1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
3 * This file is a part of LEMON, a generic C++ optimization library.
5 * Copyright (C) 2003-2010
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 /// In general, \ref NetworkSimplex and \ref CostScaling are the fastest
101 /// implementations available in LEMON for this problem.
103 /// Most of the parameters of the problem (except for the digraph)
104 /// can be given using separate functions, and the algorithm can be
105 /// executed using the \ref run() function. If some parameters are not
106 /// specified, then default values will be used.
108 /// \tparam GR The digraph type the algorithm runs on.
109 /// \tparam V The number type used for flow amounts, capacity bounds
110 /// and supply values in the algorithm. By default, it is \c int.
111 /// \tparam C The number type used for costs and potentials in the
112 /// algorithm. By default, it is the same as \c V.
113 /// \tparam TR The traits class that defines various types used by the
114 /// algorithm. By default, it is \ref CostScalingDefaultTraits
115 /// "CostScalingDefaultTraits<GR, V, C>".
116 /// In most cases, this parameter should not be set directly,
117 /// consider to use the named template parameters instead.
119 /// \warning Both \c V and \c C must be signed number types.
120 /// \warning All input data (capacities, supply values, and costs) must
122 /// \warning This algorithm does not support negative costs for
123 /// arcs having infinite upper bound.
125 /// \note %CostScaling provides three different internal methods,
126 /// from which the most efficient one is used by default.
127 /// For more information, see \ref Method.
129 template <typename GR, typename V, typename C, typename TR>
131 template < typename GR, typename V = int, typename C = V,
132 typename TR = CostScalingDefaultTraits<GR, V, C> >
138 /// The type of the digraph
139 typedef typename TR::Digraph Digraph;
140 /// The type of the flow amounts, capacity bounds and supply values
141 typedef typename TR::Value Value;
142 /// The type of the arc costs
143 typedef typename TR::Cost Cost;
145 /// \brief The large cost type
147 /// The large cost type used for internal computations.
148 /// By default, it is \c long \c long if the \c Cost type is integer,
149 /// otherwise it is \c double.
150 typedef typename TR::LargeCost LargeCost;
152 /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
157 /// \brief Problem type constants for the \c run() function.
159 /// Enum type containing the problem type constants that can be
160 /// returned by the \ref run() function of the algorithm.
162 /// The problem has no feasible solution (flow).
164 /// The problem has optimal solution (i.e. it is feasible and
165 /// bounded), and the algorithm has found optimal flow and node
166 /// potentials (primal and dual solutions).
168 /// The digraph contains an arc of negative cost and infinite
169 /// upper bound. It means that the objective function is unbounded
170 /// on that arc, however, note that it could actually be bounded
171 /// over the feasible flows, but this algroithm cannot handle
176 /// \brief Constants for selecting the internal method.
178 /// Enum type containing constants for selecting the internal method
179 /// for the \ref run() function.
181 /// \ref CostScaling provides three internal methods that differ mainly
182 /// in their base operations, which are used in conjunction with the
183 /// relabel operation.
184 /// By default, the so called \ref PARTIAL_AUGMENT
185 /// "Partial Augment-Relabel" method is used, which turned out to be
186 /// the most efficient and the most robust on various test inputs.
187 /// However, the other methods can be selected using the \ref run()
188 /// function with the proper parameter.
190 /// Local push operations are used, i.e. flow is moved only on one
191 /// admissible arc at once.
193 /// Augment operations are used, i.e. flow is moved on admissible
194 /// paths from a node with excess to a node with deficit.
196 /// Partial augment operations are used, i.e. flow is moved on
197 /// admissible paths started from a node with excess, but the
198 /// lengths of these paths are limited. This method can be viewed
199 /// as a combined version of the previous two operations.
205 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
207 typedef std::vector<int> IntVector;
208 typedef std::vector<Value> ValueVector;
209 typedef std::vector<Cost> CostVector;
210 typedef std::vector<LargeCost> LargeCostVector;
211 typedef std::vector<char> BoolVector;
212 // Note: vector<char> is used instead of vector<bool> for efficiency reasons
216 template <typename KT, typename VT>
217 class StaticVectorMap {
222 StaticVectorMap(std::vector<Value>& v) : _v(v) {}
224 const Value& operator[](const Key& key) const {
225 return _v[StaticDigraph::id(key)];
228 Value& operator[](const Key& key) {
229 return _v[StaticDigraph::id(key)];
232 void set(const Key& key, const Value& val) {
233 _v[StaticDigraph::id(key)] = val;
237 std::vector<Value>& _v;
240 typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
241 typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
245 // Data related to the underlying digraph
253 // Parameters of the problem
258 // Data structures for storing the digraph
262 IntVector _first_out;
274 ValueVector _res_cap;
275 LargeCostVector _cost;
279 std::deque<int> _active_nodes;
286 IntVector _bucket_next;
287 IntVector _bucket_prev;
291 // Data for a StaticDigraph structure
292 typedef std::pair<int, int> IntPair;
294 std::vector<IntPair> _arc_vec;
295 std::vector<LargeCost> _cost_vec;
296 LargeCostArcMap _cost_map;
297 LargeCostNodeMap _pi_map;
301 /// \brief Constant for infinite upper bounds (capacities).
303 /// Constant for infinite upper bounds (capacities).
304 /// It is \c std::numeric_limits<Value>::infinity() if available,
305 /// \c std::numeric_limits<Value>::max() otherwise.
310 /// \name Named Template Parameters
313 template <typename T>
314 struct SetLargeCostTraits : public Traits {
318 /// \brief \ref named-templ-param "Named parameter" for setting
319 /// \c LargeCost type.
321 /// \ref named-templ-param "Named parameter" for setting \c LargeCost
322 /// type, which is used for internal computations in the algorithm.
323 /// \c Cost must be convertible to \c LargeCost.
324 template <typename T>
326 : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
327 typedef CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
338 /// \brief Constructor.
340 /// The constructor of the class.
342 /// \param graph The digraph the algorithm runs on.
343 CostScaling(const GR& graph) :
344 _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
345 _cost_map(_cost_vec), _pi_map(_pi),
346 INF(std::numeric_limits<Value>::has_infinity ?
347 std::numeric_limits<Value>::infinity() :
348 std::numeric_limits<Value>::max())
350 // Check the number types
351 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
352 "The flow type of CostScaling must be signed");
353 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
354 "The cost type of CostScaling must be signed");
356 // Reset data structures
361 /// The parameters of the algorithm can be specified using these
366 /// \brief Set the lower bounds on the arcs.
368 /// This function sets the lower bounds on the arcs.
369 /// If it is not used before calling \ref run(), the lower bounds
370 /// will be set to zero on all arcs.
372 /// \param map An arc map storing the lower bounds.
373 /// Its \c Value type must be convertible to the \c Value type
374 /// of the algorithm.
376 /// \return <tt>(*this)</tt>
377 template <typename LowerMap>
378 CostScaling& lowerMap(const LowerMap& map) {
380 for (ArcIt a(_graph); a != INVALID; ++a) {
381 _lower[_arc_idf[a]] = map[a];
382 _lower[_arc_idb[a]] = map[a];
387 /// \brief Set the upper bounds (capacities) on the arcs.
389 /// This function sets the upper bounds (capacities) on the arcs.
390 /// If it is not used before calling \ref run(), the upper bounds
391 /// will be set to \ref INF on all arcs (i.e. the flow value will be
392 /// unbounded from above).
394 /// \param map An arc map storing the upper bounds.
395 /// Its \c Value type must be convertible to the \c Value type
396 /// of the algorithm.
398 /// \return <tt>(*this)</tt>
399 template<typename UpperMap>
400 CostScaling& upperMap(const UpperMap& map) {
401 for (ArcIt a(_graph); a != INVALID; ++a) {
402 _upper[_arc_idf[a]] = map[a];
407 /// \brief Set the costs of the arcs.
409 /// This function sets the costs of the arcs.
410 /// If it is not used before calling \ref run(), the costs
411 /// will be set to \c 1 on all arcs.
413 /// \param map An arc map storing the costs.
414 /// Its \c Value type must be convertible to the \c Cost type
415 /// of the algorithm.
417 /// \return <tt>(*this)</tt>
418 template<typename CostMap>
419 CostScaling& costMap(const CostMap& map) {
420 for (ArcIt a(_graph); a != INVALID; ++a) {
421 _scost[_arc_idf[a]] = map[a];
422 _scost[_arc_idb[a]] = -map[a];
427 /// \brief Set the supply values of the nodes.
429 /// This function sets the supply values of the nodes.
430 /// If neither this function nor \ref stSupply() is used before
431 /// calling \ref run(), the supply of each node will be set to zero.
433 /// \param map A node map storing the supply values.
434 /// Its \c Value type must be convertible to the \c Value type
435 /// of the algorithm.
437 /// \return <tt>(*this)</tt>
438 template<typename SupplyMap>
439 CostScaling& supplyMap(const SupplyMap& map) {
440 for (NodeIt n(_graph); n != INVALID; ++n) {
441 _supply[_node_id[n]] = map[n];
446 /// \brief Set single source and target nodes and a supply value.
448 /// This function sets a single source node and a single target node
449 /// and the required flow value.
450 /// If neither this function nor \ref supplyMap() is used before
451 /// calling \ref run(), the supply of each node will be set to zero.
453 /// Using this function has the same effect as using \ref supplyMap()
454 /// with a map in which \c k is assigned to \c s, \c -k is
455 /// assigned to \c t and all other nodes have zero supply value.
457 /// \param s The source node.
458 /// \param t The target node.
459 /// \param k The required amount of flow from node \c s to node \c t
460 /// (i.e. the supply of \c s and the demand of \c t).
462 /// \return <tt>(*this)</tt>
463 CostScaling& stSupply(const Node& s, const Node& t, Value k) {
464 for (int i = 0; i != _res_node_num; ++i) {
467 _supply[_node_id[s]] = k;
468 _supply[_node_id[t]] = -k;
474 /// \name Execution control
475 /// The algorithm can be executed using \ref run().
479 /// \brief Run the algorithm.
481 /// This function runs the algorithm.
482 /// The paramters can be specified using functions \ref lowerMap(),
483 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
486 /// CostScaling<ListDigraph> cs(graph);
487 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
488 /// .supplyMap(sup).run();
491 /// This function can be called more than once. All the given parameters
492 /// are kept for the next call, unless \ref resetParams() or \ref reset()
493 /// is used, thus only the modified parameters have to be set again.
494 /// If the underlying digraph was also modified after the construction
495 /// of the class (or the last \ref reset() call), then the \ref reset()
496 /// function must be called.
498 /// \param method The internal method that will be used in the
499 /// algorithm. For more information, see \ref Method.
500 /// \param factor The cost scaling factor. It must be larger than one.
502 /// \return \c INFEASIBLE if no feasible flow exists,
503 /// \n \c OPTIMAL if the problem has optimal solution
504 /// (i.e. it is feasible and bounded), and the algorithm has found
505 /// optimal flow and node potentials (primal and dual solutions),
506 /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
507 /// and infinite upper bound. It means that the objective function
508 /// is unbounded on that arc, however, note that it could actually be
509 /// bounded over the feasible flows, but this algroithm cannot handle
512 /// \see ProblemType, Method
513 /// \see resetParams(), reset()
514 ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
516 ProblemType pt = init();
517 if (pt != OPTIMAL) return pt;
522 /// \brief Reset all the parameters that have been given before.
524 /// This function resets all the paramaters that have been given
525 /// before using functions \ref lowerMap(), \ref upperMap(),
526 /// \ref costMap(), \ref supplyMap(), \ref stSupply().
528 /// It is useful for multiple \ref run() calls. Basically, all the given
529 /// parameters are kept for the next \ref run() call, unless
530 /// \ref resetParams() or \ref reset() is used.
531 /// If the underlying digraph was also modified after the construction
532 /// of the class or the last \ref reset() call, then the \ref reset()
533 /// function must be used, otherwise \ref resetParams() is sufficient.
537 /// CostScaling<ListDigraph> cs(graph);
540 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
541 /// .supplyMap(sup).run();
543 /// // Run again with modified cost map (resetParams() is not called,
544 /// // so only the cost map have to be set again)
546 /// cs.costMap(cost).run();
548 /// // Run again from scratch using resetParams()
549 /// // (the lower bounds will be set to zero on all arcs)
550 /// cs.resetParams();
551 /// cs.upperMap(capacity).costMap(cost)
552 /// .supplyMap(sup).run();
555 /// \return <tt>(*this)</tt>
557 /// \see reset(), run()
558 CostScaling& resetParams() {
559 for (int i = 0; i != _res_node_num; ++i) {
562 int limit = _first_out[_root];
563 for (int j = 0; j != limit; ++j) {
566 _scost[j] = _forward[j] ? 1 : -1;
568 for (int j = limit; j != _res_arc_num; ++j) {
572 _scost[_reverse[j]] = 0;
578 /// \brief Reset the internal data structures and all the parameters
579 /// that have been given before.
581 /// This function resets the internal data structures and all the
582 /// paramaters that have been given before using functions \ref lowerMap(),
583 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
585 /// It is useful for multiple \ref run() calls. By default, all the given
586 /// parameters are kept for the next \ref run() call, unless
587 /// \ref resetParams() or \ref reset() is used.
588 /// If the underlying digraph was also modified after the construction
589 /// of the class or the last \ref reset() call, then the \ref reset()
590 /// function must be used, otherwise \ref resetParams() is sufficient.
592 /// See \ref resetParams() for examples.
594 /// \return <tt>(*this)</tt>
596 /// \see resetParams(), run()
597 CostScaling& reset() {
599 _node_num = countNodes(_graph);
600 _arc_num = countArcs(_graph);
601 _res_node_num = _node_num + 1;
602 _res_arc_num = 2 * (_arc_num + _node_num);
605 _first_out.resize(_res_node_num + 1);
606 _forward.resize(_res_arc_num);
607 _source.resize(_res_arc_num);
608 _target.resize(_res_arc_num);
609 _reverse.resize(_res_arc_num);
611 _lower.resize(_res_arc_num);
612 _upper.resize(_res_arc_num);
613 _scost.resize(_res_arc_num);
614 _supply.resize(_res_node_num);
616 _res_cap.resize(_res_arc_num);
617 _cost.resize(_res_arc_num);
618 _pi.resize(_res_node_num);
619 _excess.resize(_res_node_num);
620 _next_out.resize(_res_node_num);
622 _arc_vec.reserve(_res_arc_num);
623 _cost_vec.reserve(_res_arc_num);
626 int i = 0, j = 0, k = 2 * _arc_num + _node_num;
627 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
631 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
633 for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
637 _target[j] = _node_id[_graph.runningNode(a)];
639 for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
643 _target[j] = _node_id[_graph.runningNode(a)];
656 _first_out[_res_node_num] = k;
657 for (ArcIt a(_graph); a != INVALID; ++a) {
658 int fi = _arc_idf[a];
659 int bi = _arc_idb[a];
671 /// \name Query Functions
672 /// The results of the algorithm can be obtained using these
674 /// The \ref run() function must be called before using them.
678 /// \brief Return the total cost of the found flow.
680 /// This function returns the total cost of the found flow.
681 /// Its complexity is O(e).
683 /// \note The return type of the function can be specified as a
684 /// template parameter. For example,
686 /// cs.totalCost<double>();
688 /// It is useful if the total cost cannot be stored in the \c Cost
689 /// type of the algorithm, which is the default return type of the
692 /// \pre \ref run() must be called before using this function.
693 template <typename Number>
694 Number totalCost() const {
696 for (ArcIt a(_graph); a != INVALID; ++a) {
698 c += static_cast<Number>(_res_cap[i]) *
699 (-static_cast<Number>(_scost[i]));
705 Cost totalCost() const {
706 return totalCost<Cost>();
710 /// \brief Return the flow on the given arc.
712 /// This function returns the flow on the given arc.
714 /// \pre \ref run() must be called before using this function.
715 Value flow(const Arc& a) const {
716 return _res_cap[_arc_idb[a]];
719 /// \brief Return the flow map (the primal solution).
721 /// This function copies the flow value on each arc into the given
722 /// map. The \c Value type of the algorithm must be convertible to
723 /// the \c Value type of the map.
725 /// \pre \ref run() must be called before using this function.
726 template <typename FlowMap>
727 void flowMap(FlowMap &map) const {
728 for (ArcIt a(_graph); a != INVALID; ++a) {
729 map.set(a, _res_cap[_arc_idb[a]]);
733 /// \brief Return the potential (dual value) of the given node.
735 /// This function returns the potential (dual value) of the
738 /// \pre \ref run() must be called before using this function.
739 Cost potential(const Node& n) const {
740 return static_cast<Cost>(_pi[_node_id[n]]);
743 /// \brief Return the potential map (the dual solution).
745 /// This function copies the potential (dual value) of each node
746 /// into the given map.
747 /// The \c Cost type of the algorithm must be convertible to the
748 /// \c Value type of the map.
750 /// \pre \ref run() must be called before using this function.
751 template <typename PotentialMap>
752 void potentialMap(PotentialMap &map) const {
753 for (NodeIt n(_graph); n != INVALID; ++n) {
754 map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
762 // Initialize the algorithm
764 if (_res_node_num <= 1) return INFEASIBLE;
766 // Check the sum of supply values
768 for (int i = 0; i != _root; ++i) {
769 _sum_supply += _supply[i];
771 if (_sum_supply > 0) return INFEASIBLE;
774 // Initialize vectors
775 for (int i = 0; i != _res_node_num; ++i) {
777 _excess[i] = _supply[i];
780 // Remove infinite upper bounds and check negative arcs
781 const Value MAX = std::numeric_limits<Value>::max();
784 for (int i = 0; i != _root; ++i) {
785 last_out = _first_out[i+1];
786 for (int j = _first_out[i]; j != last_out; ++j) {
788 Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
789 if (c >= MAX) return UNBOUNDED;
791 _excess[_target[j]] += c;
796 for (int i = 0; i != _root; ++i) {
797 last_out = _first_out[i+1];
798 for (int j = _first_out[i]; j != last_out; ++j) {
799 if (_forward[j] && _scost[j] < 0) {
801 if (c >= MAX) return UNBOUNDED;
803 _excess[_target[j]] += c;
808 Value ex, max_cap = 0;
809 for (int i = 0; i != _res_node_num; ++i) {
812 if (ex < 0) max_cap -= ex;
814 for (int j = 0; j != _res_arc_num; ++j) {
815 if (_upper[j] >= MAX) _upper[j] = max_cap;
818 // Initialize the large cost vector and the epsilon parameter
821 for (int i = 0; i != _root; ++i) {
822 last_out = _first_out[i+1];
823 for (int j = _first_out[i]; j != last_out; ++j) {
824 lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
826 if (lc > _epsilon) _epsilon = lc;
831 // Initialize maps for Circulation and remove non-zero lower bounds
832 ConstMap<Arc, Value> low(0);
833 typedef typename Digraph::template ArcMap<Value> ValueArcMap;
834 typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
835 ValueArcMap cap(_graph), flow(_graph);
836 ValueNodeMap sup(_graph);
837 for (NodeIt n(_graph); n != INVALID; ++n) {
838 sup[n] = _supply[_node_id[n]];
841 for (ArcIt a(_graph); a != INVALID; ++a) {
844 cap[a] = _upper[j] - c;
845 sup[_graph.source(a)] -= c;
846 sup[_graph.target(a)] += c;
849 for (ArcIt a(_graph); a != INVALID; ++a) {
850 cap[a] = _upper[_arc_idf[a]];
855 for (NodeIt n(_graph); n != INVALID; ++n) {
856 if (sup[n] > 0) ++_sup_node_num;
859 // Find a feasible flow using Circulation
860 Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
861 circ(_graph, low, cap, sup);
862 if (!circ.flowMap(flow).run()) return INFEASIBLE;
864 // Set residual capacities and handle GEQ supply type
865 if (_sum_supply < 0) {
866 for (ArcIt a(_graph); a != INVALID; ++a) {
868 _res_cap[_arc_idf[a]] = cap[a] - fa;
869 _res_cap[_arc_idb[a]] = fa;
870 sup[_graph.source(a)] -= fa;
871 sup[_graph.target(a)] += fa;
873 for (NodeIt n(_graph); n != INVALID; ++n) {
874 _excess[_node_id[n]] = sup[n];
876 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
878 int ra = _reverse[a];
879 _res_cap[a] = -_sum_supply + 1;
880 _res_cap[ra] = -_excess[u];
886 for (ArcIt a(_graph); a != INVALID; ++a) {
888 _res_cap[_arc_idf[a]] = cap[a] - fa;
889 _res_cap[_arc_idb[a]] = fa;
891 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
892 int ra = _reverse[a];
900 // Initialize data structures for buckets
901 _max_rank = _alpha * _res_node_num;
902 _buckets.resize(_max_rank);
903 _bucket_next.resize(_res_node_num + 1);
904 _bucket_prev.resize(_res_node_num + 1);
905 _rank.resize(_res_node_num + 1);
910 // Execute the algorithm and transform the results
911 void start(Method method) {
912 const int MAX_PARTIAL_PATH_LENGTH = 4;
919 startAugment(_res_node_num - 1);
921 case PARTIAL_AUGMENT:
922 startAugment(MAX_PARTIAL_PATH_LENGTH);
926 // Compute node potentials for the original costs
929 for (int j = 0; j != _res_arc_num; ++j) {
930 if (_res_cap[j] > 0) {
931 _arc_vec.push_back(IntPair(_source[j], _target[j]));
932 _cost_vec.push_back(_scost[j]);
935 _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
937 typename BellmanFord<StaticDigraph, LargeCostArcMap>
938 ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
943 // Handle non-zero lower bounds
945 int limit = _first_out[_root];
946 for (int j = 0; j != limit; ++j) {
947 if (!_forward[j]) _res_cap[j] += _lower[j];
952 // Initialize a cost scaling phase
954 // Saturate arcs not satisfying the optimality condition
955 for (int u = 0; u != _res_node_num; ++u) {
956 int last_out = _first_out[u+1];
957 LargeCost pi_u = _pi[u];
958 for (int a = _first_out[u]; a != last_out; ++a) {
959 Value delta = _res_cap[a];
962 if (_cost[a] + pi_u - _pi[v] < 0) {
966 _res_cap[_reverse[a]] += delta;
972 // Find active nodes (i.e. nodes with positive excess)
973 for (int u = 0; u != _res_node_num; ++u) {
974 if (_excess[u] > 0) _active_nodes.push_back(u);
977 // Initialize the next arcs
978 for (int u = 0; u != _res_node_num; ++u) {
979 _next_out[u] = _first_out[u];
983 // Early termination heuristic
984 bool earlyTermination() {
985 const double EARLY_TERM_FACTOR = 3.0;
987 // Build a static residual graph
990 for (int j = 0; j != _res_arc_num; ++j) {
991 if (_res_cap[j] > 0) {
992 _arc_vec.push_back(IntPair(_source[j], _target[j]));
993 _cost_vec.push_back(_cost[j] + 1);
996 _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
998 // Run Bellman-Ford algorithm to check if the current flow is optimal
999 BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
1002 int K = int(EARLY_TERM_FACTOR * std::sqrt(double(_res_node_num)));
1003 for (int i = 0; i < K && !done; ++i) {
1004 done = bf.processNextWeakRound();
1009 // Global potential update heuristic
1010 void globalUpdate() {
1011 const int bucket_end = _root + 1;
1013 // Initialize buckets
1014 for (int r = 0; r != _max_rank; ++r) {
1015 _buckets[r] = bucket_end;
1017 Value total_excess = 0;
1018 int b0 = bucket_end;
1019 for (int i = 0; i != _res_node_num; ++i) {
1020 if (_excess[i] < 0) {
1022 _bucket_next[i] = b0;
1023 _bucket_prev[b0] = i;
1026 total_excess += _excess[i];
1027 _rank[i] = _max_rank;
1030 if (total_excess == 0) return;
1033 // Search the buckets
1035 for ( ; r != _max_rank; ++r) {
1036 while (_buckets[r] != bucket_end) {
1037 // Remove the first node from the current bucket
1038 int u = _buckets[r];
1039 _buckets[r] = _bucket_next[u];
1041 // Search the incomming arcs of u
1042 LargeCost pi_u = _pi[u];
1043 int last_out = _first_out[u+1];
1044 for (int a = _first_out[u]; a != last_out; ++a) {
1045 int ra = _reverse[a];
1046 if (_res_cap[ra] > 0) {
1047 int v = _source[ra];
1048 int old_rank_v = _rank[v];
1049 if (r < old_rank_v) {
1050 // Compute the new rank of v
1051 LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
1052 int new_rank_v = old_rank_v;
1053 if (nrc < LargeCost(_max_rank)) {
1054 new_rank_v = r + 1 + static_cast<int>(nrc);
1057 // Change the rank of v
1058 if (new_rank_v < old_rank_v) {
1059 _rank[v] = new_rank_v;
1060 _next_out[v] = _first_out[v];
1062 // Remove v from its old bucket
1063 if (old_rank_v < _max_rank) {
1064 if (_buckets[old_rank_v] == v) {
1065 _buckets[old_rank_v] = _bucket_next[v];
1067 int pv = _bucket_prev[v], nv = _bucket_next[v];
1068 _bucket_next[pv] = nv;
1069 _bucket_prev[nv] = pv;
1073 // Insert v into its new bucket
1074 int nv = _buckets[new_rank_v];
1075 _bucket_next[v] = nv;
1076 _bucket_prev[nv] = v;
1077 _buckets[new_rank_v] = v;
1083 // Finish search if there are no more active nodes
1084 if (_excess[u] > 0) {
1085 total_excess -= _excess[u];
1086 if (total_excess <= 0) break;
1089 if (total_excess <= 0) break;
1093 for (int u = 0; u != _res_node_num; ++u) {
1094 int k = std::min(_rank[u], r);
1096 _pi[u] -= _epsilon * k;
1097 _next_out[u] = _first_out[u];
1102 /// Execute the algorithm performing augment and relabel operations
1103 void startAugment(int max_length) {
1104 // Paramters for heuristics
1105 const int EARLY_TERM_EPSILON_LIMIT = 1000;
1106 const double GLOBAL_UPDATE_FACTOR = 1.0;
1107 const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1108 (_res_node_num + _sup_node_num * _sup_node_num));
1109 int next_global_update_limit = global_update_skip;
1111 // Perform cost scaling phases
1113 BoolVector path_arc(_res_arc_num, false);
1114 int relabel_cnt = 0;
1115 for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1116 1 : _epsilon / _alpha )
1118 // Early termination heuristic
1119 if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1120 if (earlyTermination()) break;
1123 // Initialize current phase
1126 // Perform partial augment and relabel operations
1128 // Select an active node (FIFO selection)
1129 while (_active_nodes.size() > 0 &&
1130 _excess[_active_nodes.front()] <= 0) {
1131 _active_nodes.pop_front();
1133 if (_active_nodes.size() == 0) break;
1134 int start = _active_nodes.front();
1136 // Find an augmenting path from the start node
1138 while (int(path.size()) < max_length && _excess[tip] >= 0) {
1140 LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max();
1141 LargeCost pi_tip = _pi[tip];
1142 int last_out = _first_out[tip+1];
1143 for (int a = _next_out[tip]; a != last_out; ++a) {
1144 if (_res_cap[a] > 0) {
1146 rc = _cost[a] + pi_tip - _pi[u];
1151 goto augment; // a cycle is found, stop path search
1157 else if (rc < min_red_cost) {
1165 int ra = _reverse[path.back()];
1167 std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]);
1169 last_out = _next_out[tip];
1170 for (int a = _first_out[tip]; a != last_out; ++a) {
1171 if (_res_cap[a] > 0) {
1172 rc = _cost[a] + pi_tip - _pi[_target[a]];
1173 if (rc < min_red_cost) {
1178 _pi[tip] -= min_red_cost + _epsilon;
1179 _next_out[tip] = _first_out[tip];
1184 int pa = path.back();
1185 path_arc[pa] = false;
1193 // Augment along the found path (as much flow as possible)
1196 int pa, u, v = start;
1197 for (int i = 0; i != int(path.size()); ++i) {
1201 path_arc[pa] = false;
1202 delta = std::min(_res_cap[pa], _excess[u]);
1203 _res_cap[pa] -= delta;
1204 _res_cap[_reverse[pa]] += delta;
1205 _excess[u] -= delta;
1206 _excess[v] += delta;
1207 if (_excess[v] > 0 && _excess[v] <= delta) {
1208 _active_nodes.push_back(v);
1213 // Global update heuristic
1214 if (relabel_cnt >= next_global_update_limit) {
1216 next_global_update_limit += global_update_skip;
1224 /// Execute the algorithm performing push and relabel operations
1226 // Paramters for heuristics
1227 const int EARLY_TERM_EPSILON_LIMIT = 1000;
1228 const double GLOBAL_UPDATE_FACTOR = 2.0;
1230 const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1231 (_res_node_num + _sup_node_num * _sup_node_num));
1232 int next_global_update_limit = global_update_skip;
1234 // Perform cost scaling phases
1235 BoolVector hyper(_res_node_num, false);
1236 LargeCostVector hyper_cost(_res_node_num);
1237 int relabel_cnt = 0;
1238 for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1239 1 : _epsilon / _alpha )
1241 // Early termination heuristic
1242 if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
1243 if (earlyTermination()) break;
1246 // Initialize current phase
1249 // Perform push and relabel operations
1250 while (_active_nodes.size() > 0) {
1251 LargeCost min_red_cost, rc, pi_n;
1253 int n, t, a, last_out = _res_arc_num;
1256 // Select an active node (FIFO selection)
1257 n = _active_nodes.front();
1258 last_out = _first_out[n+1];
1261 // Perform push operations if there are admissible arcs
1262 if (_excess[n] > 0) {
1263 for (a = _next_out[n]; a != last_out; ++a) {
1264 if (_res_cap[a] > 0 &&
1265 _cost[a] + pi_n - _pi[_target[a]] < 0) {
1266 delta = std::min(_res_cap[a], _excess[n]);
1269 // Push-look-ahead heuristic
1270 Value ahead = -_excess[t];
1271 int last_out_t = _first_out[t+1];
1272 LargeCost pi_t = _pi[t];
1273 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1274 if (_res_cap[ta] > 0 &&
1275 _cost[ta] + pi_t - _pi[_target[ta]] < 0)
1276 ahead += _res_cap[ta];
1277 if (ahead >= delta) break;
1279 if (ahead < 0) ahead = 0;
1281 // Push flow along the arc
1282 if (ahead < delta && !hyper[t]) {
1283 _res_cap[a] -= ahead;
1284 _res_cap[_reverse[a]] += ahead;
1285 _excess[n] -= ahead;
1286 _excess[t] += ahead;
1287 _active_nodes.push_front(t);
1289 hyper_cost[t] = _cost[a] + pi_n - pi_t;
1293 _res_cap[a] -= delta;
1294 _res_cap[_reverse[a]] += delta;
1295 _excess[n] -= delta;
1296 _excess[t] += delta;
1297 if (_excess[t] > 0 && _excess[t] <= delta)
1298 _active_nodes.push_back(t);
1301 if (_excess[n] == 0) {
1310 // Relabel the node if it is still active (or hyper)
1311 if (_excess[n] > 0 || hyper[n]) {
1312 min_red_cost = hyper[n] ? -hyper_cost[n] :
1313 std::numeric_limits<LargeCost>::max();
1314 for (int a = _first_out[n]; a != last_out; ++a) {
1315 if (_res_cap[a] > 0) {
1316 rc = _cost[a] + pi_n - _pi[_target[a]];
1317 if (rc < min_red_cost) {
1322 _pi[n] -= min_red_cost + _epsilon;
1323 _next_out[n] = _first_out[n];
1328 // Remove nodes that are not active nor hyper
1330 while ( _active_nodes.size() > 0 &&
1331 _excess[_active_nodes.front()] <= 0 &&
1332 !hyper[_active_nodes.front()] ) {
1333 _active_nodes.pop_front();
1336 // Global update heuristic
1337 if (relabel_cnt >= next_global_update_limit) {
1339 for (int u = 0; u != _res_node_num; ++u)
1341 next_global_update_limit += global_update_skip;
1347 }; //class CostScaling
1353 #endif //LEMON_COST_SCALING_H