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_NETWORK_SIMPLEX_H
20 #define LEMON_NETWORK_SIMPLEX_H
22 /// \ingroup min_cost_flow_algs
25 /// \brief Network Simplex algorithm for finding a minimum cost flow.
31 #include <lemon/core.h>
32 #include <lemon/math.h>
36 /// \addtogroup min_cost_flow_algs
39 /// \brief Implementation of the primal Network Simplex algorithm
40 /// for finding a \ref min_cost_flow "minimum cost flow".
42 /// \ref NetworkSimplex implements the primal Network Simplex algorithm
43 /// for finding a \ref min_cost_flow "minimum cost flow"
44 /// \ref amo93networkflows, \ref dantzig63linearprog,
45 /// \ref kellyoneill91netsimplex.
46 /// This algorithm is a highly efficient specialized version of the
47 /// linear programming simplex method directly for the minimum cost
50 /// In general, %NetworkSimplex is the fastest implementation available
51 /// in LEMON for this problem.
52 /// Moreover, it supports both directions of the supply/demand inequality
53 /// constraints. For more information, see \ref SupplyType.
55 /// Most of the parameters of the problem (except for the digraph)
56 /// can be given using separate functions, and the algorithm can be
57 /// executed using the \ref run() function. If some parameters are not
58 /// specified, then default values will be used.
60 /// \tparam GR The digraph type the algorithm runs on.
61 /// \tparam V The number type used for flow amounts, capacity bounds
62 /// and supply values in the algorithm. By default, it is \c int.
63 /// \tparam C The number type used for costs and potentials in the
64 /// algorithm. By default, it is the same as \c V.
66 /// \warning Both \c V and \c C must be signed number types.
67 /// \warning All input data (capacities, supply values, and costs) must
70 /// \note %NetworkSimplex provides five different pivot rule
71 /// implementations, from which the most efficient one is used
72 /// by default. For more information, see \ref PivotRule.
73 template <typename GR, typename V = int, typename C = V>
78 /// The type of the flow amounts, capacity bounds and supply values
80 /// The type of the arc costs
85 /// \brief Problem type constants for the \c run() function.
87 /// Enum type containing the problem type constants that can be
88 /// returned by the \ref run() function of the algorithm.
90 /// The problem has no feasible solution (flow).
92 /// The problem has optimal solution (i.e. it is feasible and
93 /// bounded), and the algorithm has found optimal flow and node
94 /// potentials (primal and dual solutions).
96 /// The objective function of the problem is unbounded, i.e.
97 /// there is a directed cycle having negative total cost and
98 /// infinite upper bound.
102 /// \brief Constants for selecting the type of the supply constraints.
104 /// Enum type containing constants for selecting the supply type,
105 /// i.e. the direction of the inequalities in the supply/demand
106 /// constraints of the \ref min_cost_flow "minimum cost flow problem".
108 /// The default supply type is \c GEQ, the \c LEQ type can be
109 /// selected using \ref supplyType().
110 /// The equality form is a special case of both supply types.
112 /// This option means that there are <em>"greater or equal"</em>
113 /// supply/demand constraints in the definition of the problem.
115 /// This option means that there are <em>"less or equal"</em>
116 /// supply/demand constraints in the definition of the problem.
120 /// \brief Constants for selecting the pivot rule.
122 /// Enum type containing constants for selecting the pivot rule for
123 /// the \ref run() function.
125 /// \ref NetworkSimplex provides five different pivot rule
126 /// implementations that significantly affect the running time
127 /// of the algorithm.
128 /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
129 /// proved to be the most efficient and the most robust on various
131 /// However, another pivot rule can be selected using the \ref run()
132 /// function with the proper parameter.
135 /// The \e First \e Eligible pivot rule.
136 /// The next eligible arc is selected in a wraparound fashion
137 /// in every iteration.
140 /// The \e Best \e Eligible pivot rule.
141 /// The best eligible arc is selected in every iteration.
144 /// The \e Block \e Search pivot rule.
145 /// A specified number of arcs are examined in every iteration
146 /// in a wraparound fashion and the best eligible arc is selected
150 /// The \e Candidate \e List pivot rule.
151 /// In a major iteration a candidate list is built from eligible arcs
152 /// in a wraparound fashion and in the following minor iterations
153 /// the best eligible arc is selected from this list.
156 /// The \e Altering \e Candidate \e List pivot rule.
157 /// It is a modified version of the Candidate List method.
158 /// It keeps only the several best eligible arcs from the former
159 /// candidate list and extends this list in every iteration.
165 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
167 typedef std::vector<int> IntVector;
168 typedef std::vector<Value> ValueVector;
169 typedef std::vector<Cost> CostVector;
170 typedef std::vector<signed char> CharVector;
171 // Note: vector<signed char> is used instead of vector<ArcState> and
172 // vector<ArcDirection> for efficiency reasons
174 // State constants for arcs
181 // Direction constants for tree arcs
189 // Data related to the underlying digraph
196 // Parameters of the problem
201 // Data structures for storing the digraph
217 // Data for storing the spanning tree structure
221 IntVector _rev_thread;
223 IntVector _last_succ;
224 CharVector _pred_dir;
226 IntVector _dirty_revs;
229 // Temporary data used in the current pivot iteration
230 int in_arc, join, u_in, v_in, u_out, v_out;
237 /// \brief Constant for infinite upper bounds (capacities).
239 /// Constant for infinite upper bounds (capacities).
240 /// It is \c std::numeric_limits<Value>::infinity() if available,
241 /// \c std::numeric_limits<Value>::max() otherwise.
246 // Implementation of the First Eligible pivot rule
247 class FirstEligiblePivotRule
251 // References to the NetworkSimplex class
252 const IntVector &_source;
253 const IntVector &_target;
254 const CostVector &_cost;
255 const CharVector &_state;
256 const CostVector &_pi;
266 FirstEligiblePivotRule(NetworkSimplex &ns) :
267 _source(ns._source), _target(ns._target),
268 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
269 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
273 // Find next entering arc
274 bool findEnteringArc() {
276 for (int e = _next_arc; e != _search_arc_num; ++e) {
277 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
284 for (int e = 0; e != _next_arc; ++e) {
285 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
295 }; //class FirstEligiblePivotRule
298 // Implementation of the Best Eligible pivot rule
299 class BestEligiblePivotRule
303 // References to the NetworkSimplex class
304 const IntVector &_source;
305 const IntVector &_target;
306 const CostVector &_cost;
307 const CharVector &_state;
308 const CostVector &_pi;
315 BestEligiblePivotRule(NetworkSimplex &ns) :
316 _source(ns._source), _target(ns._target),
317 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
318 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
321 // Find next entering arc
322 bool findEnteringArc() {
324 for (int e = 0; e != _search_arc_num; ++e) {
325 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
334 }; //class BestEligiblePivotRule
337 // Implementation of the Block Search pivot rule
338 class BlockSearchPivotRule
342 // References to the NetworkSimplex class
343 const IntVector &_source;
344 const IntVector &_target;
345 const CostVector &_cost;
346 const CharVector &_state;
347 const CostVector &_pi;
358 BlockSearchPivotRule(NetworkSimplex &ns) :
359 _source(ns._source), _target(ns._target),
360 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
361 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
364 // The main parameters of the pivot rule
365 const double BLOCK_SIZE_FACTOR = 1.0;
366 const int MIN_BLOCK_SIZE = 10;
368 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
369 std::sqrt(double(_search_arc_num))),
373 // Find next entering arc
374 bool findEnteringArc() {
376 int cnt = _block_size;
378 for (e = _next_arc; e != _search_arc_num; ++e) {
379 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
385 if (min < 0) goto search_end;
389 for (e = 0; e != _next_arc; ++e) {
390 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
396 if (min < 0) goto search_end;
400 if (min >= 0) return false;
407 }; //class BlockSearchPivotRule
410 // Implementation of the Candidate List pivot rule
411 class CandidateListPivotRule
415 // References to the NetworkSimplex class
416 const IntVector &_source;
417 const IntVector &_target;
418 const CostVector &_cost;
419 const CharVector &_state;
420 const CostVector &_pi;
425 IntVector _candidates;
426 int _list_length, _minor_limit;
427 int _curr_length, _minor_count;
433 CandidateListPivotRule(NetworkSimplex &ns) :
434 _source(ns._source), _target(ns._target),
435 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
436 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
439 // The main parameters of the pivot rule
440 const double LIST_LENGTH_FACTOR = 0.25;
441 const int MIN_LIST_LENGTH = 10;
442 const double MINOR_LIMIT_FACTOR = 0.1;
443 const int MIN_MINOR_LIMIT = 3;
445 _list_length = std::max( int(LIST_LENGTH_FACTOR *
446 std::sqrt(double(_search_arc_num))),
448 _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
450 _curr_length = _minor_count = 0;
451 _candidates.resize(_list_length);
454 /// Find next entering arc
455 bool findEnteringArc() {
458 if (_curr_length > 0 && _minor_count < _minor_limit) {
459 // Minor iteration: select the best eligible arc from the
460 // current candidate list
463 for (int i = 0; i < _curr_length; ++i) {
465 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
471 _candidates[i--] = _candidates[--_curr_length];
474 if (min < 0) return true;
477 // Major iteration: build a new candidate list
480 for (e = _next_arc; e != _search_arc_num; ++e) {
481 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
483 _candidates[_curr_length++] = e;
488 if (_curr_length == _list_length) goto search_end;
491 for (e = 0; e != _next_arc; ++e) {
492 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
494 _candidates[_curr_length++] = e;
499 if (_curr_length == _list_length) goto search_end;
502 if (_curr_length == 0) return false;
510 }; //class CandidateListPivotRule
513 // Implementation of the Altering Candidate List pivot rule
514 class AlteringListPivotRule
518 // References to the NetworkSimplex class
519 const IntVector &_source;
520 const IntVector &_target;
521 const CostVector &_cost;
522 const CharVector &_state;
523 const CostVector &_pi;
528 int _block_size, _head_length, _curr_length;
530 IntVector _candidates;
531 CostVector _cand_cost;
533 // Functor class to compare arcs during sort of the candidate list
537 const CostVector &_map;
539 SortFunc(const CostVector &map) : _map(map) {}
540 bool operator()(int left, int right) {
541 return _map[left] > _map[right];
550 AlteringListPivotRule(NetworkSimplex &ns) :
551 _source(ns._source), _target(ns._target),
552 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
553 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
554 _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
556 // The main parameters of the pivot rule
557 const double BLOCK_SIZE_FACTOR = 1.0;
558 const int MIN_BLOCK_SIZE = 10;
559 const double HEAD_LENGTH_FACTOR = 0.1;
560 const int MIN_HEAD_LENGTH = 3;
562 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
563 std::sqrt(double(_search_arc_num))),
565 _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
567 _candidates.resize(_head_length + _block_size);
571 // Find next entering arc
572 bool findEnteringArc() {
573 // Check the current candidate list
576 for (int i = 0; i != _curr_length; ++i) {
578 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
582 _candidates[i--] = _candidates[--_curr_length];
587 int cnt = _block_size;
588 int limit = _head_length;
590 for (e = _next_arc; e != _search_arc_num; ++e) {
591 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
594 _candidates[_curr_length++] = e;
597 if (_curr_length > limit) goto search_end;
602 for (e = 0; e != _next_arc; ++e) {
603 _cand_cost[e] = _state[e] *
604 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
605 if (_cand_cost[e] < 0) {
606 _candidates[_curr_length++] = e;
609 if (_curr_length > limit) goto search_end;
614 if (_curr_length == 0) return false;
618 // Make heap of the candidate list (approximating a partial sort)
619 make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
622 // Pop the first element of the heap
623 _in_arc = _candidates[0];
625 pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
627 _curr_length = std::min(_head_length, _curr_length - 1);
631 }; //class AlteringListPivotRule
635 /// \brief Constructor.
637 /// The constructor of the class.
639 /// \param graph The digraph the algorithm runs on.
640 /// \param arc_mixing Indicate if the arcs will be stored in a
641 /// mixed order in the internal data structure.
642 /// In general, it leads to similar performance as using the original
643 /// arc order, but it makes the algorithm more robust and in special
644 /// cases, even significantly faster. Therefore, it is enabled by default.
645 NetworkSimplex(const GR& graph, bool arc_mixing = true) :
646 _graph(graph), _node_id(graph), _arc_id(graph),
647 _arc_mixing(arc_mixing),
648 MAX(std::numeric_limits<Value>::max()),
649 INF(std::numeric_limits<Value>::has_infinity ?
650 std::numeric_limits<Value>::infinity() : MAX)
652 // Check the number types
653 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
654 "The flow type of NetworkSimplex must be signed");
655 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
656 "The cost type of NetworkSimplex must be signed");
658 // Reset data structures
663 /// The parameters of the algorithm can be specified using these
668 /// \brief Set the lower bounds on the arcs.
670 /// This function sets the lower bounds on the arcs.
671 /// If it is not used before calling \ref run(), the lower bounds
672 /// will be set to zero on all arcs.
674 /// \param map An arc map storing the lower bounds.
675 /// Its \c Value type must be convertible to the \c Value type
676 /// of the algorithm.
678 /// \return <tt>(*this)</tt>
679 template <typename LowerMap>
680 NetworkSimplex& lowerMap(const LowerMap& map) {
682 for (ArcIt a(_graph); a != INVALID; ++a) {
683 _lower[_arc_id[a]] = map[a];
688 /// \brief Set the upper bounds (capacities) on the arcs.
690 /// This function sets the upper bounds (capacities) on the arcs.
691 /// If it is not used before calling \ref run(), the upper bounds
692 /// will be set to \ref INF on all arcs (i.e. the flow value will be
693 /// unbounded from above).
695 /// \param map An arc map storing the upper bounds.
696 /// Its \c Value type must be convertible to the \c Value type
697 /// of the algorithm.
699 /// \return <tt>(*this)</tt>
700 template<typename UpperMap>
701 NetworkSimplex& upperMap(const UpperMap& map) {
702 for (ArcIt a(_graph); a != INVALID; ++a) {
703 _upper[_arc_id[a]] = map[a];
708 /// \brief Set the costs of the arcs.
710 /// This function sets the costs of the arcs.
711 /// If it is not used before calling \ref run(), the costs
712 /// will be set to \c 1 on all arcs.
714 /// \param map An arc map storing the costs.
715 /// Its \c Value type must be convertible to the \c Cost type
716 /// of the algorithm.
718 /// \return <tt>(*this)</tt>
719 template<typename CostMap>
720 NetworkSimplex& costMap(const CostMap& map) {
721 for (ArcIt a(_graph); a != INVALID; ++a) {
722 _cost[_arc_id[a]] = map[a];
727 /// \brief Set the supply values of the nodes.
729 /// This function sets the supply values of the nodes.
730 /// If neither this function nor \ref stSupply() is used before
731 /// calling \ref run(), the supply of each node will be set to zero.
733 /// \param map A node map storing the supply values.
734 /// Its \c Value type must be convertible to the \c Value type
735 /// of the algorithm.
737 /// \return <tt>(*this)</tt>
738 template<typename SupplyMap>
739 NetworkSimplex& supplyMap(const SupplyMap& map) {
740 for (NodeIt n(_graph); n != INVALID; ++n) {
741 _supply[_node_id[n]] = map[n];
746 /// \brief Set single source and target nodes and a supply value.
748 /// This function sets a single source node and a single target node
749 /// and the required flow value.
750 /// If neither this function nor \ref supplyMap() is used before
751 /// calling \ref run(), the supply of each node will be set to zero.
753 /// Using this function has the same effect as using \ref supplyMap()
754 /// with such a map in which \c k is assigned to \c s, \c -k is
755 /// assigned to \c t and all other nodes have zero supply value.
757 /// \param s The source node.
758 /// \param t The target node.
759 /// \param k The required amount of flow from node \c s to node \c t
760 /// (i.e. the supply of \c s and the demand of \c t).
762 /// \return <tt>(*this)</tt>
763 NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
764 for (int i = 0; i != _node_num; ++i) {
767 _supply[_node_id[s]] = k;
768 _supply[_node_id[t]] = -k;
772 /// \brief Set the type of the supply constraints.
774 /// This function sets the type of the supply/demand constraints.
775 /// If it is not used before calling \ref run(), the \ref GEQ supply
776 /// type will be used.
778 /// For more information, see \ref SupplyType.
780 /// \return <tt>(*this)</tt>
781 NetworkSimplex& supplyType(SupplyType supply_type) {
782 _stype = supply_type;
788 /// \name Execution Control
789 /// The algorithm can be executed using \ref run().
793 /// \brief Run the algorithm.
795 /// This function runs the algorithm.
796 /// The paramters can be specified using functions \ref lowerMap(),
797 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
798 /// \ref supplyType().
801 /// NetworkSimplex<ListDigraph> ns(graph);
802 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
803 /// .supplyMap(sup).run();
806 /// This function can be called more than once. All the given parameters
807 /// are kept for the next call, unless \ref resetParams() or \ref reset()
808 /// is used, thus only the modified parameters have to be set again.
809 /// If the underlying digraph was also modified after the construction
810 /// of the class (or the last \ref reset() call), then the \ref reset()
811 /// function must be called.
813 /// \param pivot_rule The pivot rule that will be used during the
814 /// algorithm. For more information, see \ref PivotRule.
816 /// \return \c INFEASIBLE if no feasible flow exists,
817 /// \n \c OPTIMAL if the problem has optimal solution
818 /// (i.e. it is feasible and bounded), and the algorithm has found
819 /// optimal flow and node potentials (primal and dual solutions),
820 /// \n \c UNBOUNDED if the objective function of the problem is
821 /// unbounded, i.e. there is a directed cycle having negative total
822 /// cost and infinite upper bound.
824 /// \see ProblemType, PivotRule
825 /// \see resetParams(), reset()
826 ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
827 if (!init()) return INFEASIBLE;
828 return start(pivot_rule);
831 /// \brief Reset all the parameters that have been given before.
833 /// This function resets all the paramaters that have been given
834 /// before using functions \ref lowerMap(), \ref upperMap(),
835 /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
837 /// It is useful for multiple \ref run() calls. Basically, all the given
838 /// parameters are kept for the next \ref run() call, unless
839 /// \ref resetParams() or \ref reset() is used.
840 /// If the underlying digraph was also modified after the construction
841 /// of the class or the last \ref reset() call, then the \ref reset()
842 /// function must be used, otherwise \ref resetParams() is sufficient.
846 /// NetworkSimplex<ListDigraph> ns(graph);
849 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
850 /// .supplyMap(sup).run();
852 /// // Run again with modified cost map (resetParams() is not called,
853 /// // so only the cost map have to be set again)
855 /// ns.costMap(cost).run();
857 /// // Run again from scratch using resetParams()
858 /// // (the lower bounds will be set to zero on all arcs)
859 /// ns.resetParams();
860 /// ns.upperMap(capacity).costMap(cost)
861 /// .supplyMap(sup).run();
864 /// \return <tt>(*this)</tt>
866 /// \see reset(), run()
867 NetworkSimplex& resetParams() {
868 for (int i = 0; i != _node_num; ++i) {
871 for (int i = 0; i != _arc_num; ++i) {
881 /// \brief Reset the internal data structures and all the parameters
882 /// that have been given before.
884 /// This function resets the internal data structures and all the
885 /// paramaters that have been given before using functions \ref lowerMap(),
886 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
887 /// \ref supplyType().
889 /// It is useful for multiple \ref run() calls. Basically, all the given
890 /// parameters are kept for the next \ref run() call, unless
891 /// \ref resetParams() or \ref reset() is used.
892 /// If the underlying digraph was also modified after the construction
893 /// of the class or the last \ref reset() call, then the \ref reset()
894 /// function must be used, otherwise \ref resetParams() is sufficient.
896 /// See \ref resetParams() for examples.
898 /// \return <tt>(*this)</tt>
900 /// \see resetParams(), run()
901 NetworkSimplex& reset() {
903 _node_num = countNodes(_graph);
904 _arc_num = countArcs(_graph);
905 int all_node_num = _node_num + 1;
906 int max_arc_num = _arc_num + 2 * _node_num;
908 _source.resize(max_arc_num);
909 _target.resize(max_arc_num);
911 _lower.resize(_arc_num);
912 _upper.resize(_arc_num);
913 _cap.resize(max_arc_num);
914 _cost.resize(max_arc_num);
915 _supply.resize(all_node_num);
916 _flow.resize(max_arc_num);
917 _pi.resize(all_node_num);
919 _parent.resize(all_node_num);
920 _pred.resize(all_node_num);
921 _pred_dir.resize(all_node_num);
922 _thread.resize(all_node_num);
923 _rev_thread.resize(all_node_num);
924 _succ_num.resize(all_node_num);
925 _last_succ.resize(all_node_num);
926 _state.resize(max_arc_num);
930 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
934 // Store the arcs in a mixed order
935 const int skip = std::max(_arc_num / _node_num, 3);
937 for (ArcIt a(_graph); a != INVALID; ++a) {
939 _source[i] = _node_id[_graph.source(a)];
940 _target[i] = _node_id[_graph.target(a)];
941 if ((i += skip) >= _arc_num) i = ++j;
944 // Store the arcs in the original order
946 for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
948 _source[i] = _node_id[_graph.source(a)];
949 _target[i] = _node_id[_graph.target(a)];
960 /// \name Query Functions
961 /// The results of the algorithm can be obtained using these
963 /// The \ref run() function must be called before using them.
967 /// \brief Return the total cost of the found flow.
969 /// This function returns the total cost of the found flow.
970 /// Its complexity is O(e).
972 /// \note The return type of the function can be specified as a
973 /// template parameter. For example,
975 /// ns.totalCost<double>();
977 /// It is useful if the total cost cannot be stored in the \c Cost
978 /// type of the algorithm, which is the default return type of the
981 /// \pre \ref run() must be called before using this function.
982 template <typename Number>
983 Number totalCost() const {
985 for (ArcIt a(_graph); a != INVALID; ++a) {
987 c += Number(_flow[i]) * Number(_cost[i]);
993 Cost totalCost() const {
994 return totalCost<Cost>();
998 /// \brief Return the flow on the given arc.
1000 /// This function returns the flow on the given arc.
1002 /// \pre \ref run() must be called before using this function.
1003 Value flow(const Arc& a) const {
1004 return _flow[_arc_id[a]];
1007 /// \brief Return the flow map (the primal solution).
1009 /// This function copies the flow value on each arc into the given
1010 /// map. The \c Value type of the algorithm must be convertible to
1011 /// the \c Value type of the map.
1013 /// \pre \ref run() must be called before using this function.
1014 template <typename FlowMap>
1015 void flowMap(FlowMap &map) const {
1016 for (ArcIt a(_graph); a != INVALID; ++a) {
1017 map.set(a, _flow[_arc_id[a]]);
1021 /// \brief Return the potential (dual value) of the given node.
1023 /// This function returns the potential (dual value) of the
1026 /// \pre \ref run() must be called before using this function.
1027 Cost potential(const Node& n) const {
1028 return _pi[_node_id[n]];
1031 /// \brief Return the potential map (the dual solution).
1033 /// This function copies the potential (dual value) of each node
1034 /// into the given map.
1035 /// The \c Cost type of the algorithm must be convertible to the
1036 /// \c Value type of the map.
1038 /// \pre \ref run() must be called before using this function.
1039 template <typename PotentialMap>
1040 void potentialMap(PotentialMap &map) const {
1041 for (NodeIt n(_graph); n != INVALID; ++n) {
1042 map.set(n, _pi[_node_id[n]]);
1050 // Initialize internal data structures
1052 if (_node_num == 0) return false;
1054 // Check the sum of supply values
1056 for (int i = 0; i != _node_num; ++i) {
1057 _sum_supply += _supply[i];
1059 if ( !((_stype == GEQ && _sum_supply <= 0) ||
1060 (_stype == LEQ && _sum_supply >= 0)) ) return false;
1062 // Remove non-zero lower bounds
1064 for (int i = 0; i != _arc_num; ++i) {
1065 Value c = _lower[i];
1067 _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1069 _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1071 _supply[_source[i]] -= c;
1072 _supply[_target[i]] += c;
1075 for (int i = 0; i != _arc_num; ++i) {
1076 _cap[i] = _upper[i];
1080 // Initialize artifical cost
1082 if (std::numeric_limits<Cost>::is_exact) {
1083 ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1086 for (int i = 0; i != _arc_num; ++i) {
1087 if (_cost[i] > ART_COST) ART_COST = _cost[i];
1089 ART_COST = (ART_COST + 1) * _node_num;
1092 // Initialize arc maps
1093 for (int i = 0; i != _arc_num; ++i) {
1095 _state[i] = STATE_LOWER;
1098 // Set data for the artificial root node
1100 _parent[_root] = -1;
1103 _rev_thread[0] = _root;
1104 _succ_num[_root] = _node_num + 1;
1105 _last_succ[_root] = _root - 1;
1106 _supply[_root] = -_sum_supply;
1109 // Add artificial arcs and initialize the spanning tree data structure
1110 if (_sum_supply == 0) {
1111 // EQ supply constraints
1112 _search_arc_num = _arc_num;
1113 _all_arc_num = _arc_num + _node_num;
1114 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1118 _rev_thread[u + 1] = u;
1122 _state[e] = STATE_TREE;
1123 if (_supply[u] >= 0) {
1124 _pred_dir[u] = DIR_UP;
1128 _flow[e] = _supply[u];
1131 _pred_dir[u] = DIR_DOWN;
1135 _flow[e] = -_supply[u];
1136 _cost[e] = ART_COST;
1140 else if (_sum_supply > 0) {
1141 // LEQ supply constraints
1142 _search_arc_num = _arc_num + _node_num;
1143 int f = _arc_num + _node_num;
1144 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1147 _rev_thread[u + 1] = u;
1150 if (_supply[u] >= 0) {
1151 _pred_dir[u] = DIR_UP;
1157 _flow[e] = _supply[u];
1159 _state[e] = STATE_TREE;
1161 _pred_dir[u] = DIR_DOWN;
1167 _flow[f] = -_supply[u];
1168 _cost[f] = ART_COST;
1169 _state[f] = STATE_TREE;
1175 _state[e] = STATE_LOWER;
1182 // GEQ supply constraints
1183 _search_arc_num = _arc_num + _node_num;
1184 int f = _arc_num + _node_num;
1185 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1188 _rev_thread[u + 1] = u;
1191 if (_supply[u] <= 0) {
1192 _pred_dir[u] = DIR_DOWN;
1198 _flow[e] = -_supply[u];
1200 _state[e] = STATE_TREE;
1202 _pred_dir[u] = DIR_UP;
1208 _flow[f] = _supply[u];
1209 _state[f] = STATE_TREE;
1210 _cost[f] = ART_COST;
1216 _state[e] = STATE_LOWER;
1226 // Find the join node
1227 void findJoinNode() {
1228 int u = _source[in_arc];
1229 int v = _target[in_arc];
1231 if (_succ_num[u] < _succ_num[v]) {
1240 // Find the leaving arc of the cycle and returns true if the
1241 // leaving arc is not the same as the entering arc
1242 bool findLeavingArc() {
1243 // Initialize first and second nodes according to the direction
1246 if (_state[in_arc] == STATE_LOWER) {
1247 first = _source[in_arc];
1248 second = _target[in_arc];
1250 first = _target[in_arc];
1251 second = _source[in_arc];
1253 delta = _cap[in_arc];
1258 // Search the cycle form the first node to the join node
1259 for (int u = first; u != join; u = _parent[u]) {
1262 if (_pred_dir[u] == DIR_DOWN) {
1264 d = c >= MAX ? INF : c - d;
1273 // Search the cycle form the second node to the join node
1274 for (int u = second; u != join; u = _parent[u]) {
1277 if (_pred_dir[u] == DIR_UP) {
1279 d = c >= MAX ? INF : c - d;
1298 // Change _flow and _state vectors
1299 void changeFlow(bool change) {
1300 // Augment along the cycle
1302 Value val = _state[in_arc] * delta;
1303 _flow[in_arc] += val;
1304 for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1305 _flow[_pred[u]] -= _pred_dir[u] * val;
1307 for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1308 _flow[_pred[u]] += _pred_dir[u] * val;
1311 // Update the state of the entering and leaving arcs
1313 _state[in_arc] = STATE_TREE;
1314 _state[_pred[u_out]] =
1315 (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1317 _state[in_arc] = -_state[in_arc];
1321 // Update the tree structure
1322 void updateTreeStructure() {
1323 int old_rev_thread = _rev_thread[u_out];
1324 int old_succ_num = _succ_num[u_out];
1325 int old_last_succ = _last_succ[u_out];
1326 v_out = _parent[u_out];
1328 // Check if u_in and u_out coincide
1329 if (u_in == u_out) {
1330 // Update _parent, _pred, _pred_dir
1331 _parent[u_in] = v_in;
1332 _pred[u_in] = in_arc;
1333 _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1335 // Update _thread and _rev_thread
1336 if (_thread[v_in] != u_out) {
1337 int after = _thread[old_last_succ];
1338 _thread[old_rev_thread] = after;
1339 _rev_thread[after] = old_rev_thread;
1340 after = _thread[v_in];
1341 _thread[v_in] = u_out;
1342 _rev_thread[u_out] = v_in;
1343 _thread[old_last_succ] = after;
1344 _rev_thread[after] = old_last_succ;
1347 // Handle the case when old_rev_thread equals to v_in
1348 // (it also means that join and v_out coincide)
1349 int thread_continue = old_rev_thread == v_in ?
1350 _thread[old_last_succ] : _thread[v_in];
1352 // Update _thread and _parent along the stem nodes (i.e. the nodes
1353 // between u_in and u_out, whose parent have to be changed)
1354 int stem = u_in; // the current stem node
1355 int par_stem = v_in; // the new parent of stem
1356 int next_stem; // the next stem node
1357 int last = _last_succ[u_in]; // the last successor of stem
1358 int before, after = _thread[last];
1359 _thread[v_in] = u_in;
1360 _dirty_revs.clear();
1361 _dirty_revs.push_back(v_in);
1362 while (stem != u_out) {
1363 // Insert the next stem node into the thread list
1364 next_stem = _parent[stem];
1365 _thread[last] = next_stem;
1366 _dirty_revs.push_back(last);
1368 // Remove the subtree of stem from the thread list
1369 before = _rev_thread[stem];
1370 _thread[before] = after;
1371 _rev_thread[after] = before;
1373 // Change the parent node and shift stem nodes
1374 _parent[stem] = par_stem;
1378 // Update last and after
1379 last = _last_succ[stem] == _last_succ[par_stem] ?
1380 _rev_thread[par_stem] : _last_succ[stem];
1381 after = _thread[last];
1383 _parent[u_out] = par_stem;
1384 _thread[last] = thread_continue;
1385 _rev_thread[thread_continue] = last;
1386 _last_succ[u_out] = last;
1388 // Remove the subtree of u_out from the thread list except for
1389 // the case when old_rev_thread equals to v_in
1390 if (old_rev_thread != v_in) {
1391 _thread[old_rev_thread] = after;
1392 _rev_thread[after] = old_rev_thread;
1395 // Update _rev_thread using the new _thread values
1396 for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1397 int u = _dirty_revs[i];
1398 _rev_thread[_thread[u]] = u;
1401 // Update _pred, _pred_dir, _last_succ and _succ_num for the
1402 // stem nodes from u_out to u_in
1403 int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1404 for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
1405 _pred[u] = _pred[p];
1406 _pred_dir[u] = -_pred_dir[p];
1407 tmp_sc += _succ_num[u] - _succ_num[p];
1408 _succ_num[u] = tmp_sc;
1409 _last_succ[p] = tmp_ls;
1411 _pred[u_in] = in_arc;
1412 _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1413 _succ_num[u_in] = old_succ_num;
1416 // Update _last_succ from v_in towards the root
1417 int up_limit_out = _last_succ[join] == v_in ? join : -1;
1418 int last_succ_out = _last_succ[u_out];
1419 for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {
1420 _last_succ[u] = last_succ_out;
1423 // Update _last_succ from v_out towards the root
1424 if (join != old_rev_thread && v_in != old_rev_thread) {
1425 for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1427 _last_succ[u] = old_rev_thread;
1430 else if (last_succ_out != old_last_succ) {
1431 for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1433 _last_succ[u] = last_succ_out;
1437 // Update _succ_num from v_in to join
1438 for (int u = v_in; u != join; u = _parent[u]) {
1439 _succ_num[u] += old_succ_num;
1441 // Update _succ_num from v_out to join
1442 for (int u = v_out; u != join; u = _parent[u]) {
1443 _succ_num[u] -= old_succ_num;
1447 // Update potentials in the subtree that has been moved
1448 void updatePotential() {
1449 Cost sigma = _pi[v_in] - _pi[u_in] -
1450 _pred_dir[u_in] * _cost[in_arc];
1451 int end = _thread[_last_succ[u_in]];
1452 for (int u = u_in; u != end; u = _thread[u]) {
1457 // Heuristic initial pivots
1458 bool initialPivots() {
1459 Value curr, total = 0;
1460 std::vector<Node> supply_nodes, demand_nodes;
1461 for (NodeIt u(_graph); u != INVALID; ++u) {
1462 curr = _supply[_node_id[u]];
1465 supply_nodes.push_back(u);
1467 else if (curr < 0) {
1468 demand_nodes.push_back(u);
1471 if (_sum_supply > 0) total -= _sum_supply;
1472 if (total <= 0) return true;
1474 IntVector arc_vector;
1475 if (_sum_supply >= 0) {
1476 if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1477 // Perform a reverse graph search from the sink to the source
1478 typename GR::template NodeMap<bool> reached(_graph, false);
1479 Node s = supply_nodes[0], t = demand_nodes[0];
1480 std::vector<Node> stack;
1483 while (!stack.empty()) {
1484 Node u, v = stack.back();
1487 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1488 if (reached[u = _graph.source(a)]) continue;
1490 if (_cap[j] >= total) {
1491 arc_vector.push_back(j);
1498 // Find the min. cost incomming arc for each demand node
1499 for (int i = 0; i != int(demand_nodes.size()); ++i) {
1500 Node v = demand_nodes[i];
1501 Cost c, min_cost = std::numeric_limits<Cost>::max();
1502 Arc min_arc = INVALID;
1503 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1504 c = _cost[_arc_id[a]];
1510 if (min_arc != INVALID) {
1511 arc_vector.push_back(_arc_id[min_arc]);
1516 // Find the min. cost outgoing arc for each supply node
1517 for (int i = 0; i != int(supply_nodes.size()); ++i) {
1518 Node u = supply_nodes[i];
1519 Cost c, min_cost = std::numeric_limits<Cost>::max();
1520 Arc min_arc = INVALID;
1521 for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1522 c = _cost[_arc_id[a]];
1528 if (min_arc != INVALID) {
1529 arc_vector.push_back(_arc_id[min_arc]);
1534 // Perform heuristic initial pivots
1535 for (int i = 0; i != int(arc_vector.size()); ++i) {
1536 in_arc = arc_vector[i];
1537 if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1538 _pi[_target[in_arc]]) >= 0) continue;
1540 bool change = findLeavingArc();
1541 if (delta >= MAX) return false;
1544 updateTreeStructure();
1551 // Execute the algorithm
1552 ProblemType start(PivotRule pivot_rule) {
1553 // Select the pivot rule implementation
1554 switch (pivot_rule) {
1555 case FIRST_ELIGIBLE:
1556 return start<FirstEligiblePivotRule>();
1558 return start<BestEligiblePivotRule>();
1560 return start<BlockSearchPivotRule>();
1561 case CANDIDATE_LIST:
1562 return start<CandidateListPivotRule>();
1564 return start<AlteringListPivotRule>();
1566 return INFEASIBLE; // avoid warning
1569 template <typename PivotRuleImpl>
1570 ProblemType start() {
1571 PivotRuleImpl pivot(*this);
1573 // Perform heuristic initial pivots
1574 if (!initialPivots()) return UNBOUNDED;
1576 // Execute the Network Simplex algorithm
1577 while (pivot.findEnteringArc()) {
1579 bool change = findLeavingArc();
1580 if (delta >= MAX) return UNBOUNDED;
1583 updateTreeStructure();
1588 // Check feasibility
1589 for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1590 if (_flow[e] != 0) return INFEASIBLE;
1593 // Transform the solution and the supply map to the original form
1595 for (int i = 0; i != _arc_num; ++i) {
1596 Value c = _lower[i];
1599 _supply[_source[i]] += c;
1600 _supply[_target[i]] -= c;
1605 // Shift potentials to meet the requirements of the GEQ/LEQ type
1606 // optimality conditions
1607 if (_sum_supply == 0) {
1608 if (_stype == GEQ) {
1609 Cost max_pot = -std::numeric_limits<Cost>::max();
1610 for (int i = 0; i != _node_num; ++i) {
1611 if (_pi[i] > max_pot) max_pot = _pi[i];
1614 for (int i = 0; i != _node_num; ++i)
1618 Cost min_pot = std::numeric_limits<Cost>::max();
1619 for (int i = 0; i != _node_num; ++i) {
1620 if (_pi[i] < min_pot) min_pot = _pi[i];
1623 for (int i = 0; i != _node_num; ++i)
1632 }; //class NetworkSimplex
1638 #endif //LEMON_NETWORK_SIMPLEX_H