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 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<signed char> CharVector;
170 // Note: vector<signed char> is used instead of vector<ArcState> and
171 // vector<ArcDirection> for efficiency reasons
173 // State constants for arcs
180 // Direction constants for tree arcs
188 // Data related to the underlying digraph
195 // Parameters of the problem
200 // Data structures for storing the digraph
216 // Data for storing the spanning tree structure
220 IntVector _rev_thread;
222 IntVector _last_succ;
223 CharVector _pred_dir;
225 IntVector _dirty_revs;
228 // Temporary data used in the current pivot iteration
229 int in_arc, join, u_in, v_in, u_out, v_out;
236 /// \brief Constant for infinite upper bounds (capacities).
238 /// Constant for infinite upper bounds (capacities).
239 /// It is \c std::numeric_limits<Value>::infinity() if available,
240 /// \c std::numeric_limits<Value>::max() otherwise.
245 // Implementation of the First Eligible pivot rule
246 class FirstEligiblePivotRule
250 // References to the NetworkSimplex class
251 const IntVector &_source;
252 const IntVector &_target;
253 const CostVector &_cost;
254 const CharVector &_state;
255 const CostVector &_pi;
265 FirstEligiblePivotRule(NetworkSimplex &ns) :
266 _source(ns._source), _target(ns._target),
267 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
268 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
272 // Find next entering arc
273 bool findEnteringArc() {
275 for (int e = _next_arc; e != _search_arc_num; ++e) {
276 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
283 for (int e = 0; e != _next_arc; ++e) {
284 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
294 }; //class FirstEligiblePivotRule
297 // Implementation of the Best Eligible pivot rule
298 class BestEligiblePivotRule
302 // References to the NetworkSimplex class
303 const IntVector &_source;
304 const IntVector &_target;
305 const CostVector &_cost;
306 const CharVector &_state;
307 const CostVector &_pi;
314 BestEligiblePivotRule(NetworkSimplex &ns) :
315 _source(ns._source), _target(ns._target),
316 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
317 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
320 // Find next entering arc
321 bool findEnteringArc() {
323 for (int e = 0; e != _search_arc_num; ++e) {
324 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
333 }; //class BestEligiblePivotRule
336 // Implementation of the Block Search pivot rule
337 class BlockSearchPivotRule
341 // References to the NetworkSimplex class
342 const IntVector &_source;
343 const IntVector &_target;
344 const CostVector &_cost;
345 const CharVector &_state;
346 const CostVector &_pi;
357 BlockSearchPivotRule(NetworkSimplex &ns) :
358 _source(ns._source), _target(ns._target),
359 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
360 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
363 // The main parameters of the pivot rule
364 const double BLOCK_SIZE_FACTOR = 1.0;
365 const int MIN_BLOCK_SIZE = 10;
367 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
368 std::sqrt(double(_search_arc_num))),
372 // Find next entering arc
373 bool findEnteringArc() {
375 int cnt = _block_size;
377 for (e = _next_arc; e != _search_arc_num; ++e) {
378 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
384 if (min < 0) goto search_end;
388 for (e = 0; e != _next_arc; ++e) {
389 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
395 if (min < 0) goto search_end;
399 if (min >= 0) return false;
406 }; //class BlockSearchPivotRule
409 // Implementation of the Candidate List pivot rule
410 class CandidateListPivotRule
414 // References to the NetworkSimplex class
415 const IntVector &_source;
416 const IntVector &_target;
417 const CostVector &_cost;
418 const CharVector &_state;
419 const CostVector &_pi;
424 IntVector _candidates;
425 int _list_length, _minor_limit;
426 int _curr_length, _minor_count;
432 CandidateListPivotRule(NetworkSimplex &ns) :
433 _source(ns._source), _target(ns._target),
434 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
435 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
438 // The main parameters of the pivot rule
439 const double LIST_LENGTH_FACTOR = 0.25;
440 const int MIN_LIST_LENGTH = 10;
441 const double MINOR_LIMIT_FACTOR = 0.1;
442 const int MIN_MINOR_LIMIT = 3;
444 _list_length = std::max( int(LIST_LENGTH_FACTOR *
445 std::sqrt(double(_search_arc_num))),
447 _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
449 _curr_length = _minor_count = 0;
450 _candidates.resize(_list_length);
453 /// Find next entering arc
454 bool findEnteringArc() {
457 if (_curr_length > 0 && _minor_count < _minor_limit) {
458 // Minor iteration: select the best eligible arc from the
459 // current candidate list
462 for (int i = 0; i < _curr_length; ++i) {
464 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
470 _candidates[i--] = _candidates[--_curr_length];
473 if (min < 0) return true;
476 // Major iteration: build a new candidate list
479 for (e = _next_arc; e != _search_arc_num; ++e) {
480 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
482 _candidates[_curr_length++] = e;
487 if (_curr_length == _list_length) goto search_end;
490 for (e = 0; e != _next_arc; ++e) {
491 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
493 _candidates[_curr_length++] = e;
498 if (_curr_length == _list_length) goto search_end;
501 if (_curr_length == 0) return false;
509 }; //class CandidateListPivotRule
512 // Implementation of the Altering Candidate List pivot rule
513 class AlteringListPivotRule
517 // References to the NetworkSimplex class
518 const IntVector &_source;
519 const IntVector &_target;
520 const CostVector &_cost;
521 const CharVector &_state;
522 const CostVector &_pi;
527 int _block_size, _head_length, _curr_length;
529 IntVector _candidates;
530 CostVector _cand_cost;
532 // Functor class to compare arcs during sort of the candidate list
536 const CostVector &_map;
538 SortFunc(const CostVector &map) : _map(map) {}
539 bool operator()(int left, int right) {
540 return _map[left] > _map[right];
549 AlteringListPivotRule(NetworkSimplex &ns) :
550 _source(ns._source), _target(ns._target),
551 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
552 _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
553 _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
555 // The main parameters of the pivot rule
556 const double BLOCK_SIZE_FACTOR = 1.0;
557 const int MIN_BLOCK_SIZE = 10;
558 const double HEAD_LENGTH_FACTOR = 0.1;
559 const int MIN_HEAD_LENGTH = 3;
561 _block_size = std::max( int(BLOCK_SIZE_FACTOR *
562 std::sqrt(double(_search_arc_num))),
564 _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
566 _candidates.resize(_head_length + _block_size);
570 // Find next entering arc
571 bool findEnteringArc() {
572 // Check the current candidate list
575 for (int i = 0; i != _curr_length; ++i) {
577 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
581 _candidates[i--] = _candidates[--_curr_length];
586 int cnt = _block_size;
587 int limit = _head_length;
589 for (e = _next_arc; e != _search_arc_num; ++e) {
590 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
593 _candidates[_curr_length++] = e;
596 if (_curr_length > limit) goto search_end;
601 for (e = 0; e != _next_arc; ++e) {
602 _cand_cost[e] = _state[e] *
603 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
604 if (_cand_cost[e] < 0) {
605 _candidates[_curr_length++] = e;
608 if (_curr_length > limit) goto search_end;
613 if (_curr_length == 0) return false;
617 // Make heap of the candidate list (approximating a partial sort)
618 make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
621 // Pop the first element of the heap
622 _in_arc = _candidates[0];
624 pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
626 _curr_length = std::min(_head_length, _curr_length - 1);
630 }; //class AlteringListPivotRule
634 /// \brief Constructor.
636 /// The constructor of the class.
638 /// \param graph The digraph the algorithm runs on.
639 /// \param arc_mixing Indicate if the arcs have to be stored in a
640 /// mixed order in the internal data structure.
641 /// In special cases, it could lead to better overall performance,
642 /// but it is usually slower. Therefore it is disabled by default.
643 NetworkSimplex(const GR& graph, bool arc_mixing = false) :
644 _graph(graph), _node_id(graph), _arc_id(graph),
645 _arc_mixing(arc_mixing),
646 MAX(std::numeric_limits<Value>::max()),
647 INF(std::numeric_limits<Value>::has_infinity ?
648 std::numeric_limits<Value>::infinity() : MAX)
650 // Check the number types
651 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
652 "The flow type of NetworkSimplex must be signed");
653 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
654 "The cost type of NetworkSimplex must be signed");
656 // Reset data structures
661 /// The parameters of the algorithm can be specified using these
666 /// \brief Set the lower bounds on the arcs.
668 /// This function sets the lower bounds on the arcs.
669 /// If it is not used before calling \ref run(), the lower bounds
670 /// will be set to zero on all arcs.
672 /// \param map An arc map storing the lower bounds.
673 /// Its \c Value type must be convertible to the \c Value type
674 /// of the algorithm.
676 /// \return <tt>(*this)</tt>
677 template <typename LowerMap>
678 NetworkSimplex& lowerMap(const LowerMap& map) {
680 for (ArcIt a(_graph); a != INVALID; ++a) {
681 _lower[_arc_id[a]] = map[a];
686 /// \brief Set the upper bounds (capacities) on the arcs.
688 /// This function sets the upper bounds (capacities) on the arcs.
689 /// If it is not used before calling \ref run(), the upper bounds
690 /// will be set to \ref INF on all arcs (i.e. the flow value will be
691 /// unbounded from above).
693 /// \param map An arc map storing the upper bounds.
694 /// Its \c Value type must be convertible to the \c Value type
695 /// of the algorithm.
697 /// \return <tt>(*this)</tt>
698 template<typename UpperMap>
699 NetworkSimplex& upperMap(const UpperMap& map) {
700 for (ArcIt a(_graph); a != INVALID; ++a) {
701 _upper[_arc_id[a]] = map[a];
706 /// \brief Set the costs of the arcs.
708 /// This function sets the costs of the arcs.
709 /// If it is not used before calling \ref run(), the costs
710 /// will be set to \c 1 on all arcs.
712 /// \param map An arc map storing the costs.
713 /// Its \c Value type must be convertible to the \c Cost type
714 /// of the algorithm.
716 /// \return <tt>(*this)</tt>
717 template<typename CostMap>
718 NetworkSimplex& costMap(const CostMap& map) {
719 for (ArcIt a(_graph); a != INVALID; ++a) {
720 _cost[_arc_id[a]] = map[a];
725 /// \brief Set the supply values of the nodes.
727 /// This function sets the supply values of the nodes.
728 /// If neither this function nor \ref stSupply() is used before
729 /// calling \ref run(), the supply of each node will be set to zero.
731 /// \param map A node map storing the supply values.
732 /// Its \c Value type must be convertible to the \c Value type
733 /// of the algorithm.
735 /// \return <tt>(*this)</tt>
736 template<typename SupplyMap>
737 NetworkSimplex& supplyMap(const SupplyMap& map) {
738 for (NodeIt n(_graph); n != INVALID; ++n) {
739 _supply[_node_id[n]] = map[n];
744 /// \brief Set single source and target nodes and a supply value.
746 /// This function sets a single source node and a single target node
747 /// and the required flow value.
748 /// If neither this function nor \ref supplyMap() is used before
749 /// calling \ref run(), the supply of each node will be set to zero.
751 /// Using this function has the same effect as using \ref supplyMap()
752 /// with such a map in which \c k is assigned to \c s, \c -k is
753 /// assigned to \c t and all other nodes have zero supply value.
755 /// \param s The source node.
756 /// \param t The target node.
757 /// \param k The required amount of flow from node \c s to node \c t
758 /// (i.e. the supply of \c s and the demand of \c t).
760 /// \return <tt>(*this)</tt>
761 NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
762 for (int i = 0; i != _node_num; ++i) {
765 _supply[_node_id[s]] = k;
766 _supply[_node_id[t]] = -k;
770 /// \brief Set the type of the supply constraints.
772 /// This function sets the type of the supply/demand constraints.
773 /// If it is not used before calling \ref run(), the \ref GEQ supply
774 /// type will be used.
776 /// For more information, see \ref SupplyType.
778 /// \return <tt>(*this)</tt>
779 NetworkSimplex& supplyType(SupplyType supply_type) {
780 _stype = supply_type;
786 /// \name Execution Control
787 /// The algorithm can be executed using \ref run().
791 /// \brief Run the algorithm.
793 /// This function runs the algorithm.
794 /// The paramters can be specified using functions \ref lowerMap(),
795 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
796 /// \ref supplyType().
799 /// NetworkSimplex<ListDigraph> ns(graph);
800 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
801 /// .supplyMap(sup).run();
804 /// This function can be called more than once. All the given parameters
805 /// are kept for the next call, unless \ref resetParams() or \ref reset()
806 /// is used, thus only the modified parameters have to be set again.
807 /// If the underlying digraph was also modified after the construction
808 /// of the class (or the last \ref reset() call), then the \ref reset()
809 /// function must be called.
811 /// \param pivot_rule The pivot rule that will be used during the
812 /// algorithm. For more information, see \ref PivotRule.
814 /// \return \c INFEASIBLE if no feasible flow exists,
815 /// \n \c OPTIMAL if the problem has optimal solution
816 /// (i.e. it is feasible and bounded), and the algorithm has found
817 /// optimal flow and node potentials (primal and dual solutions),
818 /// \n \c UNBOUNDED if the objective function of the problem is
819 /// unbounded, i.e. there is a directed cycle having negative total
820 /// cost and infinite upper bound.
822 /// \see ProblemType, PivotRule
823 /// \see resetParams(), reset()
824 ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
825 if (!init()) return INFEASIBLE;
826 return start(pivot_rule);
829 /// \brief Reset all the parameters that have been given before.
831 /// This function resets all the paramaters that have been given
832 /// before using functions \ref lowerMap(), \ref upperMap(),
833 /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
835 /// It is useful for multiple \ref run() calls. Basically, all the given
836 /// parameters are kept for the next \ref run() call, unless
837 /// \ref resetParams() or \ref reset() is used.
838 /// If the underlying digraph was also modified after the construction
839 /// of the class or the last \ref reset() call, then the \ref reset()
840 /// function must be used, otherwise \ref resetParams() is sufficient.
844 /// NetworkSimplex<ListDigraph> ns(graph);
847 /// ns.lowerMap(lower).upperMap(upper).costMap(cost)
848 /// .supplyMap(sup).run();
850 /// // Run again with modified cost map (resetParams() is not called,
851 /// // so only the cost map have to be set again)
853 /// ns.costMap(cost).run();
855 /// // Run again from scratch using resetParams()
856 /// // (the lower bounds will be set to zero on all arcs)
857 /// ns.resetParams();
858 /// ns.upperMap(capacity).costMap(cost)
859 /// .supplyMap(sup).run();
862 /// \return <tt>(*this)</tt>
864 /// \see reset(), run()
865 NetworkSimplex& resetParams() {
866 for (int i = 0; i != _node_num; ++i) {
869 for (int i = 0; i != _arc_num; ++i) {
879 /// \brief Reset the internal data structures and all the parameters
880 /// that have been given before.
882 /// This function resets the internal data structures and all the
883 /// paramaters that have been given before using functions \ref lowerMap(),
884 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
885 /// \ref supplyType().
887 /// It is useful for multiple \ref run() calls. Basically, all the given
888 /// parameters are kept for the next \ref run() call, unless
889 /// \ref resetParams() or \ref reset() is used.
890 /// If the underlying digraph was also modified after the construction
891 /// of the class or the last \ref reset() call, then the \ref reset()
892 /// function must be used, otherwise \ref resetParams() is sufficient.
894 /// See \ref resetParams() for examples.
896 /// \return <tt>(*this)</tt>
898 /// \see resetParams(), run()
899 NetworkSimplex& reset() {
901 _node_num = countNodes(_graph);
902 _arc_num = countArcs(_graph);
903 int all_node_num = _node_num + 1;
904 int max_arc_num = _arc_num + 2 * _node_num;
906 _source.resize(max_arc_num);
907 _target.resize(max_arc_num);
909 _lower.resize(_arc_num);
910 _upper.resize(_arc_num);
911 _cap.resize(max_arc_num);
912 _cost.resize(max_arc_num);
913 _supply.resize(all_node_num);
914 _flow.resize(max_arc_num);
915 _pi.resize(all_node_num);
917 _parent.resize(all_node_num);
918 _pred.resize(all_node_num);
919 _pred_dir.resize(all_node_num);
920 _thread.resize(all_node_num);
921 _rev_thread.resize(all_node_num);
922 _succ_num.resize(all_node_num);
923 _last_succ.resize(all_node_num);
924 _state.resize(max_arc_num);
928 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
932 // Store the arcs in a mixed order
933 int k = std::max(int(std::sqrt(double(_arc_num))), 10);
935 for (ArcIt a(_graph); a != INVALID; ++a) {
937 _source[i] = _node_id[_graph.source(a)];
938 _target[i] = _node_id[_graph.target(a)];
939 if ((i += k) >= _arc_num) i = ++j;
942 // Store the arcs in the original order
944 for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
946 _source[i] = _node_id[_graph.source(a)];
947 _target[i] = _node_id[_graph.target(a)];
958 /// \name Query Functions
959 /// The results of the algorithm can be obtained using these
961 /// The \ref run() function must be called before using them.
965 /// \brief Return the total cost of the found flow.
967 /// This function returns the total cost of the found flow.
968 /// Its complexity is O(e).
970 /// \note The return type of the function can be specified as a
971 /// template parameter. For example,
973 /// ns.totalCost<double>();
975 /// It is useful if the total cost cannot be stored in the \c Cost
976 /// type of the algorithm, which is the default return type of the
979 /// \pre \ref run() must be called before using this function.
980 template <typename Number>
981 Number totalCost() const {
983 for (ArcIt a(_graph); a != INVALID; ++a) {
985 c += Number(_flow[i]) * Number(_cost[i]);
991 Cost totalCost() const {
992 return totalCost<Cost>();
996 /// \brief Return the flow on the given arc.
998 /// This function returns the flow on the given arc.
1000 /// \pre \ref run() must be called before using this function.
1001 Value flow(const Arc& a) const {
1002 return _flow[_arc_id[a]];
1005 /// \brief Return the flow map (the primal solution).
1007 /// This function copies the flow value on each arc into the given
1008 /// map. The \c Value type of the algorithm must be convertible to
1009 /// the \c Value type of the map.
1011 /// \pre \ref run() must be called before using this function.
1012 template <typename FlowMap>
1013 void flowMap(FlowMap &map) const {
1014 for (ArcIt a(_graph); a != INVALID; ++a) {
1015 map.set(a, _flow[_arc_id[a]]);
1019 /// \brief Return the potential (dual value) of the given node.
1021 /// This function returns the potential (dual value) of the
1024 /// \pre \ref run() must be called before using this function.
1025 Cost potential(const Node& n) const {
1026 return _pi[_node_id[n]];
1029 /// \brief Return the potential map (the dual solution).
1031 /// This function copies the potential (dual value) of each node
1032 /// into the given map.
1033 /// The \c Cost type of the algorithm must be convertible to the
1034 /// \c Value type of the map.
1036 /// \pre \ref run() must be called before using this function.
1037 template <typename PotentialMap>
1038 void potentialMap(PotentialMap &map) const {
1039 for (NodeIt n(_graph); n != INVALID; ++n) {
1040 map.set(n, _pi[_node_id[n]]);
1048 // Initialize internal data structures
1050 if (_node_num == 0) return false;
1052 // Check the sum of supply values
1054 for (int i = 0; i != _node_num; ++i) {
1055 _sum_supply += _supply[i];
1057 if ( !((_stype == GEQ && _sum_supply <= 0) ||
1058 (_stype == LEQ && _sum_supply >= 0)) ) return false;
1060 // Remove non-zero lower bounds
1062 for (int i = 0; i != _arc_num; ++i) {
1063 Value c = _lower[i];
1065 _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1067 _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1069 _supply[_source[i]] -= c;
1070 _supply[_target[i]] += c;
1073 for (int i = 0; i != _arc_num; ++i) {
1074 _cap[i] = _upper[i];
1078 // Initialize artifical cost
1080 if (std::numeric_limits<Cost>::is_exact) {
1081 ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1084 for (int i = 0; i != _arc_num; ++i) {
1085 if (_cost[i] > ART_COST) ART_COST = _cost[i];
1087 ART_COST = (ART_COST + 1) * _node_num;
1090 // Initialize arc maps
1091 for (int i = 0; i != _arc_num; ++i) {
1093 _state[i] = STATE_LOWER;
1096 // Set data for the artificial root node
1098 _parent[_root] = -1;
1101 _rev_thread[0] = _root;
1102 _succ_num[_root] = _node_num + 1;
1103 _last_succ[_root] = _root - 1;
1104 _supply[_root] = -_sum_supply;
1107 // Add artificial arcs and initialize the spanning tree data structure
1108 if (_sum_supply == 0) {
1109 // EQ supply constraints
1110 _search_arc_num = _arc_num;
1111 _all_arc_num = _arc_num + _node_num;
1112 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1116 _rev_thread[u + 1] = u;
1120 _state[e] = STATE_TREE;
1121 if (_supply[u] >= 0) {
1122 _pred_dir[u] = DIR_UP;
1126 _flow[e] = _supply[u];
1129 _pred_dir[u] = DIR_DOWN;
1133 _flow[e] = -_supply[u];
1134 _cost[e] = ART_COST;
1138 else if (_sum_supply > 0) {
1139 // LEQ supply constraints
1140 _search_arc_num = _arc_num + _node_num;
1141 int f = _arc_num + _node_num;
1142 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1145 _rev_thread[u + 1] = u;
1148 if (_supply[u] >= 0) {
1149 _pred_dir[u] = DIR_UP;
1155 _flow[e] = _supply[u];
1157 _state[e] = STATE_TREE;
1159 _pred_dir[u] = DIR_DOWN;
1165 _flow[f] = -_supply[u];
1166 _cost[f] = ART_COST;
1167 _state[f] = STATE_TREE;
1173 _state[e] = STATE_LOWER;
1180 // GEQ supply constraints
1181 _search_arc_num = _arc_num + _node_num;
1182 int f = _arc_num + _node_num;
1183 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1186 _rev_thread[u + 1] = u;
1189 if (_supply[u] <= 0) {
1190 _pred_dir[u] = DIR_DOWN;
1196 _flow[e] = -_supply[u];
1198 _state[e] = STATE_TREE;
1200 _pred_dir[u] = DIR_UP;
1206 _flow[f] = _supply[u];
1207 _state[f] = STATE_TREE;
1208 _cost[f] = ART_COST;
1214 _state[e] = STATE_LOWER;
1224 // Find the join node
1225 void findJoinNode() {
1226 int u = _source[in_arc];
1227 int v = _target[in_arc];
1229 if (_succ_num[u] < _succ_num[v]) {
1238 // Find the leaving arc of the cycle and returns true if the
1239 // leaving arc is not the same as the entering arc
1240 bool findLeavingArc() {
1241 // Initialize first and second nodes according to the direction
1244 if (_state[in_arc] == STATE_LOWER) {
1245 first = _source[in_arc];
1246 second = _target[in_arc];
1248 first = _target[in_arc];
1249 second = _source[in_arc];
1251 delta = _cap[in_arc];
1256 // Search the cycle form the first node to the join node
1257 for (int u = first; u != join; u = _parent[u]) {
1260 if (_pred_dir[u] == DIR_DOWN) {
1262 d = c >= MAX ? INF : c - d;
1271 // Search the cycle form the second node to the join node
1272 for (int u = second; u != join; u = _parent[u]) {
1275 if (_pred_dir[u] == DIR_UP) {
1277 d = c >= MAX ? INF : c - d;
1296 // Change _flow and _state vectors
1297 void changeFlow(bool change) {
1298 // Augment along the cycle
1300 Value val = _state[in_arc] * delta;
1301 _flow[in_arc] += val;
1302 for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1303 _flow[_pred[u]] -= _pred_dir[u] * val;
1305 for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1306 _flow[_pred[u]] += _pred_dir[u] * val;
1309 // Update the state of the entering and leaving arcs
1311 _state[in_arc] = STATE_TREE;
1312 _state[_pred[u_out]] =
1313 (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1315 _state[in_arc] = -_state[in_arc];
1319 // Update the tree structure
1320 void updateTreeStructure() {
1321 int old_rev_thread = _rev_thread[u_out];
1322 int old_succ_num = _succ_num[u_out];
1323 int old_last_succ = _last_succ[u_out];
1324 v_out = _parent[u_out];
1326 // Check if u_in and u_out coincide
1327 if (u_in == u_out) {
1328 // Update _parent, _pred, _pred_dir
1329 _parent[u_in] = v_in;
1330 _pred[u_in] = in_arc;
1331 _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1333 // Update _thread and _rev_thread
1334 if (_thread[v_in] != u_out) {
1335 int after = _thread[old_last_succ];
1336 _thread[old_rev_thread] = after;
1337 _rev_thread[after] = old_rev_thread;
1338 after = _thread[v_in];
1339 _thread[v_in] = u_out;
1340 _rev_thread[u_out] = v_in;
1341 _thread[old_last_succ] = after;
1342 _rev_thread[after] = old_last_succ;
1345 // Handle the case when old_rev_thread equals to v_in
1346 // (it also means that join and v_out coincide)
1347 int thread_continue = old_rev_thread == v_in ?
1348 _thread[old_last_succ] : _thread[v_in];
1350 // Update _thread and _parent along the stem nodes (i.e. the nodes
1351 // between u_in and u_out, whose parent have to be changed)
1352 int stem = u_in; // the current stem node
1353 int par_stem = v_in; // the new parent of stem
1354 int next_stem; // the next stem node
1355 int last = _last_succ[u_in]; // the last successor of stem
1356 int before, after = _thread[last];
1357 _thread[v_in] = u_in;
1358 _dirty_revs.clear();
1359 _dirty_revs.push_back(v_in);
1360 while (stem != u_out) {
1361 // Insert the next stem node into the thread list
1362 next_stem = _parent[stem];
1363 _thread[last] = next_stem;
1364 _dirty_revs.push_back(last);
1366 // Remove the subtree of stem from the thread list
1367 before = _rev_thread[stem];
1368 _thread[before] = after;
1369 _rev_thread[after] = before;
1371 // Change the parent node and shift stem nodes
1372 _parent[stem] = par_stem;
1376 // Update last and after
1377 last = _last_succ[stem] == _last_succ[par_stem] ?
1378 _rev_thread[par_stem] : _last_succ[stem];
1379 after = _thread[last];
1381 _parent[u_out] = par_stem;
1382 _thread[last] = thread_continue;
1383 _rev_thread[thread_continue] = last;
1384 _last_succ[u_out] = last;
1386 // Remove the subtree of u_out from the thread list except for
1387 // the case when old_rev_thread equals to v_in
1388 if (old_rev_thread != v_in) {
1389 _thread[old_rev_thread] = after;
1390 _rev_thread[after] = old_rev_thread;
1393 // Update _rev_thread using the new _thread values
1394 for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1395 int u = _dirty_revs[i];
1396 _rev_thread[_thread[u]] = u;
1399 // Update _pred, _pred_dir, _last_succ and _succ_num for the
1400 // stem nodes from u_out to u_in
1401 int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1402 for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
1403 _pred[u] = _pred[p];
1404 _pred_dir[u] = -_pred_dir[p];
1405 tmp_sc += _succ_num[u] - _succ_num[p];
1406 _succ_num[u] = tmp_sc;
1407 _last_succ[p] = tmp_ls;
1409 _pred[u_in] = in_arc;
1410 _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1411 _succ_num[u_in] = old_succ_num;
1414 // Update _last_succ from v_in towards the root
1415 int up_limit_out = _last_succ[join] == v_in ? join : -1;
1416 int last_succ_out = _last_succ[u_out];
1417 for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {
1418 _last_succ[u] = last_succ_out;
1421 // Update _last_succ from v_out towards the root
1422 if (join != old_rev_thread && v_in != old_rev_thread) {
1423 for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1425 _last_succ[u] = old_rev_thread;
1428 else if (last_succ_out != old_last_succ) {
1429 for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1431 _last_succ[u] = last_succ_out;
1435 // Update _succ_num from v_in to join
1436 for (int u = v_in; u != join; u = _parent[u]) {
1437 _succ_num[u] += old_succ_num;
1439 // Update _succ_num from v_out to join
1440 for (int u = v_out; u != join; u = _parent[u]) {
1441 _succ_num[u] -= old_succ_num;
1445 // Update potentials in the subtree that has been moved
1446 void updatePotential() {
1447 Cost sigma = _pi[v_in] - _pi[u_in] -
1448 _pred_dir[u_in] * _cost[in_arc];
1449 int end = _thread[_last_succ[u_in]];
1450 for (int u = u_in; u != end; u = _thread[u]) {
1455 // Heuristic initial pivots
1456 bool initialPivots() {
1457 Value curr, total = 0;
1458 std::vector<Node> supply_nodes, demand_nodes;
1459 for (NodeIt u(_graph); u != INVALID; ++u) {
1460 curr = _supply[_node_id[u]];
1463 supply_nodes.push_back(u);
1465 else if (curr < 0) {
1466 demand_nodes.push_back(u);
1469 if (_sum_supply > 0) total -= _sum_supply;
1470 if (total <= 0) return true;
1472 IntVector arc_vector;
1473 if (_sum_supply >= 0) {
1474 if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1475 // Perform a reverse graph search from the sink to the source
1476 typename GR::template NodeMap<bool> reached(_graph, false);
1477 Node s = supply_nodes[0], t = demand_nodes[0];
1478 std::vector<Node> stack;
1481 while (!stack.empty()) {
1482 Node u, v = stack.back();
1485 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1486 if (reached[u = _graph.source(a)]) continue;
1488 if (_cap[j] >= total) {
1489 arc_vector.push_back(j);
1496 // Find the min. cost incomming arc for each demand node
1497 for (int i = 0; i != int(demand_nodes.size()); ++i) {
1498 Node v = demand_nodes[i];
1499 Cost c, min_cost = std::numeric_limits<Cost>::max();
1500 Arc min_arc = INVALID;
1501 for (InArcIt a(_graph, v); a != INVALID; ++a) {
1502 c = _cost[_arc_id[a]];
1508 if (min_arc != INVALID) {
1509 arc_vector.push_back(_arc_id[min_arc]);
1514 // Find the min. cost outgoing arc for each supply node
1515 for (int i = 0; i != int(supply_nodes.size()); ++i) {
1516 Node u = supply_nodes[i];
1517 Cost c, min_cost = std::numeric_limits<Cost>::max();
1518 Arc min_arc = INVALID;
1519 for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1520 c = _cost[_arc_id[a]];
1526 if (min_arc != INVALID) {
1527 arc_vector.push_back(_arc_id[min_arc]);
1532 // Perform heuristic initial pivots
1533 for (int i = 0; i != int(arc_vector.size()); ++i) {
1534 in_arc = arc_vector[i];
1535 if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1536 _pi[_target[in_arc]]) >= 0) continue;
1538 bool change = findLeavingArc();
1539 if (delta >= MAX) return false;
1542 updateTreeStructure();
1549 // Execute the algorithm
1550 ProblemType start(PivotRule pivot_rule) {
1551 // Select the pivot rule implementation
1552 switch (pivot_rule) {
1553 case FIRST_ELIGIBLE:
1554 return start<FirstEligiblePivotRule>();
1556 return start<BestEligiblePivotRule>();
1558 return start<BlockSearchPivotRule>();
1559 case CANDIDATE_LIST:
1560 return start<CandidateListPivotRule>();
1562 return start<AlteringListPivotRule>();
1564 return INFEASIBLE; // avoid warning
1567 template <typename PivotRuleImpl>
1568 ProblemType start() {
1569 PivotRuleImpl pivot(*this);
1571 // Perform heuristic initial pivots
1572 if (!initialPivots()) return UNBOUNDED;
1574 // Execute the Network Simplex algorithm
1575 while (pivot.findEnteringArc()) {
1577 bool change = findLeavingArc();
1578 if (delta >= MAX) return UNBOUNDED;
1581 updateTreeStructure();
1586 // Check feasibility
1587 for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1588 if (_flow[e] != 0) return INFEASIBLE;
1591 // Transform the solution and the supply map to the original form
1593 for (int i = 0; i != _arc_num; ++i) {
1594 Value c = _lower[i];
1597 _supply[_source[i]] += c;
1598 _supply[_target[i]] -= c;
1603 // Shift potentials to meet the requirements of the GEQ/LEQ type
1604 // optimality conditions
1605 if (_sum_supply == 0) {
1606 if (_stype == GEQ) {
1607 Cost max_pot = -std::numeric_limits<Cost>::max();
1608 for (int i = 0; i != _node_num; ++i) {
1609 if (_pi[i] > max_pot) max_pot = _pi[i];
1612 for (int i = 0; i != _node_num; ++i)
1616 Cost min_pot = std::numeric_limits<Cost>::max();
1617 for (int i = 0; i != _node_num; ++i) {
1618 if (_pi[i] < min_pot) min_pot = _pi[i];
1621 for (int i = 0; i != _node_num; ++i)
1630 }; //class NetworkSimplex
1636 #endif //LEMON_NETWORK_SIMPLEX_H