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 solving this problem.
102 /// (For more information, see \ref min_cost_flow_algs "the module page".)
104 /// Most of the parameters of the problem (except for the digraph)
105 /// can be given using separate functions, and the algorithm can be
106 /// executed using the \ref run() function. If some parameters are not
107 /// specified, then default values will be used.
109 /// \tparam GR The digraph type the algorithm runs on.
110 /// \tparam V The number type used for flow amounts, capacity bounds
111 /// and supply values in the algorithm. By default, it is \c int.
112 /// \tparam C The number type used for costs and potentials in the
113 /// algorithm. By default, it is the same as \c V.
114 /// \tparam TR The traits class that defines various types used by the
115 /// algorithm. By default, it is \ref CostScalingDefaultTraits
116 /// "CostScalingDefaultTraits<GR, V, C>".
117 /// In most cases, this parameter should not be set directly,
118 /// consider to use the named template parameters instead.
120 /// \warning Both \c V and \c C must be signed number types.
121 /// \warning All input data (capacities, supply values, and costs) must
123 /// \warning This algorithm does not support negative costs for
124 /// arcs having infinite upper bound.
126 /// \note %CostScaling provides three different internal methods,
127 /// from which the most efficient one is used by default.
128 /// For more information, see \ref Method.
130 template <typename GR, typename V, typename C, typename TR>
132 template < typename GR, typename V = int, typename C = V,
133 typename TR = CostScalingDefaultTraits<GR, V, C> >
139 /// The type of the digraph
140 typedef typename TR::Digraph Digraph;
141 /// The type of the flow amounts, capacity bounds and supply values
142 typedef typename TR::Value Value;
143 /// The type of the arc costs
144 typedef typename TR::Cost Cost;
146 /// \brief The large cost type
148 /// The large cost type used for internal computations.
149 /// By default, it is \c long \c long if the \c Cost type is integer,
150 /// otherwise it is \c double.
151 typedef typename TR::LargeCost LargeCost;
153 /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
158 /// \brief Problem type constants for the \c run() function.
160 /// Enum type containing the problem type constants that can be
161 /// returned by the \ref run() function of the algorithm.
163 /// The problem has no feasible solution (flow).
165 /// The problem has optimal solution (i.e. it is feasible and
166 /// bounded), and the algorithm has found optimal flow and node
167 /// potentials (primal and dual solutions).
169 /// The digraph contains an arc of negative cost and infinite
170 /// upper bound. It means that the objective function is unbounded
171 /// on that arc, however, note that it could actually be bounded
172 /// over the feasible flows, but this algroithm cannot handle
177 /// \brief Constants for selecting the internal method.
179 /// Enum type containing constants for selecting the internal method
180 /// for the \ref run() function.
182 /// \ref CostScaling provides three internal methods that differ mainly
183 /// in their base operations, which are used in conjunction with the
184 /// relabel operation.
185 /// By default, the so called \ref PARTIAL_AUGMENT
186 /// "Partial Augment-Relabel" method is used, which turned out to be
187 /// the most efficient and the most robust on various test inputs.
188 /// However, the other methods can be selected using the \ref run()
189 /// function with the proper parameter.
191 /// Local push operations are used, i.e. flow is moved only on one
192 /// admissible arc at once.
194 /// Augment operations are used, i.e. flow is moved on admissible
195 /// paths from a node with excess to a node with deficit.
197 /// Partial augment operations are used, i.e. flow is moved on
198 /// admissible paths started from a node with excess, but the
199 /// lengths of these paths are limited. This method can be viewed
200 /// as a combined version of the previous two operations.
206 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
208 typedef std::vector<int> IntVector;
209 typedef std::vector<Value> ValueVector;
210 typedef std::vector<Cost> CostVector;
211 typedef std::vector<LargeCost> LargeCostVector;
212 typedef std::vector<char> BoolVector;
213 // Note: vector<char> is used instead of vector<bool> for efficiency reasons
217 template <typename KT, typename VT>
218 class StaticVectorMap {
223 StaticVectorMap(std::vector<Value>& v) : _v(v) {}
225 const Value& operator[](const Key& key) const {
226 return _v[StaticDigraph::id(key)];
229 Value& operator[](const Key& key) {
230 return _v[StaticDigraph::id(key)];
233 void set(const Key& key, const Value& val) {
234 _v[StaticDigraph::id(key)] = val;
238 std::vector<Value>& _v;
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;
293 /// \brief Constant for infinite upper bounds (capacities).
295 /// Constant for infinite upper bounds (capacities).
296 /// It is \c std::numeric_limits<Value>::infinity() if available,
297 /// \c std::numeric_limits<Value>::max() otherwise.
302 /// \name Named Template Parameters
305 template <typename T>
306 struct SetLargeCostTraits : public Traits {
310 /// \brief \ref named-templ-param "Named parameter" for setting
311 /// \c LargeCost type.
313 /// \ref named-templ-param "Named parameter" for setting \c LargeCost
314 /// type, which is used for internal computations in the algorithm.
315 /// \c Cost must be convertible to \c LargeCost.
316 template <typename T>
318 : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
319 typedef CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
330 /// \brief Constructor.
332 /// The constructor of the class.
334 /// \param graph The digraph the algorithm runs on.
335 CostScaling(const GR& graph) :
336 _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
337 INF(std::numeric_limits<Value>::has_infinity ?
338 std::numeric_limits<Value>::infinity() :
339 std::numeric_limits<Value>::max())
341 // Check the number types
342 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
343 "The flow type of CostScaling must be signed");
344 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
345 "The cost type of CostScaling must be signed");
347 // Reset data structures
352 /// The parameters of the algorithm can be specified using these
357 /// \brief Set the lower bounds on the arcs.
359 /// This function sets the lower bounds on the arcs.
360 /// If it is not used before calling \ref run(), the lower bounds
361 /// will be set to zero on all arcs.
363 /// \param map An arc map storing the lower bounds.
364 /// Its \c Value type must be convertible to the \c Value type
365 /// of the algorithm.
367 /// \return <tt>(*this)</tt>
368 template <typename LowerMap>
369 CostScaling& lowerMap(const LowerMap& map) {
371 for (ArcIt a(_graph); a != INVALID; ++a) {
372 _lower[_arc_idf[a]] = map[a];
377 /// \brief Set the upper bounds (capacities) on the arcs.
379 /// This function sets the upper bounds (capacities) on the arcs.
380 /// If it is not used before calling \ref run(), the upper bounds
381 /// will be set to \ref INF on all arcs (i.e. the flow value will be
382 /// unbounded from above).
384 /// \param map An arc map storing the upper bounds.
385 /// Its \c Value type must be convertible to the \c Value type
386 /// of the algorithm.
388 /// \return <tt>(*this)</tt>
389 template<typename UpperMap>
390 CostScaling& upperMap(const UpperMap& map) {
391 for (ArcIt a(_graph); a != INVALID; ++a) {
392 _upper[_arc_idf[a]] = map[a];
397 /// \brief Set the costs of the arcs.
399 /// This function sets the costs of the arcs.
400 /// If it is not used before calling \ref run(), the costs
401 /// will be set to \c 1 on all arcs.
403 /// \param map An arc map storing the costs.
404 /// Its \c Value type must be convertible to the \c Cost type
405 /// of the algorithm.
407 /// \return <tt>(*this)</tt>
408 template<typename CostMap>
409 CostScaling& costMap(const CostMap& map) {
410 for (ArcIt a(_graph); a != INVALID; ++a) {
411 _scost[_arc_idf[a]] = map[a];
412 _scost[_arc_idb[a]] = -map[a];
417 /// \brief Set the supply values of the nodes.
419 /// This function sets the supply values of the nodes.
420 /// If neither this function nor \ref stSupply() is used before
421 /// calling \ref run(), the supply of each node will be set to zero.
423 /// \param map A node map storing the supply values.
424 /// Its \c Value type must be convertible to the \c Value type
425 /// of the algorithm.
427 /// \return <tt>(*this)</tt>
428 template<typename SupplyMap>
429 CostScaling& supplyMap(const SupplyMap& map) {
430 for (NodeIt n(_graph); n != INVALID; ++n) {
431 _supply[_node_id[n]] = map[n];
436 /// \brief Set single source and target nodes and a supply value.
438 /// This function sets a single source node and a single target node
439 /// and the required flow value.
440 /// If neither this function nor \ref supplyMap() is used before
441 /// calling \ref run(), the supply of each node will be set to zero.
443 /// Using this function has the same effect as using \ref supplyMap()
444 /// with a map in which \c k is assigned to \c s, \c -k is
445 /// assigned to \c t and all other nodes have zero supply value.
447 /// \param s The source node.
448 /// \param t The target node.
449 /// \param k The required amount of flow from node \c s to node \c t
450 /// (i.e. the supply of \c s and the demand of \c t).
452 /// \return <tt>(*this)</tt>
453 CostScaling& stSupply(const Node& s, const Node& t, Value k) {
454 for (int i = 0; i != _res_node_num; ++i) {
457 _supply[_node_id[s]] = k;
458 _supply[_node_id[t]] = -k;
464 /// \name Execution control
465 /// The algorithm can be executed using \ref run().
469 /// \brief Run the algorithm.
471 /// This function runs the algorithm.
472 /// The paramters can be specified using functions \ref lowerMap(),
473 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
476 /// CostScaling<ListDigraph> cs(graph);
477 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
478 /// .supplyMap(sup).run();
481 /// This function can be called more than once. All the given parameters
482 /// are kept for the next call, unless \ref resetParams() or \ref reset()
483 /// is used, thus only the modified parameters have to be set again.
484 /// If the underlying digraph was also modified after the construction
485 /// of the class (or the last \ref reset() call), then the \ref reset()
486 /// function must be called.
488 /// \param method The internal method that will be used in the
489 /// algorithm. For more information, see \ref Method.
490 /// \param factor The cost scaling factor. It must be at least two.
492 /// \return \c INFEASIBLE if no feasible flow exists,
493 /// \n \c OPTIMAL if the problem has optimal solution
494 /// (i.e. it is feasible and bounded), and the algorithm has found
495 /// optimal flow and node potentials (primal and dual solutions),
496 /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
497 /// and infinite upper bound. It means that the objective function
498 /// is unbounded on that arc, however, note that it could actually be
499 /// bounded over the feasible flows, but this algroithm cannot handle
502 /// \see ProblemType, Method
503 /// \see resetParams(), reset()
504 ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 16) {
505 LEMON_ASSERT(factor >= 2, "The scaling factor must be at least 2");
507 ProblemType pt = init();
508 if (pt != OPTIMAL) return pt;
513 /// \brief Reset all the parameters that have been given before.
515 /// This function resets all the paramaters that have been given
516 /// before using functions \ref lowerMap(), \ref upperMap(),
517 /// \ref costMap(), \ref supplyMap(), \ref stSupply().
519 /// It is useful for multiple \ref run() calls. Basically, all the given
520 /// parameters are kept for the next \ref run() call, unless
521 /// \ref resetParams() or \ref reset() is used.
522 /// If the underlying digraph was also modified after the construction
523 /// of the class or the last \ref reset() call, then the \ref reset()
524 /// function must be used, otherwise \ref resetParams() is sufficient.
528 /// CostScaling<ListDigraph> cs(graph);
531 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
532 /// .supplyMap(sup).run();
534 /// // Run again with modified cost map (resetParams() is not called,
535 /// // so only the cost map have to be set again)
537 /// cs.costMap(cost).run();
539 /// // Run again from scratch using resetParams()
540 /// // (the lower bounds will be set to zero on all arcs)
541 /// cs.resetParams();
542 /// cs.upperMap(capacity).costMap(cost)
543 /// .supplyMap(sup).run();
546 /// \return <tt>(*this)</tt>
548 /// \see reset(), run()
549 CostScaling& resetParams() {
550 for (int i = 0; i != _res_node_num; ++i) {
553 int limit = _first_out[_root];
554 for (int j = 0; j != limit; ++j) {
557 _scost[j] = _forward[j] ? 1 : -1;
559 for (int j = limit; j != _res_arc_num; ++j) {
563 _scost[_reverse[j]] = 0;
569 /// \brief Reset the internal data structures and all the parameters
570 /// that have been given before.
572 /// This function resets the internal data structures and all the
573 /// paramaters that have been given before using functions \ref lowerMap(),
574 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
576 /// It is useful for multiple \ref run() calls. By default, all the given
577 /// parameters are kept for the next \ref run() call, unless
578 /// \ref resetParams() or \ref reset() is used.
579 /// If the underlying digraph was also modified after the construction
580 /// of the class or the last \ref reset() call, then the \ref reset()
581 /// function must be used, otherwise \ref resetParams() is sufficient.
583 /// See \ref resetParams() for examples.
585 /// \return <tt>(*this)</tt>
587 /// \see resetParams(), run()
588 CostScaling& reset() {
590 _node_num = countNodes(_graph);
591 _arc_num = countArcs(_graph);
592 _res_node_num = _node_num + 1;
593 _res_arc_num = 2 * (_arc_num + _node_num);
596 _first_out.resize(_res_node_num + 1);
597 _forward.resize(_res_arc_num);
598 _source.resize(_res_arc_num);
599 _target.resize(_res_arc_num);
600 _reverse.resize(_res_arc_num);
602 _lower.resize(_res_arc_num);
603 _upper.resize(_res_arc_num);
604 _scost.resize(_res_arc_num);
605 _supply.resize(_res_node_num);
607 _res_cap.resize(_res_arc_num);
608 _cost.resize(_res_arc_num);
609 _pi.resize(_res_node_num);
610 _excess.resize(_res_node_num);
611 _next_out.resize(_res_node_num);
614 int i = 0, j = 0, k = 2 * _arc_num + _node_num;
615 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
619 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
621 for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
625 _target[j] = _node_id[_graph.runningNode(a)];
627 for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
631 _target[j] = _node_id[_graph.runningNode(a)];
644 _first_out[_res_node_num] = k;
645 for (ArcIt a(_graph); a != INVALID; ++a) {
646 int fi = _arc_idf[a];
647 int bi = _arc_idb[a];
659 /// \name Query Functions
660 /// The results of the algorithm can be obtained using these
662 /// The \ref run() function must be called before using them.
666 /// \brief Return the total cost of the found flow.
668 /// This function returns the total cost of the found flow.
669 /// Its complexity is O(e).
671 /// \note The return type of the function can be specified as a
672 /// template parameter. For example,
674 /// cs.totalCost<double>();
676 /// It is useful if the total cost cannot be stored in the \c Cost
677 /// type of the algorithm, which is the default return type of the
680 /// \pre \ref run() must be called before using this function.
681 template <typename Number>
682 Number totalCost() const {
684 for (ArcIt a(_graph); a != INVALID; ++a) {
686 c += static_cast<Number>(_res_cap[i]) *
687 (-static_cast<Number>(_scost[i]));
693 Cost totalCost() const {
694 return totalCost<Cost>();
698 /// \brief Return the flow on the given arc.
700 /// This function returns the flow on the given arc.
702 /// \pre \ref run() must be called before using this function.
703 Value flow(const Arc& a) const {
704 return _res_cap[_arc_idb[a]];
707 /// \brief Copy the flow values (the primal solution) into the
710 /// This function copies the flow value on each arc into the given
711 /// map. The \c Value type of the algorithm must be convertible to
712 /// the \c Value type of the map.
714 /// \pre \ref run() must be called before using this function.
715 template <typename FlowMap>
716 void flowMap(FlowMap &map) const {
717 for (ArcIt a(_graph); a != INVALID; ++a) {
718 map.set(a, _res_cap[_arc_idb[a]]);
722 /// \brief Return the potential (dual value) of the given node.
724 /// This function returns the potential (dual value) of the
727 /// \pre \ref run() must be called before using this function.
728 Cost potential(const Node& n) const {
729 return static_cast<Cost>(_pi[_node_id[n]]);
732 /// \brief Copy the potential values (the dual solution) into the
735 /// This function copies the potential (dual value) of each node
736 /// into the given map.
737 /// The \c Cost type of the algorithm must be convertible to the
738 /// \c Value type of the map.
740 /// \pre \ref run() must be called before using this function.
741 template <typename PotentialMap>
742 void potentialMap(PotentialMap &map) const {
743 for (NodeIt n(_graph); n != INVALID; ++n) {
744 map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
752 // Initialize the algorithm
754 if (_res_node_num <= 1) return INFEASIBLE;
756 // Check the sum of supply values
758 for (int i = 0; i != _root; ++i) {
759 _sum_supply += _supply[i];
761 if (_sum_supply > 0) return INFEASIBLE;
763 // Check lower and upper bounds
764 LEMON_DEBUG(checkBoundMaps(),
765 "Upper bounds must be greater or equal to the lower bounds");
768 // Initialize vectors
769 for (int i = 0; i != _res_node_num; ++i) {
771 _excess[i] = _supply[i];
774 // Remove infinite upper bounds and check negative arcs
775 const Value MAX = std::numeric_limits<Value>::max();
778 for (int i = 0; i != _root; ++i) {
779 last_out = _first_out[i+1];
780 for (int j = _first_out[i]; j != last_out; ++j) {
782 Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
783 if (c >= MAX) return UNBOUNDED;
785 _excess[_target[j]] += c;
790 for (int i = 0; i != _root; ++i) {
791 last_out = _first_out[i+1];
792 for (int j = _first_out[i]; j != last_out; ++j) {
793 if (_forward[j] && _scost[j] < 0) {
795 if (c >= MAX) return UNBOUNDED;
797 _excess[_target[j]] += c;
802 Value ex, max_cap = 0;
803 for (int i = 0; i != _res_node_num; ++i) {
806 if (ex < 0) max_cap -= ex;
808 for (int j = 0; j != _res_arc_num; ++j) {
809 if (_upper[j] >= MAX) _upper[j] = max_cap;
812 // Initialize the large cost vector and the epsilon parameter
815 for (int i = 0; i != _root; ++i) {
816 last_out = _first_out[i+1];
817 for (int j = _first_out[i]; j != last_out; ++j) {
818 lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
820 if (lc > _epsilon) _epsilon = lc;
825 // Initialize maps for Circulation and remove non-zero lower bounds
826 ConstMap<Arc, Value> low(0);
827 typedef typename Digraph::template ArcMap<Value> ValueArcMap;
828 typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
829 ValueArcMap cap(_graph), flow(_graph);
830 ValueNodeMap sup(_graph);
831 for (NodeIt n(_graph); n != INVALID; ++n) {
832 sup[n] = _supply[_node_id[n]];
835 for (ArcIt a(_graph); a != INVALID; ++a) {
838 cap[a] = _upper[j] - c;
839 sup[_graph.source(a)] -= c;
840 sup[_graph.target(a)] += c;
843 for (ArcIt a(_graph); a != INVALID; ++a) {
844 cap[a] = _upper[_arc_idf[a]];
849 for (NodeIt n(_graph); n != INVALID; ++n) {
850 if (sup[n] > 0) ++_sup_node_num;
853 // Find a feasible flow using Circulation
854 Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
855 circ(_graph, low, cap, sup);
856 if (!circ.flowMap(flow).run()) return INFEASIBLE;
858 // Set residual capacities and handle GEQ supply type
859 if (_sum_supply < 0) {
860 for (ArcIt a(_graph); a != INVALID; ++a) {
862 _res_cap[_arc_idf[a]] = cap[a] - fa;
863 _res_cap[_arc_idb[a]] = fa;
864 sup[_graph.source(a)] -= fa;
865 sup[_graph.target(a)] += fa;
867 for (NodeIt n(_graph); n != INVALID; ++n) {
868 _excess[_node_id[n]] = sup[n];
870 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
872 int ra = _reverse[a];
873 _res_cap[a] = -_sum_supply + 1;
874 _res_cap[ra] = -_excess[u];
880 for (ArcIt a(_graph); a != INVALID; ++a) {
882 _res_cap[_arc_idf[a]] = cap[a] - fa;
883 _res_cap[_arc_idb[a]] = fa;
885 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
886 int ra = _reverse[a];
894 // Initialize data structures for buckets
895 _max_rank = _alpha * _res_node_num;
896 _buckets.resize(_max_rank);
897 _bucket_next.resize(_res_node_num + 1);
898 _bucket_prev.resize(_res_node_num + 1);
899 _rank.resize(_res_node_num + 1);
904 // Check if the upper bound is greater than or equal to the lower bound
905 // on each forward arc.
906 bool checkBoundMaps() {
907 for (int j = 0; j != _res_arc_num; ++j) {
908 if (_forward[j] && _upper[j] < _lower[j]) return false;
913 // Execute the algorithm and transform the results
914 void start(Method method) {
915 const int MAX_PARTIAL_PATH_LENGTH = 4;
922 startAugment(_res_node_num - 1);
924 case PARTIAL_AUGMENT:
925 startAugment(MAX_PARTIAL_PATH_LENGTH);
929 // Compute node potentials (dual solution)
930 for (int i = 0; i != _res_node_num; ++i) {
931 _pi[i] = static_cast<Cost>(_pi[i] / (_res_node_num * _alpha));
934 for (int i = 0; optimal && i != _res_node_num; ++i) {
935 LargeCost pi_i = _pi[i];
936 int last_out = _first_out[i+1];
937 for (int j = _first_out[i]; j != last_out; ++j) {
938 if (_res_cap[j] > 0 && _scost[j] + pi_i - _pi[_target[j]] < 0) {
946 // Compute node potentials for the original costs with BellmanFord
947 // (if it is necessary)
948 typedef std::pair<int, int> IntPair;
950 std::vector<IntPair> arc_vec;
951 std::vector<LargeCost> cost_vec;
952 LargeCostArcMap cost_map(cost_vec);
956 for (int j = 0; j != _res_arc_num; ++j) {
957 if (_res_cap[j] > 0) {
958 int u = _source[j], v = _target[j];
959 arc_vec.push_back(IntPair(u, v));
960 cost_vec.push_back(_scost[j] + _pi[u] - _pi[v]);
963 sgr.build(_res_node_num, arc_vec.begin(), arc_vec.end());
965 typename BellmanFord<StaticDigraph, LargeCostArcMap>::Create
970 for (int i = 0; i != _res_node_num; ++i) {
971 _pi[i] += bf.dist(sgr.node(i));
975 // Shift potentials to meet the requirements of the GEQ type
976 // optimality conditions
977 LargeCost max_pot = _pi[_root];
978 for (int i = 0; i != _res_node_num; ++i) {
979 if (_pi[i] > max_pot) max_pot = _pi[i];
982 for (int i = 0; i != _res_node_num; ++i) {
987 // Handle non-zero lower bounds
989 int limit = _first_out[_root];
990 for (int j = 0; j != limit; ++j) {
991 if (_forward[j]) _res_cap[_reverse[j]] += _lower[j];
996 // Initialize a cost scaling phase
998 // Saturate arcs not satisfying the optimality condition
999 for (int u = 0; u != _res_node_num; ++u) {
1000 int last_out = _first_out[u+1];
1001 LargeCost pi_u = _pi[u];
1002 for (int a = _first_out[u]; a != last_out; ++a) {
1003 Value delta = _res_cap[a];
1006 if (_cost[a] + pi_u - _pi[v] < 0) {
1007 _excess[u] -= delta;
1008 _excess[v] += delta;
1010 _res_cap[_reverse[a]] += delta;
1016 // Find active nodes (i.e. nodes with positive excess)
1017 for (int u = 0; u != _res_node_num; ++u) {
1018 if (_excess[u] > 0) _active_nodes.push_back(u);
1021 // Initialize the next arcs
1022 for (int u = 0; u != _res_node_num; ++u) {
1023 _next_out[u] = _first_out[u];
1027 // Price (potential) refinement heuristic
1028 bool priceRefinement() {
1030 // Stack for stroing the topological order
1031 IntVector stack(_res_node_num);
1035 while (topologicalSort(stack, stack_top)) {
1037 // Compute node ranks in the acyclic admissible network and
1038 // store the nodes in buckets
1039 for (int i = 0; i != _res_node_num; ++i) {
1042 const int bucket_end = _root + 1;
1043 for (int r = 0; r != _max_rank; ++r) {
1044 _buckets[r] = bucket_end;
1047 for ( ; stack_top >= 0; --stack_top) {
1048 int u = stack[stack_top], v;
1049 int rank_u = _rank[u];
1051 LargeCost rc, pi_u = _pi[u];
1052 int last_out = _first_out[u+1];
1053 for (int a = _first_out[u]; a != last_out; ++a) {
1054 if (_res_cap[a] > 0) {
1056 rc = _cost[a] + pi_u - _pi[v];
1058 LargeCost nrc = static_cast<LargeCost>((-rc - 0.5) / _epsilon);
1059 if (nrc < LargeCost(_max_rank)) {
1060 int new_rank_v = rank_u + static_cast<int>(nrc);
1061 if (new_rank_v > _rank[v]) {
1062 _rank[v] = new_rank_v;
1070 top_rank = std::max(top_rank, rank_u);
1071 int bfirst = _buckets[rank_u];
1072 _bucket_next[u] = bfirst;
1073 _bucket_prev[bfirst] = u;
1074 _buckets[rank_u] = u;
1078 // Check if the current flow is epsilon-optimal
1079 if (top_rank == 0) {
1083 // Process buckets in top-down order
1084 for (int rank = top_rank; rank > 0; --rank) {
1085 while (_buckets[rank] != bucket_end) {
1086 // Remove the first node from the current bucket
1087 int u = _buckets[rank];
1088 _buckets[rank] = _bucket_next[u];
1090 // Search the outgoing arcs of u
1091 LargeCost rc, pi_u = _pi[u];
1092 int last_out = _first_out[u+1];
1093 int v, old_rank_v, new_rank_v;
1094 for (int a = _first_out[u]; a != last_out; ++a) {
1095 if (_res_cap[a] > 0) {
1097 old_rank_v = _rank[v];
1099 if (old_rank_v < rank) {
1101 // Compute the new rank of node v
1102 rc = _cost[a] + pi_u - _pi[v];
1106 LargeCost nrc = rc / _epsilon;
1108 if (nrc < LargeCost(_max_rank)) {
1109 new_rank_v = rank - 1 - static_cast<int>(nrc);
1113 // Change the rank of node v
1114 if (new_rank_v > old_rank_v) {
1115 _rank[v] = new_rank_v;
1117 // Remove v from its old bucket
1118 if (old_rank_v > 0) {
1119 if (_buckets[old_rank_v] == v) {
1120 _buckets[old_rank_v] = _bucket_next[v];
1122 int pv = _bucket_prev[v], nv = _bucket_next[v];
1123 _bucket_next[pv] = nv;
1124 _bucket_prev[nv] = pv;
1128 // Insert v into its new bucket
1129 int nv = _buckets[new_rank_v];
1130 _bucket_next[v] = nv;
1131 _bucket_prev[nv] = v;
1132 _buckets[new_rank_v] = v;
1138 // Refine potential of node u
1139 _pi[u] -= rank * _epsilon;
1148 // Find and cancel cycles in the admissible network and
1149 // determine topological order using DFS
1150 bool topologicalSort(IntVector &stack, int &stack_top) {
1151 const int MAX_CYCLE_CANCEL = 1;
1153 BoolVector reached(_res_node_num, false);
1154 BoolVector processed(_res_node_num, false);
1155 IntVector pred(_res_node_num);
1156 for (int i = 0; i != _res_node_num; ++i) {
1157 _next_out[i] = _first_out[i];
1162 for (int start = 0; start != _res_node_num; ++start) {
1163 if (reached[start]) continue;
1165 // Start DFS search from this start node
1169 // Check the outgoing arcs of the current tip node
1170 reached[tip] = true;
1171 LargeCost pi_tip = _pi[tip];
1172 int a, last_out = _first_out[tip+1];
1173 for (a = _next_out[tip]; a != last_out; ++a) {
1174 if (_res_cap[a] > 0) {
1176 if (_cost[a] + pi_tip - _pi[v] < 0) {
1178 // A new node is reached
1184 last_out = _first_out[tip+1];
1187 else if (!processed[v]) {
1192 // Find the minimum residual capacity along the cycle
1193 Value d, delta = _res_cap[a];
1194 int u, delta_node = tip;
1195 for (u = tip; u != v; ) {
1197 d = _res_cap[_next_out[u]];
1204 // Augment along the cycle
1205 _res_cap[a] -= delta;
1206 _res_cap[_reverse[a]] += delta;
1207 for (u = tip; u != v; ) {
1209 int ca = _next_out[u];
1210 _res_cap[ca] -= delta;
1211 _res_cap[_reverse[ca]] += delta;
1214 // Check the maximum number of cycle canceling
1215 if (cycle_cnt >= MAX_CYCLE_CANCEL) {
1219 // Roll back search to delta_node
1220 if (delta_node != tip) {
1221 for (u = tip; u != delta_node; u = pred[u]) {
1225 a = _next_out[tip] + 1;
1226 last_out = _first_out[tip+1];
1234 // Step back to the previous node
1235 if (a == last_out) {
1236 processed[tip] = true;
1237 stack[++stack_top] = tip;
1240 // Finish DFS from the current start node
1249 return (cycle_cnt == 0);
1252 // Global potential update heuristic
1253 void globalUpdate() {
1254 const int bucket_end = _root + 1;
1256 // Initialize buckets
1257 for (int r = 0; r != _max_rank; ++r) {
1258 _buckets[r] = bucket_end;
1260 Value total_excess = 0;
1261 int b0 = bucket_end;
1262 for (int i = 0; i != _res_node_num; ++i) {
1263 if (_excess[i] < 0) {
1265 _bucket_next[i] = b0;
1266 _bucket_prev[b0] = i;
1269 total_excess += _excess[i];
1270 _rank[i] = _max_rank;
1273 if (total_excess == 0) return;
1276 // Search the buckets
1278 for ( ; r != _max_rank; ++r) {
1279 while (_buckets[r] != bucket_end) {
1280 // Remove the first node from the current bucket
1281 int u = _buckets[r];
1282 _buckets[r] = _bucket_next[u];
1284 // Search the incomming arcs of u
1285 LargeCost pi_u = _pi[u];
1286 int last_out = _first_out[u+1];
1287 for (int a = _first_out[u]; a != last_out; ++a) {
1288 int ra = _reverse[a];
1289 if (_res_cap[ra] > 0) {
1290 int v = _source[ra];
1291 int old_rank_v = _rank[v];
1292 if (r < old_rank_v) {
1293 // Compute the new rank of v
1294 LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
1295 int new_rank_v = old_rank_v;
1296 if (nrc < LargeCost(_max_rank)) {
1297 new_rank_v = r + 1 + static_cast<int>(nrc);
1300 // Change the rank of v
1301 if (new_rank_v < old_rank_v) {
1302 _rank[v] = new_rank_v;
1303 _next_out[v] = _first_out[v];
1305 // Remove v from its old bucket
1306 if (old_rank_v < _max_rank) {
1307 if (_buckets[old_rank_v] == v) {
1308 _buckets[old_rank_v] = _bucket_next[v];
1310 int pv = _bucket_prev[v], nv = _bucket_next[v];
1311 _bucket_next[pv] = nv;
1312 _bucket_prev[nv] = pv;
1316 // Insert v into its new bucket
1317 int nv = _buckets[new_rank_v];
1318 _bucket_next[v] = nv;
1319 _bucket_prev[nv] = v;
1320 _buckets[new_rank_v] = v;
1326 // Finish search if there are no more active nodes
1327 if (_excess[u] > 0) {
1328 total_excess -= _excess[u];
1329 if (total_excess <= 0) break;
1332 if (total_excess <= 0) break;
1336 for (int u = 0; u != _res_node_num; ++u) {
1337 int k = std::min(_rank[u], r);
1339 _pi[u] -= _epsilon * k;
1340 _next_out[u] = _first_out[u];
1345 /// Execute the algorithm performing augment and relabel operations
1346 void startAugment(int max_length) {
1347 // Paramters for heuristics
1348 const int PRICE_REFINEMENT_LIMIT = 2;
1349 const double GLOBAL_UPDATE_FACTOR = 1.0;
1350 const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1351 (_res_node_num + _sup_node_num * _sup_node_num));
1352 int next_global_update_limit = global_update_skip;
1354 // Perform cost scaling phases
1356 BoolVector path_arc(_res_arc_num, false);
1357 int relabel_cnt = 0;
1358 int eps_phase_cnt = 0;
1359 for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1360 1 : _epsilon / _alpha )
1364 // Price refinement heuristic
1365 if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
1366 if (priceRefinement()) continue;
1369 // Initialize current phase
1372 // Perform partial augment and relabel operations
1374 // Select an active node (FIFO selection)
1375 while (_active_nodes.size() > 0 &&
1376 _excess[_active_nodes.front()] <= 0) {
1377 _active_nodes.pop_front();
1379 if (_active_nodes.size() == 0) break;
1380 int start = _active_nodes.front();
1382 // Find an augmenting path from the start node
1384 while (int(path.size()) < max_length && _excess[tip] >= 0) {
1386 LargeCost rc, min_red_cost = std::numeric_limits<LargeCost>::max();
1387 LargeCost pi_tip = _pi[tip];
1388 int last_out = _first_out[tip+1];
1389 for (int a = _next_out[tip]; a != last_out; ++a) {
1390 if (_res_cap[a] > 0) {
1392 rc = _cost[a] + pi_tip - _pi[u];
1397 goto augment; // a cycle is found, stop path search
1403 else if (rc < min_red_cost) {
1411 int ra = _reverse[path.back()];
1413 std::min(min_red_cost, _cost[ra] + pi_tip - _pi[_target[ra]]);
1415 last_out = _next_out[tip];
1416 for (int a = _first_out[tip]; a != last_out; ++a) {
1417 if (_res_cap[a] > 0) {
1418 rc = _cost[a] + pi_tip - _pi[_target[a]];
1419 if (rc < min_red_cost) {
1424 _pi[tip] -= min_red_cost + _epsilon;
1425 _next_out[tip] = _first_out[tip];
1430 int pa = path.back();
1431 path_arc[pa] = false;
1439 // Augment along the found path (as much flow as possible)
1442 int pa, u, v = start;
1443 for (int i = 0; i != int(path.size()); ++i) {
1447 path_arc[pa] = false;
1448 delta = std::min(_res_cap[pa], _excess[u]);
1449 _res_cap[pa] -= delta;
1450 _res_cap[_reverse[pa]] += delta;
1451 _excess[u] -= delta;
1452 _excess[v] += delta;
1453 if (_excess[v] > 0 && _excess[v] <= delta) {
1454 _active_nodes.push_back(v);
1459 // Global update heuristic
1460 if (relabel_cnt >= next_global_update_limit) {
1462 next_global_update_limit += global_update_skip;
1470 /// Execute the algorithm performing push and relabel operations
1472 // Paramters for heuristics
1473 const int PRICE_REFINEMENT_LIMIT = 2;
1474 const double GLOBAL_UPDATE_FACTOR = 2.0;
1476 const int global_update_skip = static_cast<int>(GLOBAL_UPDATE_FACTOR *
1477 (_res_node_num + _sup_node_num * _sup_node_num));
1478 int next_global_update_limit = global_update_skip;
1480 // Perform cost scaling phases
1481 BoolVector hyper(_res_node_num, false);
1482 LargeCostVector hyper_cost(_res_node_num);
1483 int relabel_cnt = 0;
1484 int eps_phase_cnt = 0;
1485 for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
1486 1 : _epsilon / _alpha )
1490 // Price refinement heuristic
1491 if (eps_phase_cnt >= PRICE_REFINEMENT_LIMIT) {
1492 if (priceRefinement()) continue;
1495 // Initialize current phase
1498 // Perform push and relabel operations
1499 while (_active_nodes.size() > 0) {
1500 LargeCost min_red_cost, rc, pi_n;
1502 int n, t, a, last_out = _res_arc_num;
1505 // Select an active node (FIFO selection)
1506 n = _active_nodes.front();
1507 last_out = _first_out[n+1];
1510 // Perform push operations if there are admissible arcs
1511 if (_excess[n] > 0) {
1512 for (a = _next_out[n]; a != last_out; ++a) {
1513 if (_res_cap[a] > 0 &&
1514 _cost[a] + pi_n - _pi[_target[a]] < 0) {
1515 delta = std::min(_res_cap[a], _excess[n]);
1518 // Push-look-ahead heuristic
1519 Value ahead = -_excess[t];
1520 int last_out_t = _first_out[t+1];
1521 LargeCost pi_t = _pi[t];
1522 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
1523 if (_res_cap[ta] > 0 &&
1524 _cost[ta] + pi_t - _pi[_target[ta]] < 0)
1525 ahead += _res_cap[ta];
1526 if (ahead >= delta) break;
1528 if (ahead < 0) ahead = 0;
1530 // Push flow along the arc
1531 if (ahead < delta && !hyper[t]) {
1532 _res_cap[a] -= ahead;
1533 _res_cap[_reverse[a]] += ahead;
1534 _excess[n] -= ahead;
1535 _excess[t] += ahead;
1536 _active_nodes.push_front(t);
1538 hyper_cost[t] = _cost[a] + pi_n - pi_t;
1542 _res_cap[a] -= delta;
1543 _res_cap[_reverse[a]] += delta;
1544 _excess[n] -= delta;
1545 _excess[t] += delta;
1546 if (_excess[t] > 0 && _excess[t] <= delta)
1547 _active_nodes.push_back(t);
1550 if (_excess[n] == 0) {
1559 // Relabel the node if it is still active (or hyper)
1560 if (_excess[n] > 0 || hyper[n]) {
1561 min_red_cost = hyper[n] ? -hyper_cost[n] :
1562 std::numeric_limits<LargeCost>::max();
1563 for (int a = _first_out[n]; a != last_out; ++a) {
1564 if (_res_cap[a] > 0) {
1565 rc = _cost[a] + pi_n - _pi[_target[a]];
1566 if (rc < min_red_cost) {
1571 _pi[n] -= min_red_cost + _epsilon;
1572 _next_out[n] = _first_out[n];
1577 // Remove nodes that are not active nor hyper
1579 while ( _active_nodes.size() > 0 &&
1580 _excess[_active_nodes.front()] <= 0 &&
1581 !hyper[_active_nodes.front()] ) {
1582 _active_nodes.pop_front();
1585 // Global update heuristic
1586 if (relabel_cnt >= next_global_update_limit) {
1588 for (int u = 0; u != _res_node_num; ++u)
1590 next_global_update_limit += global_update_skip;
1596 }; //class CostScaling
1602 #endif //LEMON_COST_SCALING_H