) {
893 if (_res_cap[j] > 0) {
894 _arc_vec.push_back(IntPair(_source[j], _target[j]));
895 _cost_vec.push_back(_scost[j]);
898 _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
900 typename BellmanFord<StaticDigraph, LargeCostArcMap>
901 ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
906 // Handle non-zero lower bounds
908 int limit = _first_out[_root];
909 for (int j = 0; j != limit; ++j) {
910 if (!_forward[j]) _res_cap[j] += _lower[j];
915 /// Execute the algorithm performing augment and relabel operations
916 void startAugment(int max_length = std::numeric_limits<int>::max()) {
917 // Paramters for heuristics
918 const int BF_HEURISTIC_EPSILON_BOUND = 1000;
919 const int BF_HEURISTIC_BOUND_FACTOR = 3;
921 // Perform cost scaling phases
922 IntVector pred_arc(_res_node_num);
923 std::vector<int> path_nodes;
924 for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
925 1 : _epsilon / _alpha )
927 // "Early Termination" heuristic: use Bellman-Ford algorithm
928 // to check if the current flow is optimal
929 if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
932 for (int j = 0; j != _res_arc_num; ++j) {
933 if (_res_cap[j] > 0) {
934 _arc_vec.push_back(IntPair(_source[j], _target[j]));
935 _cost_vec.push_back(_cost[j] + 1);
938 _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
940 BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
943 int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
944 for (int i = 0; i < K && !done; ++i)
945 done = bf.processNextWeakRound();
949 // Saturate arcs not satisfying the optimality condition
950 for (int a = 0; a != _res_arc_num; ++a) {
951 if (_res_cap[a] > 0 &&
952 _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
953 Value delta = _res_cap[a];
954 _excess[_source[a]] -= delta;
955 _excess[_target[a]] += delta;
957 _res_cap[_reverse[a]] += delta;
961 // Find active nodes (i.e. nodes with positive excess)
962 for (int u = 0; u != _res_node_num; ++u) {
963 if (_excess[u] > 0) _active_nodes.push_back(u);
966 // Initialize the next arcs
967 for (int u = 0; u != _res_node_num; ++u) {
968 _next_out[u] = _first_out[u];
971 // Perform partial augment and relabel operations
973 // Select an active node (FIFO selection)
974 while (_active_nodes.size() > 0 &&
975 _excess[_active_nodes.front()] <= 0) {
976 _active_nodes.pop_front();
978 if (_active_nodes.size() == 0) break;
979 int start = _active_nodes.front();
981 path_nodes.push_back(start);
983 // Find an augmenting path from the start node
985 while (_excess[tip] >= 0 &&
986 int(path_nodes.size()) <= max_length) {
988 LargeCost min_red_cost, rc;
989 int last_out = _sum_supply < 0 ?
990 _first_out[tip+1] : _first_out[tip+1] - 1;
991 for (int a = _next_out[tip]; a != last_out; ++a) {
992 if (_res_cap[a] > 0 &&
993 _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
998 path_nodes.push_back(tip);
1004 min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
1005 for (int a = _first_out[tip]; a != last_out; ++a) {
1006 rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
1007 if (_res_cap[a] > 0 && rc < min_red_cost) {
1011 _pi[tip] -= min_red_cost + _epsilon;
1013 // Reset the next arc of tip
1014 _next_out[tip] = _first_out[tip];
1018 path_nodes.pop_back();
1019 tip = path_nodes.back();
1025 // Augment along the found path (as much flow as possible)
1027 int u, v = path_nodes.front(), pa;
1028 for (int i = 1; i < int(path_nodes.size()); ++i) {
1032 delta = std::min(_res_cap[pa], _excess[u]);
1033 _res_cap[pa] -= delta;
1034 _res_cap[_reverse[pa]] += delta;
1035 _excess[u] -= delta;
1036 _excess[v] += delta;
1037 if (_excess[v] > 0 && _excess[v] <= delta)
1038 _active_nodes.push_back(v);
1044 /// Execute the algorithm performing push and relabel operations
1046 // Paramters for heuristics
1047 const int BF_HEURISTIC_EPSILON_BOUND = 1000;
1048 const int BF_HEURISTIC_BOUND_FACTOR = 3;
1050 // Perform cost scaling phases
1051 BoolVector hyper(_res_node_num, false);
1052 for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1053 1 : _epsilon / _alpha )
1055 // "Early Termination" heuristic: use Bellman-Ford algorithm
1056 // to check if the current flow is optimal
1057 if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
1060 for (int j = 0; j != _res_arc_num; ++j) {
1061 if (_res_cap[j] > 0) {
1062 _arc_vec.push_back(IntPair(_source[j], _target[j]));
1063 _cost_vec.push_back(_cost[j] + 1);
1066 _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
1068 BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
1071 int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
1072 for (int i = 0; i < K && !done; ++i)
1073 done = bf.processNextWeakRound();
1077 // Saturate arcs not satisfying the optimality condition
1078 for (int a = 0; a != _res_arc_num; ++a) {
1079 if (_res_cap[a] > 0 &&
1080 _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
1081 Value delta = _res_cap[a];
1082 _excess[_source[a]] -= delta;
1083 _excess[_target[a]] += delta;
1085 _res_cap[_reverse[a]] += delta;
1089 // Find active nodes (i.e. nodes with positive excess)
1090 for (int u = 0; u != _res_node_num; ++u) {
1091 if (_excess[u] > 0) _active_nodes.push_back(u);
1094 // Initialize the next arcs
1095 for (int u = 0; u != _res_node_num; ++u) {
1096 _next_out[u] = _first_out[u];
1099 // Perform push and relabel operations
1100 while (_active_nodes.size() > 0) {
1101 LargeCost min_red_cost, rc;
1103 int n, t, a, last_out = _res_arc_num;
1105 // Select an active node (FIFO selection)
1107 n = _active_nodes.front();
1108 last_out = _sum_supply < 0 ?
1109 _first_out[n+1] : _first_out[n+1] - 1;
1111 // Perform push operations if there are admissible arcs
1112 if (_excess[n] > 0) {
1113 for (a = _next_out[n]; a != last_out; ++a) {
1114 if (_res_cap[a] > 0 &&
1115 _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
1116 delta = std::min(_res_cap[a], _excess[n]);
1119 // Push-look-ahead heuristic
1120 Value ahead = -_excess[t];
1121 int last_out_t = _sum_supply < 0 ?
1122 _first_out[t+1] : _first_out[t+1] - 1;
1123 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1124 if (_res_cap[ta] > 0 &&
1125 _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0)
1126 ahead += _res_cap[ta];
1127 if (ahead >= delta) break;
1129 if (ahead < 0) ahead = 0;
1131 // Push flow along the arc
1132 if (ahead < delta) {
1133 _res_cap[a] -= ahead;
1134 _res_cap[_reverse[a]] += ahead;
1135 _excess[n] -= ahead;
1136 _excess[t] += ahead;
1137 _active_nodes.push_front(t);
1142 _res_cap[a] -= delta;
1143 _res_cap[_reverse[a]] += delta;
1144 _excess[n] -= delta;
1145 _excess[t] += delta;
1146 if (_excess[t] > 0 && _excess[t] <= delta)
1147 _active_nodes.push_back(t);
1150 if (_excess[n] == 0) {
1159 // Relabel the node if it is still active (or hyper)
1160 if (_excess[n] > 0 || hyper[n]) {
1161 min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
1162 for (int a = _first_out[n]; a != last_out; ++a) {
1163 rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
1164 if (_res_cap[a] > 0 && rc < min_red_cost) {
1168 _pi[n] -= min_red_cost + _epsilon;
1171 // Reset the next arc
1172 _next_out[n] = _first_out[n];
1175 // Remove nodes that are not active nor hyper
1177 while ( _active_nodes.size() > 0 &&
1178 _excess[_active_nodes.front()] <= 0 &&
1179 !hyper[_active_nodes.front()] ) {
1180 _active_nodes.pop_front();
1186 }; //class CostScaling
1192 #endif //LEMON_COST_SCALING_H