1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
3 * This file is a part of LEMON, a generic C++ optimization library.
5 * Copyright (C) 2003-2009
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 number types must be signed and all input data must
69 /// \note %NetworkSimplex provides five different pivot rule
70 /// implementations, from which the most efficient one is used
71 /// by default. For more information, see \ref PivotRule.
72 template <typename GR, typename V = int, typename C = V>
77 /// The type of the flow amounts, capacity bounds and supply values
79 /// The type of the arc costs
84 /// \brief Problem type constants for the \c run() function.
86 /// Enum type containing the problem type constants that can be
87 /// returned by the \ref run() function of the algorithm.
89 /// The problem has no feasible solution (flow).
91 /// The problem has optimal solution (i.e. it is feasible and
92 /// bounded), and the algorithm has found optimal flow and node
93 /// potentials (primal and dual solutions).
95 /// The objective function of the problem is unbounded, i.e.
96 /// there is a directed cycle having negative total cost and
97 /// infinite upper bound.
101 /// \brief Constants for selecting the type of the supply constraints.
103 /// Enum type containing constants for selecting the supply type,
104 /// i.e. the direction of the inequalities in the supply/demand
105 /// constraints of the \ref min_cost_flow "minimum cost flow problem".
107 /// The default supply type is \c GEQ, the \c LEQ type can be
108 /// selected using \ref supplyType().
109 /// The equality form is a special case of both supply types.
111 /// This option means that there are <em>"greater or equal"</em>
112 /// supply/demand constraints in the definition of the problem.
114 /// This option means that there are <em>"less or equal"</em>
115 /// supply/demand constraints in the definition of the problem.
119 /// \brief Constants for selecting the pivot rule.
121 /// Enum type containing constants for selecting the pivot rule for
122 /// the \ref run() function.
124 /// \ref NetworkSimplex provides five different pivot rule
125 /// implementations that significantly affect the running time
126 /// of the algorithm.
127 /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
128 /// proved to be the most efficient and the most robust on various
130 /// However, another pivot rule can be selected using the \ref run()
131 /// function with the proper parameter.
134 /// The \e First \e Eligible pivot rule.
135 /// The next eligible arc is selected in a wraparound fashion
136 /// in every iteration.
139 /// The \e Best \e Eligible pivot rule.
140 /// The best eligible arc is selected in every iteration.
143 /// The \e Block \e Search pivot rule.
144 /// A specified number of arcs are examined in every iteration
145 /// in a wraparound fashion and the best eligible arc is selected
149 /// The \e Candidate \e List pivot rule.
150 /// In a major iteration a candidate list is built from eligible arcs
151 /// in a wraparound fashion and in the following minor iterations
152 /// the best eligible arc is selected from this list.
155 /// The \e Altering \e Candidate \e List pivot rule.
156 /// It is a modified version of the Candidate List method.
157 /// It keeps only the several best eligible arcs from the former
158 /// candidate list and extends this list in every iteration.
164 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
166 typedef std::vector<int> IntVector;
167 typedef std::vector<Value> ValueVector;
168 typedef std::vector<Cost> CostVector;
169 typedef std::vector<char> BoolVector;
170 // Note: vector<char> is used instead of vector<bool> for efficiency reasons
172 // State constants for arcs
181 // Data related to the underlying digraph
188 // Parameters of the problem
193 // Data structures for storing the digraph
208 // Data for storing the spanning tree structure
212 IntVector _rev_thread;
214 IntVector _last_succ;
215 IntVector _dirty_revs;
220 // Temporary data used in the current pivot iteration
221 int in_arc, join, u_in, v_in, u_out, v_out;
222 int first, second, right, last;
223 int stem, par_stem, new_stem;
230 /// \brief Constant for infinite upper bounds (capacities).
232 /// Constant for infinite upper bounds (capacities).
233 /// It is \c std::numeric_limits<Value>::infinity() if available,
234 /// \c std::numeric_limits<Value>::max() otherwise.
239 // Implementation of the First Eligible pivot rule
240 class FirstEligiblePivotRule
244 // References to the NetworkSimplex class
245 const IntVector &_source;
246 const IntVector &_target;
247 const CostVector &_cost;
248 const BoolVector &_state;
249 const CostVector &_pi;
259 FirstEligiblePivotRule(NetworkSimplex &ns) :
260 _source(ns._source), _target(ns._target),
261 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
262 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
266 // Find next entering arc
267 bool findEnteringArc() {
269 for (int e = _next_arc; e != _search_arc_num; ++e) {
270 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
277 for (int e = 0; e != _next_arc; ++e) {
278 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
288 }; //class FirstEligiblePivotRule
291 // Implementation of the Best Eligible pivot rule
292 class BestEligiblePivotRule
296 // References to the NetworkSimplex class
297 const IntVector &_source;
298 const IntVector &_target;
299 const CostVector &_cost;
300 const BoolVector &_state;
301 const CostVector &_pi;
308 BestEligiblePivotRule(NetworkSimplex &ns) :
309 _source(ns._source), _target(ns._target),
310 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
311 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
314 // Find next entering arc
315 bool findEnteringArc() {
317 for (int e = 0; e != _search_arc_num; ++e) {
318 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
327 }; //class BestEligiblePivotRule
330 // Implementation of the Block Search pivot rule
331 class BlockSearchPivotRule
335 // References to the NetworkSimplex class
336 const IntVector &_source;
337 const IntVector &_target;
338 const CostVector &_cost;
339 const BoolVector &_state;
340 const CostVector &_pi;
351 BlockSearchPivotRule(NetworkSimplex &ns) :
352 _source(ns._source), _target(ns._target),
353 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
354 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
357 // The main parameters of the pivot rule
358 const double BLOCK_SIZE_FACTOR = 1.0;
359 const int MIN_BLOCK_SIZE = 10;
361 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
362 std::sqrt(double(_search_arc_num))),
366 // Find next entering arc
367 bool findEnteringArc() {
369 int cnt = _block_size;
371 for (e = _next_arc; e != _search_arc_num; ++e) {
372 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
378 if (min < 0) goto search_end;
382 for (e = 0; e != _next_arc; ++e) {
383 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
389 if (min < 0) goto search_end;
393 if (min >= 0) return false;
400 }; //class BlockSearchPivotRule
403 // Implementation of the Candidate List pivot rule
404 class CandidateListPivotRule
408 // References to the NetworkSimplex class
409 const IntVector &_source;
410 const IntVector &_target;
411 const CostVector &_cost;
412 const BoolVector &_state;
413 const CostVector &_pi;
418 IntVector _candidates;
419 int _list_length, _minor_limit;
420 int _curr_length, _minor_count;
426 CandidateListPivotRule(NetworkSimplex &ns) :
427 _source(ns._source), _target(ns._target),
428 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
429 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
432 // The main parameters of the pivot rule
433 const double LIST_LENGTH_FACTOR = 0.25;
434 const int MIN_LIST_LENGTH = 10;
435 const double MINOR_LIMIT_FACTOR = 0.1;
436 const int MIN_MINOR_LIMIT = 3;
438 _list_length = std::max( int(LIST_LENGTH_FACTOR *
439 std::sqrt(double(_search_arc_num))),
441 _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
443 _curr_length = _minor_count = 0;
444 _candidates.resize(_list_length);
447 /// Find next entering arc
448 bool findEnteringArc() {
451 if (_curr_length > 0 && _minor_count < _minor_limit) {
452 // Minor iteration: select the best eligible arc from the
453 // current candidate list
456 for (int i = 0; i < _curr_length; ++i) {
458 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
464 _candidates[i--] = _candidates[--_curr_length];
467 if (min < 0) return true;
470 // Major iteration: build a new candidate list
473 for (e = _next_arc; e != _search_arc_num; ++e) {
474 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
476 _candidates[_curr_length++] = e;
481 if (_curr_length == _list_length) goto search_end;
484 for (e = 0; e != _next_arc; ++e) {
485 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
487 _candidates[_curr_length++] = e;
492 if (_curr_length == _list_length) goto search_end;
495 if (_curr_length == 0) return false;
503 }; //class CandidateListPivotRule
506 // Implementation of the Altering Candidate List pivot rule
507 class AlteringListPivotRule
511 // References to the NetworkSimplex class
512 const IntVector &_source;
513 const IntVector &_target;
514 const CostVector &_cost;
515 const BoolVector &_state;
516 const CostVector &_pi;
521 int _block_size, _head_length, _curr_length;
523 IntVector _candidates;
524 CostVector _cand_cost;
526 // Functor class to compare arcs during sort of the candidate list
530 const CostVector &_map;
532 SortFunc(const CostVector &map) : _map(map) {}
533 bool operator()(int left, int right) {
534 return _map[left] > _map[right];
543 AlteringListPivotRule(NetworkSimplex &ns) :
544 _source(ns._source), _target(ns._target),
545 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
546 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
547 _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
549 // The main parameters of the pivot rule
550 const double BLOCK_SIZE_FACTOR = 1.0;
551 const int MIN_BLOCK_SIZE = 10;
552 const double HEAD_LENGTH_FACTOR = 0.1;
553 const int MIN_HEAD_LENGTH = 3;
555 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
556 std::sqrt(double(_search_arc_num))),
558 _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
560 _candidates.resize(_head_length + _block_size);
564 // Find next entering arc
565 bool findEnteringArc() {
566 // Check the current candidate list
568 for (int i = 0; i != _curr_length; ++i) {
570 _cand_cost[e] = _state[e] *
571 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
572 if (_cand_cost[e] >= 0) {
573 _candidates[i--] = _candidates[--_curr_length];
578 int cnt = _block_size;
579 int limit = _head_length;
581 for (e = _next_arc; e != _search_arc_num; ++e) {
582 _cand_cost[e] = _state[e] *
583 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
584 if (_cand_cost[e] < 0) {
585 _candidates[_curr_length++] = e;
588 if (_curr_length > limit) goto search_end;
593 for (e = 0; e != _next_arc; ++e) {
594 _cand_cost[e] = _state[e] *
595 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
596 if (_cand_cost[e] < 0) {
597 _candidates[_curr_length++] = e;
600 if (_curr_length > limit) goto search_end;
605 if (_curr_length == 0) return false;
609 // Make heap of the candidate list (approximating a partial sort)
610 make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
613 // Pop the first element of the heap
614 _in_arc = _candidates[0];
616 pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
618 _curr_length = std::min(_head_length, _curr_length - 1);
622 }; //class AlteringListPivotRule
626 /// \brief Constructor.
628 /// The constructor of the class.
630 /// \param graph The digraph the algorithm runs on.
631 /// \param arc_mixing Indicate if the arcs have to be stored in a
632 /// mixed order in the internal data structure.
633 /// In special cases, it could lead to better overall performance,
634 /// but it is usually slower. Therefore it is disabled by default.
635 NetworkSimplex(const GR& graph, bool arc_mixing = false) :
636 _graph(graph), _node_id(graph), _arc_id(graph),
637 MAX(std::numeric_limits<Value>::max()),
638 INF(std::numeric_limits<Value>::has_infinity ?
639 std::numeric_limits<Value>::infinity() : MAX)
641 // Check the number types
642 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
643 "The flow type of NetworkSimplex must be signed");
644 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
645 "The cost type of NetworkSimplex must be signed");
648 _node_num = countNodes(_graph);
649 _arc_num = countArcs(_graph);
650 int all_node_num = _node_num + 1;
651 int max_arc_num = _arc_num + 2 * _node_num;
653 _source.resize(max_arc_num);
654 _target.resize(max_arc_num);
656 _lower.resize(_arc_num);
657 _upper.resize(_arc_num);
658 _cap.resize(max_arc_num);
659 _cost.resize(max_arc_num);
660 _supply.resize(all_node_num);
661 _flow.resize(max_arc_num);
662 _pi.resize(all_node_num);
664 _parent.resize(all_node_num);
665 _pred.resize(all_node_num);
666 _forward.resize(all_node_num);
667 _thread.resize(all_node_num);
668 _rev_thread.resize(all_node_num);
669 _succ_num.resize(all_node_num);
670 _last_succ.resize(all_node_num);
671 _state.resize(max_arc_num);
675 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
679 // Store the arcs in a mixed order
680 int k = std::max(int(std::sqrt(double(_arc_num))), 10);
682 for (ArcIt a(_graph); a != INVALID; ++a) {
684 _source[i] = _node_id[_graph.source(a)];
685 _target[i] = _node_id[_graph.target(a)];
686 if ((i += k) >= _arc_num) i = ++j;
689 // Store the arcs in the original order
691 for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
693 _source[i] = _node_id[_graph.source(a)];
694 _target[i] = _node_id[_graph.target(a)];
703 /// The parameters of the algorithm can be specified using these
708 /// \brief Set the lower bounds on the arcs.
710 /// This function sets the lower bounds on the arcs.
711 /// If it is not used before calling \ref run(), the lower bounds
712 /// will be set to zero on all arcs.
714 /// \param map An arc map storing the lower bounds.
715 /// Its \c Value type must be convertible to the \c Value type
716 /// of the algorithm.
718 /// \return <tt>(*this)</tt>
719 template <typename LowerMap>
720 NetworkSimplex& lowerMap(const LowerMap& map) {
722 for (ArcIt a(_graph); a != INVALID; ++a) {
723 _lower[_arc_id[a]] = map[a];
728 /// \brief Set the upper bounds (capacities) on the arcs.
730 /// This function sets the upper bounds (capacities) on the arcs.
731 /// If it is not used before calling \ref run(), the upper bounds
732 /// will be set to \ref INF on all arcs (i.e. the flow value will be
733 /// unbounded from above).
735 /// \param map An arc map storing the upper bounds.
736 /// Its \c Value type must be convertible to the \c Value type
737 /// of the algorithm.
739 /// \return <tt>(*this)</tt>
740 template<typename UpperMap>
741 NetworkSimplex& upperMap(const UpperMap& map) {
742 for (ArcIt a(_graph); a != INVALID; ++a) {
743 _upper[_arc_id[a]] = map[a];
748 /// \brief Set the costs of the arcs.
750 /// This function sets the costs of the arcs.
751 /// If it is not used before calling \ref run(), the costs
752 /// will be set to \c 1 on all arcs.
754 /// \param map An arc map storing the costs.
755 /// Its \c Value type must be convertible to the \c Cost type
756 /// of the algorithm.
758 /// \return <tt>(*this)</tt>
759 template<typename CostMap>
760 NetworkSimplex& costMap(const CostMap& map) {
761 for (ArcIt a(_graph); a != INVALID; ++a) {
762 _cost[_arc_id[a]] = map[a];
767 /// \brief Set the supply values of the nodes.
769 /// This function sets the supply values of the nodes.
770 /// If neither this function nor \ref stSupply() is used before
771 /// calling \ref run(), the supply of each node will be set to zero.
773 /// \param map A node map storing the supply values.
774 /// Its \c Value type must be convertible to the \c Value type
775 /// of the algorithm.
777 /// \return <tt>(*this)</tt>
778 template<typename SupplyMap>
779 NetworkSimplex& supplyMap(const SupplyMap& map) {
780 for (NodeIt n(_graph); n != INVALID; ++n) {
781 _supply[_node_id[n]] = map[n];
786 /// \brief Set single source and target nodes and a supply value.
788 /// This function sets a single source node and a single target node
789 /// and the required flow value.
790 /// If neither this function nor \ref supplyMap() is used before
791 /// calling \ref run(), the supply of each node will be set to zero.
793 /// Using this function has the same effect as using \ref supplyMap()
794 /// with such a map in which \c k is assigned to \c s, \c -k is
795 /// assigned to \c t and all other nodes have zero supply value.
797 /// \param s The source node.
798 /// \param t The target node.
799 /// \param k The required amount of flow from node \c s to node \c t
800 /// (i.e. the supply of \c s and the demand of \c t).
802 /// \return <tt>(*this)</tt>
803 NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
804 for (int i = 0; i != _node_num; ++i) {
807 _supply[_node_id[s]] = k;
808 _supply[_node_id[t]] = -k;
812 /// \brief Set the type of the supply constraints.
814 /// This function sets the type of the supply/demand constraints.
815 /// If it is not used before calling \ref run(), the \ref GEQ supply
816 /// type will be used.
818 /// For more information, see \ref SupplyType.
820 /// \return <tt>(*this)</tt>
821 NetworkSimplex& supplyType(SupplyType supply_type) {
822 _stype = supply_type;
828 /// \name Execution Control
829 /// The algorithm can be executed using \ref run().
833 /// \brief Run the algorithm.
835 /// This function runs the algorithm.
836 /// The paramters can be specified using functions \ref lowerMap(),
837 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
838 /// \ref supplyType().
841 /// NetworkSimplex<ListDigraph> ns(graph);
842 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
843 /// .supplyMap(sup).run();
846 /// This function can be called more than once. All the parameters
847 /// that have been given are kept for the next call, unless
848 /// \ref reset() is called, thus only the modified parameters
849 /// have to be set again. See \ref reset() for examples.
850 /// However, the underlying digraph must not be modified after this
851 /// class have been constructed, since it copies and extends the graph.
853 /// \param pivot_rule The pivot rule that will be used during the
854 /// algorithm. For more information, see \ref PivotRule.
856 /// \return \c INFEASIBLE if no feasible flow exists,
857 /// \n \c OPTIMAL if the problem has optimal solution
858 /// (i.e. it is feasible and bounded), and the algorithm has found
859 /// optimal flow and node potentials (primal and dual solutions),
860 /// \n \c UNBOUNDED if the objective function of the problem is
861 /// unbounded, i.e. there is a directed cycle having negative total
862 /// cost and infinite upper bound.
864 /// \see ProblemType, PivotRule
865 ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
866 if (!init()) return INFEASIBLE;
867 return start(pivot_rule);
870 /// \brief Reset all the parameters that have been given before.
872 /// This function resets all the paramaters that have been given
873 /// before using functions \ref lowerMap(), \ref upperMap(),
874 /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
876 /// It is useful for multiple run() calls. If this function is not
877 /// used, all the parameters given before are kept for the next
879 /// However, the underlying digraph must not be modified after this
880 /// class have been constructed, since it copies and extends the graph.
884 /// NetworkSimplex<ListDigraph> ns(graph);
887 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
888 /// .supplyMap(sup).run();
890 /// // Run again with modified cost map (reset() is not called,
891 /// // so only the cost map have to be set again)
893 /// ns.costMap(cost).run();
895 /// // Run again from scratch using reset()
896 /// // (the lower bounds will be set to zero on all arcs)
898 /// ns.upperMap(capacity).costMap(cost)
899 /// .supplyMap(sup).run();
902 /// \return <tt>(*this)</tt>
903 NetworkSimplex& reset() {
904 for (int i = 0; i != _node_num; ++i) {
907 for (int i = 0; i != _arc_num; ++i) {
919 /// \name Query Functions
920 /// The results of the algorithm can be obtained using these
922 /// The \ref run() function must be called before using them.
926 /// \brief Return the total cost of the found flow.
928 /// This function returns the total cost of the found flow.
929 /// Its complexity is O(e).
931 /// \note The return type of the function can be specified as a
932 /// template parameter. For example,
934 /// ns.totalCost<double>();
936 /// It is useful if the total cost cannot be stored in the \c Cost
937 /// type of the algorithm, which is the default return type of the
940 /// \pre \ref run() must be called before using this function.
941 template <typename Number>
942 Number totalCost() const {
944 for (ArcIt a(_graph); a != INVALID; ++a) {
946 c += Number(_flow[i]) * Number(_cost[i]);
952 Cost totalCost() const {
953 return totalCost<Cost>();
957 /// \brief Return the flow on the given arc.
959 /// This function returns the flow on the given arc.
961 /// \pre \ref run() must be called before using this function.
962 Value flow(const Arc& a) const {
963 return _flow[_arc_id[a]];
966 /// \brief Return the flow map (the primal solution).
968 /// This function copies the flow value on each arc into the given
969 /// map. The \c Value type of the algorithm must be convertible to
970 /// the \c Value type of the map.
972 /// \pre \ref run() must be called before using this function.
973 template <typename FlowMap>
974 void flowMap(FlowMap &map) const {
975 for (ArcIt a(_graph); a != INVALID; ++a) {
976 map.set(a, _flow[_arc_id[a]]);
980 /// \brief Return the potential (dual value) of the given node.
982 /// This function returns the potential (dual value) of the
985 /// \pre \ref run() must be called before using this function.
986 Cost potential(const Node& n) const {
987 return _pi[_node_id[n]];
990 /// \brief Return the potential map (the dual solution).
992 /// This function copies the potential (dual value) of each node
993 /// into the given map.
994 /// The \c Cost type of the algorithm must be convertible to the
995 /// \c Value type of the map.
997 /// \pre \ref run() must be called before using this function.
998 template <typename PotentialMap>
999 void potentialMap(PotentialMap &map) const {
1000 for (NodeIt n(_graph); n != INVALID; ++n) {
1001 map.set(n, _pi[_node_id[n]]);
1009 // Initialize internal data structures
1011 if (_node_num == 0) return false;
1013 // Check the sum of supply values
1015 for (int i = 0; i != _node_num; ++i) {
1016 _sum_supply += _supply[i];
1018 if ( !((_stype == GEQ && _sum_supply <= 0) ||
1019 (_stype == LEQ && _sum_supply >= 0)) ) return false;
1021 // Remove non-zero lower bounds
1023 for (int i = 0; i != _arc_num; ++i) {
1024 Value c = _lower[i];
1026 _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1028 _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1030 _supply[_source[i]] -= c;
1031 _supply[_target[i]] += c;
1034 for (int i = 0; i != _arc_num; ++i) {
1035 _cap[i] = _upper[i];
1039 // Initialize artifical cost
1041 if (std::numeric_limits<Cost>::is_exact) {
1042 ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1044 ART_COST = std::numeric_limits<Cost>::min();
1045 for (int i = 0; i != _arc_num; ++i) {
1046 if (_cost[i] > ART_COST) ART_COST = _cost[i];
1048 ART_COST = (ART_COST + 1) * _node_num;
1051 // Initialize arc maps
1052 for (int i = 0; i != _arc_num; ++i) {
1054 _state[i] = STATE_LOWER;
1057 // Set data for the artificial root node
1059 _parent[_root] = -1;
1062 _rev_thread[0] = _root;
1063 _succ_num[_root] = _node_num + 1;
1064 _last_succ[_root] = _root - 1;
1065 _supply[_root] = -_sum_supply;
1068 // Add artificial arcs and initialize the spanning tree data structure
1069 if (_sum_supply == 0) {
1070 // EQ supply constraints
1071 _search_arc_num = _arc_num;
1072 _all_arc_num = _arc_num + _node_num;
1073 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1077 _rev_thread[u + 1] = u;
1081 _state[e] = STATE_TREE;
1082 if (_supply[u] >= 0) {
1087 _flow[e] = _supply[u];
1090 _forward[u] = false;
1094 _flow[e] = -_supply[u];
1095 _cost[e] = ART_COST;
1099 else if (_sum_supply > 0) {
1100 // LEQ supply constraints
1101 _search_arc_num = _arc_num + _node_num;
1102 int f = _arc_num + _node_num;
1103 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1106 _rev_thread[u + 1] = u;
1109 if (_supply[u] >= 0) {
1116 _flow[e] = _supply[u];
1118 _state[e] = STATE_TREE;
1120 _forward[u] = false;
1126 _flow[f] = -_supply[u];
1127 _cost[f] = ART_COST;
1128 _state[f] = STATE_TREE;
1134 _state[e] = STATE_LOWER;
1141 // GEQ 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 _forward[u] = false;
1157 _flow[e] = -_supply[u];
1159 _state[e] = STATE_TREE;
1167 _flow[f] = _supply[u];
1168 _state[f] = STATE_TREE;
1169 _cost[f] = ART_COST;
1175 _state[e] = STATE_LOWER;
1185 // Find the join node
1186 void findJoinNode() {
1187 int u = _source[in_arc];
1188 int v = _target[in_arc];
1190 if (_succ_num[u] < _succ_num[v]) {
1199 // Find the leaving arc of the cycle and returns true if the
1200 // leaving arc is not the same as the entering arc
1201 bool findLeavingArc() {
1202 // Initialize first and second nodes according to the direction
1204 if (_state[in_arc] == STATE_LOWER) {
1205 first = _source[in_arc];
1206 second = _target[in_arc];
1208 first = _target[in_arc];
1209 second = _source[in_arc];
1211 delta = _cap[in_arc];
1216 // Search the cycle along the path form the first node to the root
1217 for (int u = first; u != join; u = _parent[u]) {
1220 _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
1227 // Search the cycle along the path form the second node to the root
1228 for (int u = second; u != join; u = _parent[u]) {
1231 (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
1249 // Change _flow and _state vectors
1250 void changeFlow(bool change) {
1251 // Augment along the cycle
1253 Value val = _state[in_arc] * delta;
1254 _flow[in_arc] += val;
1255 for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1256 _flow[_pred[u]] += _forward[u] ? -val : val;
1258 for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1259 _flow[_pred[u]] += _forward[u] ? val : -val;
1262 // Update the state of the entering and leaving arcs
1264 _state[in_arc] = STATE_TREE;
1265 _state[_pred[u_out]] =
1266 (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1268 _state[in_arc] = -_state[in_arc];
1272 // Update the tree structure
1273 void updateTreeStructure() {
1275 int old_rev_thread = _rev_thread[u_out];
1276 int old_succ_num = _succ_num[u_out];
1277 int old_last_succ = _last_succ[u_out];
1278 v_out = _parent[u_out];
1280 u = _last_succ[u_in]; // the last successor of u_in
1281 right = _thread[u]; // the node after it
1283 // Handle the case when old_rev_thread equals to v_in
1284 // (it also means that join and v_out coincide)
1285 if (old_rev_thread == v_in) {
1286 last = _thread[_last_succ[u_out]];
1288 last = _thread[v_in];
1291 // Update _thread and _parent along the stem nodes (i.e. the nodes
1292 // between u_in and u_out, whose parent have to be changed)
1293 _thread[v_in] = stem = u_in;
1294 _dirty_revs.clear();
1295 _dirty_revs.push_back(v_in);
1297 while (stem != u_out) {
1298 // Insert the next stem node into the thread list
1299 new_stem = _parent[stem];
1300 _thread[u] = new_stem;
1301 _dirty_revs.push_back(u);
1303 // Remove the subtree of stem from the thread list
1304 w = _rev_thread[stem];
1306 _rev_thread[right] = w;
1308 // Change the parent node and shift stem nodes
1309 _parent[stem] = par_stem;
1313 // Update u and right
1314 u = _last_succ[stem] == _last_succ[par_stem] ?
1315 _rev_thread[par_stem] : _last_succ[stem];
1318 _parent[u_out] = par_stem;
1320 _rev_thread[last] = u;
1321 _last_succ[u_out] = u;
1323 // Remove the subtree of u_out from the thread list except for
1324 // the case when old_rev_thread equals to v_in
1325 // (it also means that join and v_out coincide)
1326 if (old_rev_thread != v_in) {
1327 _thread[old_rev_thread] = right;
1328 _rev_thread[right] = old_rev_thread;
1331 // Update _rev_thread using the new _thread values
1332 for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1334 _rev_thread[_thread[u]] = u;
1337 // Update _pred, _forward, _last_succ and _succ_num for the
1338 // stem nodes from u_out to u_in
1339 int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1343 _pred[u] = _pred[w];
1344 _forward[u] = !_forward[w];
1345 tmp_sc += _succ_num[u] - _succ_num[w];
1346 _succ_num[u] = tmp_sc;
1347 _last_succ[w] = tmp_ls;
1350 _pred[u_in] = in_arc;
1351 _forward[u_in] = (u_in == _source[in_arc]);
1352 _succ_num[u_in] = old_succ_num;
1354 // Set limits for updating _last_succ form v_in and v_out
1356 int up_limit_in = -1;
1357 int up_limit_out = -1;
1358 if (_last_succ[join] == v_in) {
1359 up_limit_out = join;
1364 // Update _last_succ from v_in towards the root
1365 for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1367 _last_succ[u] = _last_succ[u_out];
1369 // Update _last_succ from v_out towards the root
1370 if (join != old_rev_thread && v_in != old_rev_thread) {
1371 for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1373 _last_succ[u] = old_rev_thread;
1376 for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1378 _last_succ[u] = _last_succ[u_out];
1382 // Update _succ_num from v_in to join
1383 for (u = v_in; u != join; u = _parent[u]) {
1384 _succ_num[u] += old_succ_num;
1386 // Update _succ_num from v_out to join
1387 for (u = v_out; u != join; u = _parent[u]) {
1388 _succ_num[u] -= old_succ_num;
1392 // Update potentials
1393 void updatePotential() {
1394 Cost sigma = _forward[u_in] ?
1395 _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1396 _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1397 // Update potentials in the subtree, which has been moved
1398 int end = _thread[_last_succ[u_in]];
1399 for (int u = u_in; u != end; u = _thread[u]) {
1404 // Heuristic initial pivots
1405 bool initialPivots() {
1406 Value curr, total = 0;
1407 std::vector<Node> supply_nodes, demand_nodes;
1408 for (NodeIt u(_graph); u != INVALID; ++u) {
1409 curr = _supply[_node_id[u]];
1412 supply_nodes.push_back(u);
1414 else if (curr < 0) {
1415 demand_nodes.push_back(u);
1418 if (_sum_supply > 0) total -= _sum_supply;
1419 if (total <= 0) return true;
1421 IntVector arc_vector;
1422 if (_sum_supply >= 0) {
1423 if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1424 // Perform a reverse graph search from the sink to the source
1425 typename GR::template NodeMap<bool> reached(_graph, false);
1426 Node s = supply_nodes[0], t = demand_nodes[0];
1427 std::vector<Node> stack;
1430 while (!stack.empty()) {
1431 Node u, v = stack.back();
1434 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1435 if (reached[u = _graph.source(a)]) continue;
1437 if (_cap[j] >= total) {
1438 arc_vector.push_back(j);
1445 // Find the min. cost incomming arc for each demand node
1446 for (int i = 0; i != int(demand_nodes.size()); ++i) {
1447 Node v = demand_nodes[i];
1448 Cost c, min_cost = std::numeric_limits<Cost>::max();
1449 Arc min_arc = INVALID;
1450 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1451 c = _cost[_arc_id[a]];
1457 if (min_arc != INVALID) {
1458 arc_vector.push_back(_arc_id[min_arc]);
1463 // Find the min. cost outgoing arc for each supply node
1464 for (int i = 0; i != int(supply_nodes.size()); ++i) {
1465 Node u = supply_nodes[i];
1466 Cost c, min_cost = std::numeric_limits<Cost>::max();
1467 Arc min_arc = INVALID;
1468 for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1469 c = _cost[_arc_id[a]];
1475 if (min_arc != INVALID) {
1476 arc_vector.push_back(_arc_id[min_arc]);
1481 // Perform heuristic initial pivots
1482 for (int i = 0; i != int(arc_vector.size()); ++i) {
1483 in_arc = arc_vector[i];
1484 if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1485 _pi[_target[in_arc]]) >= 0) continue;
1487 bool change = findLeavingArc();
1488 if (delta >= MAX) return false;
1491 updateTreeStructure();
1498 // Execute the algorithm
1499 ProblemType start(PivotRule pivot_rule) {
1500 // Select the pivot rule implementation
1501 switch (pivot_rule) {
1502 case FIRST_ELIGIBLE:
1503 return start<FirstEligiblePivotRule>();
1505 return start<BestEligiblePivotRule>();
1507 return start<BlockSearchPivotRule>();
1508 case CANDIDATE_LIST:
1509 return start<CandidateListPivotRule>();
1511 return start<AlteringListPivotRule>();
1513 return INFEASIBLE; // avoid warning
1516 template <typename PivotRuleImpl>
1517 ProblemType start() {
1518 PivotRuleImpl pivot(*this);
1520 // Perform heuristic initial pivots
1521 if (!initialPivots()) return UNBOUNDED;
1523 // Execute the Network Simplex algorithm
1524 while (pivot.findEnteringArc()) {
1526 bool change = findLeavingArc();
1527 if (delta >= MAX) return UNBOUNDED;
1530 updateTreeStructure();
1535 // Check feasibility
1536 for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1537 if (_flow[e] != 0) return INFEASIBLE;
1540 // Transform the solution and the supply map to the original form
1542 for (int i = 0; i != _arc_num; ++i) {
1543 Value c = _lower[i];
1546 _supply[_source[i]] += c;
1547 _supply[_target[i]] -= c;
1552 // Shift potentials to meet the requirements of the GEQ/LEQ type
1553 // optimality conditions
1554 if (_sum_supply == 0) {
1555 if (_stype == GEQ) {
1556 Cost max_pot = std::numeric_limits<Cost>::min();
1557 for (int i = 0; i != _node_num; ++i) {
1558 if (_pi[i] > max_pot) max_pot = _pi[i];
1561 for (int i = 0; i != _node_num; ++i)
1565 Cost min_pot = std::numeric_limits<Cost>::max();
1566 for (int i = 0; i != _node_num; ++i) {
1567 if (_pi[i] < min_pot) min_pot = _pi[i];
1570 for (int i = 0; i != _node_num; ++i)
1579 }; //class NetworkSimplex
1585 #endif //LEMON_NETWORK_SIMPLEX_H