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, \ref NetworkSimplex and \ref CostScaling are the fastest
51 /// implementations available in LEMON for solving this problem.
52 /// (For more information, see \ref min_cost_flow_algs "the module page".)
53 /// Furthermore, this class supports both directions of the supply/demand
54 /// inequality constraints. For more information, see \ref SupplyType.
56 /// Most of the parameters of the problem (except for the digraph)
57 /// can be given using separate functions, and the algorithm can be
58 /// executed using the \ref run() function. If some parameters are not
59 /// specified, then default values will be used.
61 /// \tparam GR The digraph type the algorithm runs on.
62 /// \tparam V The number type used for flow amounts, capacity bounds
63 /// and supply values in the algorithm. By default, it is \c int.
64 /// \tparam C The number type used for costs and potentials in the
65 /// algorithm. By default, it is the same as \c V.
67 /// \warning Both \c V and \c C must be signed number types.
68 /// \warning All input data (capacities, supply values, and costs) must
71 /// \note %NetworkSimplex provides five different pivot rule
72 /// implementations, from which the most efficient one is used
73 /// by default. For more information, see \ref PivotRule.
74 template <typename GR, typename V = int, typename C = V>
79 /// The type of the flow amounts, capacity bounds and supply values
81 /// The type of the arc costs
86 /// \brief Problem type constants for the \c run() function.
88 /// Enum type containing the problem type constants that can be
89 /// returned by the \ref run() function of the algorithm.
91 /// The problem has no feasible solution (flow).
93 /// The problem has optimal solution (i.e. it is feasible and
94 /// bounded), and the algorithm has found optimal flow and node
95 /// potentials (primal and dual solutions).
97 /// The objective function of the problem is unbounded, i.e.
98 /// there is a directed cycle having negative total cost and
99 /// infinite upper bound.
103 /// \brief Constants for selecting the type of the supply constraints.
105 /// Enum type containing constants for selecting the supply type,
106 /// i.e. the direction of the inequalities in the supply/demand
107 /// constraints of the \ref min_cost_flow "minimum cost flow problem".
109 /// The default supply type is \c GEQ, the \c LEQ type can be
110 /// selected using \ref supplyType().
111 /// The equality form is a special case of both supply types.
113 /// This option means that there are <em>"greater or equal"</em>
114 /// supply/demand constraints in the definition of the problem.
116 /// This option means that there are <em>"less or equal"</em>
117 /// supply/demand constraints in the definition of the problem.
121 /// \brief Constants for selecting the pivot rule.
123 /// Enum type containing constants for selecting the pivot rule for
124 /// the \ref run() function.
126 /// \ref NetworkSimplex provides five different pivot rule
127 /// implementations that significantly affect the running time
128 /// of the algorithm.
129 /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
130 /// turend out to be the most efficient and the most robust on various
132 /// However, another pivot rule can be selected using the \ref run()
133 /// function with the proper parameter.
136 /// The \e First \e Eligible pivot rule.
137 /// The next eligible arc is selected in a wraparound fashion
138 /// in every iteration.
141 /// The \e Best \e Eligible pivot rule.
142 /// The best eligible arc is selected in every iteration.
145 /// The \e Block \e Search pivot rule.
146 /// A specified number of arcs are examined in every iteration
147 /// in a wraparound fashion and the best eligible arc is selected
151 /// The \e Candidate \e List pivot rule.
152 /// In a major iteration a candidate list is built from eligible arcs
153 /// in a wraparound fashion and in the following minor iterations
154 /// the best eligible arc is selected from this list.
157 /// The \e Altering \e Candidate \e List pivot rule.
158 /// It is a modified version of the Candidate List method.
159 /// It keeps only the several best eligible arcs from the former
160 /// candidate list and extends this list in every iteration.
166 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
168 typedef std::vector<int> IntVector;
169 typedef std::vector<Value> ValueVector;
170 typedef std::vector<Cost> CostVector;
171 typedef std::vector<signed char> CharVector;
172 // Note: vector<signed char> is used instead of vector<ArcState> and
173 // vector<ArcDirection> for efficiency reasons
175 // State constants for arcs
182 // Direction constants for tree arcs
190 // Data related to the underlying digraph
197 // Parameters of the problem
202 // Data structures for storing the digraph
218 // Data for storing the spanning tree structure
222 IntVector _rev_thread;
224 IntVector _last_succ;
225 CharVector _pred_dir;
227 IntVector _dirty_revs;
230 // Temporary data used in the current pivot iteration
231 int in_arc, join, u_in, v_in, u_out, v_out;
238 /// \brief Constant for infinite upper bounds (capacities).
240 /// Constant for infinite upper bounds (capacities).
241 /// It is \c std::numeric_limits<Value>::infinity() if available,
242 /// \c std::numeric_limits<Value>::max() otherwise.
247 // Implementation of the First Eligible pivot rule
248 class FirstEligiblePivotRule
252 // References to the NetworkSimplex class
253 const IntVector &_source;
254 const IntVector &_target;
255 const CostVector &_cost;
256 const CharVector &_state;
257 const CostVector &_pi;
267 FirstEligiblePivotRule(NetworkSimplex &ns) :
268 _source(ns._source), _target(ns._target),
269 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
270 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
274 // Find next entering arc
275 bool findEnteringArc() {
277 for (int e = _next_arc; e != _search_arc_num; ++e) {
278 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
285 for (int e = 0; e != _next_arc; ++e) {
286 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
296 }; //class FirstEligiblePivotRule
299 // Implementation of the Best Eligible pivot rule
300 class BestEligiblePivotRule
304 // References to the NetworkSimplex class
305 const IntVector &_source;
306 const IntVector &_target;
307 const CostVector &_cost;
308 const CharVector &_state;
309 const CostVector &_pi;
316 BestEligiblePivotRule(NetworkSimplex &ns) :
317 _source(ns._source), _target(ns._target),
318 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
319 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
322 // Find next entering arc
323 bool findEnteringArc() {
325 for (int e = 0; e != _search_arc_num; ++e) {
326 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
335 }; //class BestEligiblePivotRule
338 // Implementation of the Block Search pivot rule
339 class BlockSearchPivotRule
343 // References to the NetworkSimplex class
344 const IntVector &_source;
345 const IntVector &_target;
346 const CostVector &_cost;
347 const CharVector &_state;
348 const CostVector &_pi;
359 BlockSearchPivotRule(NetworkSimplex &ns) :
360 _source(ns._source), _target(ns._target),
361 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
362 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
365 // The main parameters of the pivot rule
366 const double BLOCK_SIZE_FACTOR = 1.0;
367 const int MIN_BLOCK_SIZE = 10;
369 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
370 std::sqrt(double(_search_arc_num))),
374 // Find next entering arc
375 bool findEnteringArc() {
377 int cnt = _block_size;
379 for (e = _next_arc; e != _search_arc_num; ++e) {
380 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
386 if (min < 0) goto search_end;
390 for (e = 0; e != _next_arc; ++e) {
391 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
397 if (min < 0) goto search_end;
401 if (min >= 0) return false;
408 }; //class BlockSearchPivotRule
411 // Implementation of the Candidate List pivot rule
412 class CandidateListPivotRule
416 // References to the NetworkSimplex class
417 const IntVector &_source;
418 const IntVector &_target;
419 const CostVector &_cost;
420 const CharVector &_state;
421 const CostVector &_pi;
426 IntVector _candidates;
427 int _list_length, _minor_limit;
428 int _curr_length, _minor_count;
434 CandidateListPivotRule(NetworkSimplex &ns) :
435 _source(ns._source), _target(ns._target),
436 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
437 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
440 // The main parameters of the pivot rule
441 const double LIST_LENGTH_FACTOR = 0.25;
442 const int MIN_LIST_LENGTH = 10;
443 const double MINOR_LIMIT_FACTOR = 0.1;
444 const int MIN_MINOR_LIMIT = 3;
446 _list_length = std::max( int(LIST_LENGTH_FACTOR *
447 std::sqrt(double(_search_arc_num))),
449 _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
451 _curr_length = _minor_count = 0;
452 _candidates.resize(_list_length);
455 /// Find next entering arc
456 bool findEnteringArc() {
459 if (_curr_length > 0 && _minor_count < _minor_limit) {
460 // Minor iteration: select the best eligible arc from the
461 // current candidate list
464 for (int i = 0; i < _curr_length; ++i) {
466 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
472 _candidates[i--] = _candidates[--_curr_length];
475 if (min < 0) return true;
478 // Major iteration: build a new candidate list
481 for (e = _next_arc; e != _search_arc_num; ++e) {
482 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
484 _candidates[_curr_length++] = e;
489 if (_curr_length == _list_length) goto search_end;
492 for (e = 0; e != _next_arc; ++e) {
493 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
495 _candidates[_curr_length++] = e;
500 if (_curr_length == _list_length) goto search_end;
503 if (_curr_length == 0) return false;
511 }; //class CandidateListPivotRule
514 // Implementation of the Altering Candidate List pivot rule
515 class AlteringListPivotRule
519 // References to the NetworkSimplex class
520 const IntVector &_source;
521 const IntVector &_target;
522 const CostVector &_cost;
523 const CharVector &_state;
524 const CostVector &_pi;
529 int _block_size, _head_length, _curr_length;
531 IntVector _candidates;
532 CostVector _cand_cost;
534 // Functor class to compare arcs during sort of the candidate list
538 const CostVector &_map;
540 SortFunc(const CostVector &map) : _map(map) {}
541 bool operator()(int left, int right) {
542 return _map[left] > _map[right];
551 AlteringListPivotRule(NetworkSimplex &ns) :
552 _source(ns._source), _target(ns._target),
553 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
554 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
555 _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
557 // The main parameters of the pivot rule
558 const double BLOCK_SIZE_FACTOR = 1.0;
559 const int MIN_BLOCK_SIZE = 10;
560 const double HEAD_LENGTH_FACTOR = 0.1;
561 const int MIN_HEAD_LENGTH = 3;
563 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
564 std::sqrt(double(_search_arc_num))),
566 _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
568 _candidates.resize(_head_length + _block_size);
572 // Find next entering arc
573 bool findEnteringArc() {
574 // Check the current candidate list
577 for (int i = 0; i != _curr_length; ++i) {
579 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
583 _candidates[i--] = _candidates[--_curr_length];
588 int cnt = _block_size;
589 int limit = _head_length;
591 for (e = _next_arc; e != _search_arc_num; ++e) {
592 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
595 _candidates[_curr_length++] = e;
598 if (_curr_length > limit) goto search_end;
603 for (e = 0; e != _next_arc; ++e) {
604 _cand_cost[e] = _state[e] *
605 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
606 if (_cand_cost[e] < 0) {
607 _candidates[_curr_length++] = e;
610 if (_curr_length > limit) goto search_end;
615 if (_curr_length == 0) return false;
619 // Make heap of the candidate list (approximating a partial sort)
620 make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
623 // Pop the first element of the heap
624 _in_arc = _candidates[0];
626 pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
628 _curr_length = std::min(_head_length, _curr_length - 1);
632 }; //class AlteringListPivotRule
636 /// \brief Constructor.
638 /// The constructor of the class.
640 /// \param graph The digraph the algorithm runs on.
641 /// \param arc_mixing Indicate if the arcs will be stored in a
642 /// mixed order in the internal data structure.
643 /// In general, it leads to similar performance as using the original
644 /// arc order, but it makes the algorithm more robust and in special
645 /// cases, even significantly faster. Therefore, it is enabled by default.
646 NetworkSimplex(const GR& graph, bool arc_mixing = true) :
647 _graph(graph), _node_id(graph), _arc_id(graph),
648 _arc_mixing(arc_mixing),
649 MAX(std::numeric_limits<Value>::max()),
650 INF(std::numeric_limits<Value>::has_infinity ?
651 std::numeric_limits<Value>::infinity() : MAX)
653 // Check the number types
654 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
655 "The flow type of NetworkSimplex must be signed");
656 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
657 "The cost type of NetworkSimplex must be signed");
659 // Reset data structures
664 /// The parameters of the algorithm can be specified using these
669 /// \brief Set the lower bounds on the arcs.
671 /// This function sets the lower bounds on the arcs.
672 /// If it is not used before calling \ref run(), the lower bounds
673 /// will be set to zero on all arcs.
675 /// \param map An arc map storing the lower bounds.
676 /// Its \c Value type must be convertible to the \c Value type
677 /// of the algorithm.
679 /// \return <tt>(*this)</tt>
680 template <typename LowerMap>
681 NetworkSimplex& lowerMap(const LowerMap& map) {
683 for (ArcIt a(_graph); a != INVALID; ++a) {
684 _lower[_arc_id[a]] = map[a];
689 /// \brief Set the upper bounds (capacities) on the arcs.
691 /// This function sets the upper bounds (capacities) on the arcs.
692 /// If it is not used before calling \ref run(), the upper bounds
693 /// will be set to \ref INF on all arcs (i.e. the flow value will be
694 /// unbounded from above).
696 /// \param map An arc map storing the upper bounds.
697 /// Its \c Value type must be convertible to the \c Value type
698 /// of the algorithm.
700 /// \return <tt>(*this)</tt>
701 template<typename UpperMap>
702 NetworkSimplex& upperMap(const UpperMap& map) {
703 for (ArcIt a(_graph); a != INVALID; ++a) {
704 _upper[_arc_id[a]] = map[a];
709 /// \brief Set the costs of the arcs.
711 /// This function sets the costs of the arcs.
712 /// If it is not used before calling \ref run(), the costs
713 /// will be set to \c 1 on all arcs.
715 /// \param map An arc map storing the costs.
716 /// Its \c Value type must be convertible to the \c Cost type
717 /// of the algorithm.
719 /// \return <tt>(*this)</tt>
720 template<typename CostMap>
721 NetworkSimplex& costMap(const CostMap& map) {
722 for (ArcIt a(_graph); a != INVALID; ++a) {
723 _cost[_arc_id[a]] = map[a];
728 /// \brief Set the supply values of the nodes.
730 /// This function sets the supply values of the nodes.
731 /// If neither this function nor \ref stSupply() is used before
732 /// calling \ref run(), the supply of each node will be set to zero.
734 /// \param map A node map storing the supply values.
735 /// Its \c Value type must be convertible to the \c Value type
736 /// of the algorithm.
738 /// \return <tt>(*this)</tt>
741 template<typename SupplyMap>
742 NetworkSimplex& supplyMap(const SupplyMap& map) {
743 for (NodeIt n(_graph); n != INVALID; ++n) {
744 _supply[_node_id[n]] = map[n];
749 /// \brief Set single source and target nodes and a supply value.
751 /// This function sets a single source node and a single target node
752 /// and the required flow value.
753 /// If neither this function nor \ref supplyMap() is used before
754 /// calling \ref run(), the supply of each node will be set to zero.
756 /// Using this function has the same effect as using \ref supplyMap()
757 /// with a map in which \c k is assigned to \c s, \c -k is
758 /// assigned to \c t and all other nodes have zero supply value.
760 /// \param s The source node.
761 /// \param t The target node.
762 /// \param k The required amount of flow from node \c s to node \c t
763 /// (i.e. the supply of \c s and the demand of \c t).
765 /// \return <tt>(*this)</tt>
766 NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
767 for (int i = 0; i != _node_num; ++i) {
770 _supply[_node_id[s]] = k;
771 _supply[_node_id[t]] = -k;
775 /// \brief Set the type of the supply constraints.
777 /// This function sets the type of the supply/demand constraints.
778 /// If it is not used before calling \ref run(), the \ref GEQ supply
779 /// type will be used.
781 /// For more information, see \ref SupplyType.
783 /// \return <tt>(*this)</tt>
784 NetworkSimplex& supplyType(SupplyType supply_type) {
785 _stype = supply_type;
791 /// \name Execution Control
792 /// The algorithm can be executed using \ref run().
796 /// \brief Run the algorithm.
798 /// This function runs the algorithm.
799 /// The paramters can be specified using functions \ref lowerMap(),
800 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
801 /// \ref supplyType().
804 /// NetworkSimplex<ListDigraph> ns(graph);
805 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
806 /// .supplyMap(sup).run();
809 /// This function can be called more than once. All the given parameters
810 /// are kept for the next call, unless \ref resetParams() or \ref reset()
811 /// is used, thus only the modified parameters have to be set again.
812 /// If the underlying digraph was also modified after the construction
813 /// of the class (or the last \ref reset() call), then the \ref reset()
814 /// function must be called.
816 /// \param pivot_rule The pivot rule that will be used during the
817 /// algorithm. For more information, see \ref PivotRule.
819 /// \return \c INFEASIBLE if no feasible flow exists,
820 /// \n \c OPTIMAL if the problem has optimal solution
821 /// (i.e. it is feasible and bounded), and the algorithm has found
822 /// optimal flow and node potentials (primal and dual solutions),
823 /// \n \c UNBOUNDED if the objective function of the problem is
824 /// unbounded, i.e. there is a directed cycle having negative total
825 /// cost and infinite upper bound.
827 /// \see ProblemType, PivotRule
828 /// \see resetParams(), reset()
829 ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
830 if (!init()) return INFEASIBLE;
831 return start(pivot_rule);
834 /// \brief Reset all the parameters that have been given before.
836 /// This function resets all the paramaters that have been given
837 /// before using functions \ref lowerMap(), \ref upperMap(),
838 /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
840 /// It is useful for multiple \ref run() calls. Basically, all the given
841 /// parameters are kept for the next \ref run() call, unless
842 /// \ref resetParams() or \ref reset() is used.
843 /// If the underlying digraph was also modified after the construction
844 /// of the class or the last \ref reset() call, then the \ref reset()
845 /// function must be used, otherwise \ref resetParams() is sufficient.
849 /// NetworkSimplex<ListDigraph> ns(graph);
852 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
853 /// .supplyMap(sup).run();
855 /// // Run again with modified cost map (resetParams() is not called,
856 /// // so only the cost map have to be set again)
858 /// ns.costMap(cost).run();
860 /// // Run again from scratch using resetParams()
861 /// // (the lower bounds will be set to zero on all arcs)
862 /// ns.resetParams();
863 /// ns.upperMap(capacity).costMap(cost)
864 /// .supplyMap(sup).run();
867 /// \return <tt>(*this)</tt>
869 /// \see reset(), run()
870 NetworkSimplex& resetParams() {
871 for (int i = 0; i != _node_num; ++i) {
874 for (int i = 0; i != _arc_num; ++i) {
884 /// \brief Reset the internal data structures and all the parameters
885 /// that have been given before.
887 /// This function resets the internal data structures and all the
888 /// paramaters that have been given before using functions \ref lowerMap(),
889 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
890 /// \ref supplyType().
892 /// It is useful for multiple \ref run() calls. Basically, all the given
893 /// parameters are kept for the next \ref run() call, unless
894 /// \ref resetParams() or \ref reset() is used.
895 /// If the underlying digraph was also modified after the construction
896 /// of the class or the last \ref reset() call, then the \ref reset()
897 /// function must be used, otherwise \ref resetParams() is sufficient.
899 /// See \ref resetParams() for examples.
901 /// \return <tt>(*this)</tt>
903 /// \see resetParams(), run()
904 NetworkSimplex& reset() {
906 _node_num = countNodes(_graph);
907 _arc_num = countArcs(_graph);
908 int all_node_num = _node_num + 1;
909 int max_arc_num = _arc_num + 2 * _node_num;
911 _source.resize(max_arc_num);
912 _target.resize(max_arc_num);
914 _lower.resize(_arc_num);
915 _upper.resize(_arc_num);
916 _cap.resize(max_arc_num);
917 _cost.resize(max_arc_num);
918 _supply.resize(all_node_num);
919 _flow.resize(max_arc_num);
920 _pi.resize(all_node_num);
922 _parent.resize(all_node_num);
923 _pred.resize(all_node_num);
924 _pred_dir.resize(all_node_num);
925 _thread.resize(all_node_num);
926 _rev_thread.resize(all_node_num);
927 _succ_num.resize(all_node_num);
928 _last_succ.resize(all_node_num);
929 _state.resize(max_arc_num);
933 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
937 // Store the arcs in a mixed order
938 const int skip = std::max(_arc_num / _node_num, 3);
940 for (ArcIt a(_graph); a != INVALID; ++a) {
942 _source[i] = _node_id[_graph.source(a)];
943 _target[i] = _node_id[_graph.target(a)];
944 if ((i += skip) >= _arc_num) i = ++j;
947 // Store the arcs in the original order
949 for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
951 _source[i] = _node_id[_graph.source(a)];
952 _target[i] = _node_id[_graph.target(a)];
963 /// \name Query Functions
964 /// The results of the algorithm can be obtained using these
966 /// The \ref run() function must be called before using them.
970 /// \brief Return the total cost of the found flow.
972 /// This function returns the total cost of the found flow.
973 /// Its complexity is O(e).
975 /// \note The return type of the function can be specified as a
976 /// template parameter. For example,
978 /// ns.totalCost<double>();
980 /// It is useful if the total cost cannot be stored in the \c Cost
981 /// type of the algorithm, which is the default return type of the
984 /// \pre \ref run() must be called before using this function.
985 template <typename Number>
986 Number totalCost() const {
988 for (ArcIt a(_graph); a != INVALID; ++a) {
990 c += Number(_flow[i]) * Number(_cost[i]);
996 Cost totalCost() const {
997 return totalCost<Cost>();
1001 /// \brief Return the flow on the given arc.
1003 /// This function returns the flow on the given arc.
1005 /// \pre \ref run() must be called before using this function.
1006 Value flow(const Arc& a) const {
1007 return _flow[_arc_id[a]];
1010 /// \brief Copy the flow values (the primal solution) into the
1013 /// This function copies the flow value on each arc into the given
1014 /// map. The \c Value type of the algorithm must be convertible to
1015 /// the \c Value type of the map.
1017 /// \pre \ref run() must be called before using this function.
1018 template <typename FlowMap>
1019 void flowMap(FlowMap &map) const {
1020 for (ArcIt a(_graph); a != INVALID; ++a) {
1021 map.set(a, _flow[_arc_id[a]]);
1025 /// \brief Return the potential (dual value) of the given node.
1027 /// This function returns the potential (dual value) of the
1030 /// \pre \ref run() must be called before using this function.
1031 Cost potential(const Node& n) const {
1032 return _pi[_node_id[n]];
1035 /// \brief Copy the potential values (the dual solution) into the
1038 /// This function copies the potential (dual value) of each node
1039 /// into the given map.
1040 /// The \c Cost type of the algorithm must be convertible to the
1041 /// \c Value type of the map.
1043 /// \pre \ref run() must be called before using this function.
1044 template <typename PotentialMap>
1045 void potentialMap(PotentialMap &map) const {
1046 for (NodeIt n(_graph); n != INVALID; ++n) {
1047 map.set(n, _pi[_node_id[n]]);
1055 // Initialize internal data structures
1057 if (_node_num == 0) return false;
1059 // Check the sum of supply values
1061 for (int i = 0; i != _node_num; ++i) {
1062 _sum_supply += _supply[i];
1064 if ( !((_stype == GEQ && _sum_supply <= 0) ||
1065 (_stype == LEQ && _sum_supply >= 0)) ) return false;
1067 // Remove non-zero lower bounds
1069 for (int i = 0; i != _arc_num; ++i) {
1070 Value c = _lower[i];
1072 _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1074 _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1076 _supply[_source[i]] -= c;
1077 _supply[_target[i]] += c;
1080 for (int i = 0; i != _arc_num; ++i) {
1081 _cap[i] = _upper[i];
1085 // Initialize artifical cost
1087 if (std::numeric_limits<Cost>::is_exact) {
1088 ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1091 for (int i = 0; i != _arc_num; ++i) {
1092 if (_cost[i] > ART_COST) ART_COST = _cost[i];
1094 ART_COST = (ART_COST + 1) * _node_num;
1097 // Initialize arc maps
1098 for (int i = 0; i != _arc_num; ++i) {
1100 _state[i] = STATE_LOWER;
1103 // Set data for the artificial root node
1105 _parent[_root] = -1;
1108 _rev_thread[0] = _root;
1109 _succ_num[_root] = _node_num + 1;
1110 _last_succ[_root] = _root - 1;
1111 _supply[_root] = -_sum_supply;
1114 // Add artificial arcs and initialize the spanning tree data structure
1115 if (_sum_supply == 0) {
1116 // EQ supply constraints
1117 _search_arc_num = _arc_num;
1118 _all_arc_num = _arc_num + _node_num;
1119 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1123 _rev_thread[u + 1] = u;
1127 _state[e] = STATE_TREE;
1128 if (_supply[u] >= 0) {
1129 _pred_dir[u] = DIR_UP;
1133 _flow[e] = _supply[u];
1136 _pred_dir[u] = DIR_DOWN;
1140 _flow[e] = -_supply[u];
1141 _cost[e] = ART_COST;
1145 else if (_sum_supply > 0) {
1146 // LEQ supply constraints
1147 _search_arc_num = _arc_num + _node_num;
1148 int f = _arc_num + _node_num;
1149 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1152 _rev_thread[u + 1] = u;
1155 if (_supply[u] >= 0) {
1156 _pred_dir[u] = DIR_UP;
1162 _flow[e] = _supply[u];
1164 _state[e] = STATE_TREE;
1166 _pred_dir[u] = DIR_DOWN;
1172 _flow[f] = -_supply[u];
1173 _cost[f] = ART_COST;
1174 _state[f] = STATE_TREE;
1180 _state[e] = STATE_LOWER;
1187 // GEQ supply constraints
1188 _search_arc_num = _arc_num + _node_num;
1189 int f = _arc_num + _node_num;
1190 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1193 _rev_thread[u + 1] = u;
1196 if (_supply[u] <= 0) {
1197 _pred_dir[u] = DIR_DOWN;
1203 _flow[e] = -_supply[u];
1205 _state[e] = STATE_TREE;
1207 _pred_dir[u] = DIR_UP;
1213 _flow[f] = _supply[u];
1214 _state[f] = STATE_TREE;
1215 _cost[f] = ART_COST;
1221 _state[e] = STATE_LOWER;
1231 // Find the join node
1232 void findJoinNode() {
1233 int u = _source[in_arc];
1234 int v = _target[in_arc];
1236 if (_succ_num[u] < _succ_num[v]) {
1245 // Find the leaving arc of the cycle and returns true if the
1246 // leaving arc is not the same as the entering arc
1247 bool findLeavingArc() {
1248 // Initialize first and second nodes according to the direction
1251 if (_state[in_arc] == STATE_LOWER) {
1252 first = _source[in_arc];
1253 second = _target[in_arc];
1255 first = _target[in_arc];
1256 second = _source[in_arc];
1258 delta = _cap[in_arc];
1263 // Search the cycle form the first node to the join node
1264 for (int u = first; u != join; u = _parent[u]) {
1267 if (_pred_dir[u] == DIR_DOWN) {
1269 d = c >= MAX ? INF : c - d;
1278 // Search the cycle form the second node to the join node
1279 for (int u = second; u != join; u = _parent[u]) {
1282 if (_pred_dir[u] == DIR_UP) {
1284 d = c >= MAX ? INF : c - d;
1303 // Change _flow and _state vectors
1304 void changeFlow(bool change) {
1305 // Augment along the cycle
1307 Value val = _state[in_arc] * delta;
1308 _flow[in_arc] += val;
1309 for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1310 _flow[_pred[u]] -= _pred_dir[u] * val;
1312 for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1313 _flow[_pred[u]] += _pred_dir[u] * val;
1316 // Update the state of the entering and leaving arcs
1318 _state[in_arc] = STATE_TREE;
1319 _state[_pred[u_out]] =
1320 (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1322 _state[in_arc] = -_state[in_arc];
1326 // Update the tree structure
1327 void updateTreeStructure() {
1328 int old_rev_thread = _rev_thread[u_out];
1329 int old_succ_num = _succ_num[u_out];
1330 int old_last_succ = _last_succ[u_out];
1331 v_out = _parent[u_out];
1333 // Check if u_in and u_out coincide
1334 if (u_in == u_out) {
1335 // Update _parent, _pred, _pred_dir
1336 _parent[u_in] = v_in;
1337 _pred[u_in] = in_arc;
1338 _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1340 // Update _thread and _rev_thread
1341 if (_thread[v_in] != u_out) {
1342 int after = _thread[old_last_succ];
1343 _thread[old_rev_thread] = after;
1344 _rev_thread[after] = old_rev_thread;
1345 after = _thread[v_in];
1346 _thread[v_in] = u_out;
1347 _rev_thread[u_out] = v_in;
1348 _thread[old_last_succ] = after;
1349 _rev_thread[after] = old_last_succ;
1352 // Handle the case when old_rev_thread equals to v_in
1353 // (it also means that join and v_out coincide)
1354 int thread_continue = old_rev_thread == v_in ?
1355 _thread[old_last_succ] : _thread[v_in];
1357 // Update _thread and _parent along the stem nodes (i.e. the nodes
1358 // between u_in and u_out, whose parent have to be changed)
1359 int stem = u_in; // the current stem node
1360 int par_stem = v_in; // the new parent of stem
1361 int next_stem; // the next stem node
1362 int last = _last_succ[u_in]; // the last successor of stem
1363 int before, after = _thread[last];
1364 _thread[v_in] = u_in;
1365 _dirty_revs.clear();
1366 _dirty_revs.push_back(v_in);
1367 while (stem != u_out) {
1368 // Insert the next stem node into the thread list
1369 next_stem = _parent[stem];
1370 _thread[last] = next_stem;
1371 _dirty_revs.push_back(last);
1373 // Remove the subtree of stem from the thread list
1374 before = _rev_thread[stem];
1375 _thread[before] = after;
1376 _rev_thread[after] = before;
1378 // Change the parent node and shift stem nodes
1379 _parent[stem] = par_stem;
1383 // Update last and after
1384 last = _last_succ[stem] == _last_succ[par_stem] ?
1385 _rev_thread[par_stem] : _last_succ[stem];
1386 after = _thread[last];
1388 _parent[u_out] = par_stem;
1389 _thread[last] = thread_continue;
1390 _rev_thread[thread_continue] = last;
1391 _last_succ[u_out] = last;
1393 // Remove the subtree of u_out from the thread list except for
1394 // the case when old_rev_thread equals to v_in
1395 if (old_rev_thread != v_in) {
1396 _thread[old_rev_thread] = after;
1397 _rev_thread[after] = old_rev_thread;
1400 // Update _rev_thread using the new _thread values
1401 for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1402 int u = _dirty_revs[i];
1403 _rev_thread[_thread[u]] = u;
1406 // Update _pred, _pred_dir, _last_succ and _succ_num for the
1407 // stem nodes from u_out to u_in
1408 int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1409 for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
1410 _pred[u] = _pred[p];
1411 _pred_dir[u] = -_pred_dir[p];
1412 tmp_sc += _succ_num[u] - _succ_num[p];
1413 _succ_num[u] = tmp_sc;
1414 _last_succ[p] = tmp_ls;
1416 _pred[u_in] = in_arc;
1417 _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1418 _succ_num[u_in] = old_succ_num;
1421 // Update _last_succ from v_in towards the root
1422 int up_limit_out = _last_succ[join] == v_in ? join : -1;
1423 int last_succ_out = _last_succ[u_out];
1424 for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {
1425 _last_succ[u] = last_succ_out;
1428 // Update _last_succ from v_out towards the root
1429 if (join != old_rev_thread && v_in != old_rev_thread) {
1430 for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1432 _last_succ[u] = old_rev_thread;
1435 else if (last_succ_out != old_last_succ) {
1436 for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1438 _last_succ[u] = last_succ_out;
1442 // Update _succ_num from v_in to join
1443 for (int u = v_in; u != join; u = _parent[u]) {
1444 _succ_num[u] += old_succ_num;
1446 // Update _succ_num from v_out to join
1447 for (int u = v_out; u != join; u = _parent[u]) {
1448 _succ_num[u] -= old_succ_num;
1452 // Update potentials in the subtree that has been moved
1453 void updatePotential() {
1454 Cost sigma = _pi[v_in] - _pi[u_in] -
1455 _pred_dir[u_in] * _cost[in_arc];
1456 int end = _thread[_last_succ[u_in]];
1457 for (int u = u_in; u != end; u = _thread[u]) {
1462 // Heuristic initial pivots
1463 bool initialPivots() {
1464 Value curr, total = 0;
1465 std::vector<Node> supply_nodes, demand_nodes;
1466 for (NodeIt u(_graph); u != INVALID; ++u) {
1467 curr = _supply[_node_id[u]];
1470 supply_nodes.push_back(u);
1472 else if (curr < 0) {
1473 demand_nodes.push_back(u);
1476 if (_sum_supply > 0) total -= _sum_supply;
1477 if (total <= 0) return true;
1479 IntVector arc_vector;
1480 if (_sum_supply >= 0) {
1481 if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1482 // Perform a reverse graph search from the sink to the source
1483 typename GR::template NodeMap<bool> reached(_graph, false);
1484 Node s = supply_nodes[0], t = demand_nodes[0];
1485 std::vector<Node> stack;
1488 while (!stack.empty()) {
1489 Node u, v = stack.back();
1492 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1493 if (reached[u = _graph.source(a)]) continue;
1495 if (_cap[j] >= total) {
1496 arc_vector.push_back(j);
1503 // Find the min. cost incomming arc for each demand node
1504 for (int i = 0; i != int(demand_nodes.size()); ++i) {
1505 Node v = demand_nodes[i];
1506 Cost c, min_cost = std::numeric_limits<Cost>::max();
1507 Arc min_arc = INVALID;
1508 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1509 c = _cost[_arc_id[a]];
1515 if (min_arc != INVALID) {
1516 arc_vector.push_back(_arc_id[min_arc]);
1521 // Find the min. cost outgoing arc for each supply node
1522 for (int i = 0; i != int(supply_nodes.size()); ++i) {
1523 Node u = supply_nodes[i];
1524 Cost c, min_cost = std::numeric_limits<Cost>::max();
1525 Arc min_arc = INVALID;
1526 for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1527 c = _cost[_arc_id[a]];
1533 if (min_arc != INVALID) {
1534 arc_vector.push_back(_arc_id[min_arc]);
1539 // Perform heuristic initial pivots
1540 for (int i = 0; i != int(arc_vector.size()); ++i) {
1541 in_arc = arc_vector[i];
1542 if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1543 _pi[_target[in_arc]]) >= 0) continue;
1545 bool change = findLeavingArc();
1546 if (delta >= MAX) return false;
1549 updateTreeStructure();
1556 // Execute the algorithm
1557 ProblemType start(PivotRule pivot_rule) {
1558 // Select the pivot rule implementation
1559 switch (pivot_rule) {
1560 case FIRST_ELIGIBLE:
1561 return start<FirstEligiblePivotRule>();
1563 return start<BestEligiblePivotRule>();
1565 return start<BlockSearchPivotRule>();
1566 case CANDIDATE_LIST:
1567 return start<CandidateListPivotRule>();
1569 return start<AlteringListPivotRule>();
1571 return INFEASIBLE; // avoid warning
1574 template <typename PivotRuleImpl>
1575 ProblemType start() {
1576 PivotRuleImpl pivot(*this);
1578 // Perform heuristic initial pivots
1579 if (!initialPivots()) return UNBOUNDED;
1581 // Execute the Network Simplex algorithm
1582 while (pivot.findEnteringArc()) {
1584 bool change = findLeavingArc();
1585 if (delta >= MAX) return UNBOUNDED;
1588 updateTreeStructure();
1593 // Check feasibility
1594 for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1595 if (_flow[e] != 0) return INFEASIBLE;
1598 // Transform the solution and the supply map to the original form
1600 for (int i = 0; i != _arc_num; ++i) {
1601 Value c = _lower[i];
1604 _supply[_source[i]] += c;
1605 _supply[_target[i]] -= c;
1610 // Shift potentials to meet the requirements of the GEQ/LEQ type
1611 // optimality conditions
1612 if (_sum_supply == 0) {
1613 if (_stype == GEQ) {
1614 Cost max_pot = -std::numeric_limits<Cost>::max();
1615 for (int i = 0; i != _node_num; ++i) {
1616 if (_pi[i] > max_pot) max_pot = _pi[i];
1619 for (int i = 0; i != _node_num; ++i)
1623 Cost min_pot = std::numeric_limits<Cost>::max();
1624 for (int i = 0; i != _node_num; ++i) {
1625 if (_pi[i] < min_pot) min_pot = _pi[i];
1628 for (int i = 0; i != _node_num; ++i)
1637 }; //class NetworkSimplex
1643 #endif //LEMON_NETWORK_SIMPLEX_H