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
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
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".
45 /// \tparam GR The digraph type the algorithm runs on.
46 /// \tparam V The value type used in the algorithm.
47 /// By default it is \c int.
49 /// \warning \c V must be a signed integer type.
51 /// \note %NetworkSimplex provides five different pivot rule
52 /// implementations. For more information see \ref PivotRule.
53 template <typename GR, typename V = int>
58 /// The value type of the algorithm
60 /// The type of the flow map
61 typedef typename GR::template ArcMap<Value> FlowMap;
62 /// The type of the potential map
63 typedef typename GR::template NodeMap<Value> PotentialMap;
67 /// \brief Enum type for selecting the pivot rule.
69 /// Enum type for selecting the pivot rule for the \ref run()
72 /// \ref NetworkSimplex provides five different pivot rule
73 /// implementations that significantly affect the running time
75 /// By default \ref BLOCK_SEARCH "Block Search" is used, which
76 /// proved to be the most efficient and the most robust on various
77 /// test inputs according to our benchmark tests.
78 /// However another pivot rule can be selected using the \ref run()
79 /// function with the proper parameter.
82 /// The First Eligible pivot rule.
83 /// The next eligible arc is selected in a wraparound fashion
84 /// in every iteration.
87 /// The Best Eligible pivot rule.
88 /// The best eligible arc is selected in every iteration.
91 /// The Block Search pivot rule.
92 /// A specified number of arcs are examined in every iteration
93 /// in a wraparound fashion and the best eligible arc is selected
97 /// The Candidate List pivot rule.
98 /// In a major iteration a candidate list is built from eligible arcs
99 /// in a wraparound fashion and in the following minor iterations
100 /// the best eligible arc is selected from this list.
103 /// The Altering Candidate List pivot rule.
104 /// It is a modified version of the Candidate List method.
105 /// It keeps only the several best eligible arcs from the former
106 /// candidate list and extends this list in every iteration.
112 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
114 typedef typename GR::template ArcMap<Value> ValueArcMap;
115 typedef typename GR::template NodeMap<Value> ValueNodeMap;
117 typedef std::vector<Arc> ArcVector;
118 typedef std::vector<Node> NodeVector;
119 typedef std::vector<int> IntVector;
120 typedef std::vector<bool> BoolVector;
121 typedef std::vector<Value> ValueVector;
123 // State constants for arcs
132 // Data related to the underlying digraph
137 // Parameters of the problem
138 ValueArcMap *_plower;
139 ValueArcMap *_pupper;
141 ValueNodeMap *_psupply;
143 Node _psource, _ptarget;
148 PotentialMap *_potential_map;
150 bool _local_potential;
152 // Data structures for storing the digraph
165 // Data for storing the spanning tree structure
169 IntVector _rev_thread;
171 IntVector _last_succ;
172 IntVector _dirty_revs;
177 // Temporary data used in the current pivot iteration
178 int in_arc, join, u_in, v_in, u_out, v_out;
179 int first, second, right, last;
180 int stem, par_stem, new_stem;
185 // Implementation of the First Eligible pivot rule
186 class FirstEligiblePivotRule
190 // References to the NetworkSimplex class
191 const IntVector &_source;
192 const IntVector &_target;
193 const ValueVector &_cost;
194 const IntVector &_state;
195 const ValueVector &_pi;
205 FirstEligiblePivotRule(NetworkSimplex &ns) :
206 _source(ns._source), _target(ns._target),
207 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
208 _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
211 // Find next entering arc
212 bool findEnteringArc() {
214 for (int e = _next_arc; e < _arc_num; ++e) {
215 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
222 for (int e = 0; e < _next_arc; ++e) {
223 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
233 }; //class FirstEligiblePivotRule
236 // Implementation of the Best Eligible pivot rule
237 class BestEligiblePivotRule
241 // References to the NetworkSimplex class
242 const IntVector &_source;
243 const IntVector &_target;
244 const ValueVector &_cost;
245 const IntVector &_state;
246 const ValueVector &_pi;
253 BestEligiblePivotRule(NetworkSimplex &ns) :
254 _source(ns._source), _target(ns._target),
255 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
256 _in_arc(ns.in_arc), _arc_num(ns._arc_num)
259 // Find next entering arc
260 bool findEnteringArc() {
262 for (int e = 0; e < _arc_num; ++e) {
263 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
272 }; //class BestEligiblePivotRule
275 // Implementation of the Block Search pivot rule
276 class BlockSearchPivotRule
280 // References to the NetworkSimplex class
281 const IntVector &_source;
282 const IntVector &_target;
283 const ValueVector &_cost;
284 const IntVector &_state;
285 const ValueVector &_pi;
296 BlockSearchPivotRule(NetworkSimplex &ns) :
297 _source(ns._source), _target(ns._target),
298 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
299 _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
301 // The main parameters of the pivot rule
302 const double BLOCK_SIZE_FACTOR = 2.0;
303 const int MIN_BLOCK_SIZE = 10;
305 _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_arc_num)),
309 // Find next entering arc
310 bool findEnteringArc() {
312 int cnt = _block_size;
313 int e, min_arc = _next_arc;
314 for (e = _next_arc; e < _arc_num; ++e) {
315 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
325 if (min == 0 || cnt > 0) {
326 for (e = 0; e < _next_arc; ++e) {
327 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
338 if (min >= 0) return false;
344 }; //class BlockSearchPivotRule
347 // Implementation of the Candidate List pivot rule
348 class CandidateListPivotRule
352 // References to the NetworkSimplex class
353 const IntVector &_source;
354 const IntVector &_target;
355 const ValueVector &_cost;
356 const IntVector &_state;
357 const ValueVector &_pi;
362 IntVector _candidates;
363 int _list_length, _minor_limit;
364 int _curr_length, _minor_count;
370 CandidateListPivotRule(NetworkSimplex &ns) :
371 _source(ns._source), _target(ns._target),
372 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
373 _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)
375 // The main parameters of the pivot rule
376 const double LIST_LENGTH_FACTOR = 1.0;
377 const int MIN_LIST_LENGTH = 10;
378 const double MINOR_LIMIT_FACTOR = 0.1;
379 const int MIN_MINOR_LIMIT = 3;
381 _list_length = std::max( int(LIST_LENGTH_FACTOR * sqrt(_arc_num)),
383 _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
385 _curr_length = _minor_count = 0;
386 _candidates.resize(_list_length);
389 /// Find next entering arc
390 bool findEnteringArc() {
392 int e, min_arc = _next_arc;
393 if (_curr_length > 0 && _minor_count < _minor_limit) {
394 // Minor iteration: select the best eligible arc from the
395 // current candidate list
398 for (int i = 0; i < _curr_length; ++i) {
400 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
406 _candidates[i--] = _candidates[--_curr_length];
415 // Major iteration: build a new candidate list
418 for (e = _next_arc; e < _arc_num; ++e) {
419 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
421 _candidates[_curr_length++] = e;
426 if (_curr_length == _list_length) break;
429 if (_curr_length < _list_length) {
430 for (e = 0; e < _next_arc; ++e) {
431 c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
433 _candidates[_curr_length++] = e;
438 if (_curr_length == _list_length) break;
442 if (_curr_length == 0) return false;
449 }; //class CandidateListPivotRule
452 // Implementation of the Altering Candidate List pivot rule
453 class AlteringListPivotRule
457 // References to the NetworkSimplex class
458 const IntVector &_source;
459 const IntVector &_target;
460 const ValueVector &_cost;
461 const IntVector &_state;
462 const ValueVector &_pi;
467 int _block_size, _head_length, _curr_length;
469 IntVector _candidates;
470 ValueVector _cand_cost;
472 // Functor class to compare arcs during sort of the candidate list
476 const ValueVector &_map;
478 SortFunc(const ValueVector &map) : _map(map) {}
479 bool operator()(int left, int right) {
480 return _map[left] > _map[right];
489 AlteringListPivotRule(NetworkSimplex &ns) :
490 _source(ns._source), _target(ns._target),
491 _cost(ns._cost), _state(ns._state), _pi(ns._pi),
492 _in_arc(ns.in_arc), _arc_num(ns._arc_num),
493 _next_arc(0), _cand_cost(ns._arc_num), _sort_func(_cand_cost)
495 // The main parameters of the pivot rule
496 const double BLOCK_SIZE_FACTOR = 1.5;
497 const int MIN_BLOCK_SIZE = 10;
498 const double HEAD_LENGTH_FACTOR = 0.1;
499 const int MIN_HEAD_LENGTH = 3;
501 _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_arc_num)),
503 _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
505 _candidates.resize(_head_length + _block_size);
509 // Find next entering arc
510 bool findEnteringArc() {
511 // Check the current candidate list
513 for (int i = 0; i < _curr_length; ++i) {
515 _cand_cost[e] = _state[e] *
516 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
517 if (_cand_cost[e] >= 0) {
518 _candidates[i--] = _candidates[--_curr_length];
523 int cnt = _block_size;
525 int limit = _head_length;
527 for (int e = _next_arc; e < _arc_num; ++e) {
528 _cand_cost[e] = _state[e] *
529 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
530 if (_cand_cost[e] < 0) {
531 _candidates[_curr_length++] = e;
535 if (_curr_length > limit) break;
540 if (_curr_length <= limit) {
541 for (int e = 0; e < _next_arc; ++e) {
542 _cand_cost[e] = _state[e] *
543 (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
544 if (_cand_cost[e] < 0) {
545 _candidates[_curr_length++] = e;
549 if (_curr_length > limit) break;
555 if (_curr_length == 0) return false;
556 _next_arc = last_arc + 1;
558 // Make heap of the candidate list (approximating a partial sort)
559 make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
562 // Pop the first element of the heap
563 _in_arc = _candidates[0];
564 pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
566 _curr_length = std::min(_head_length, _curr_length - 1);
570 }; //class AlteringListPivotRule
574 /// \brief Constructor.
578 /// \param graph The digraph the algorithm runs on.
579 NetworkSimplex(const GR& graph) :
581 _plower(NULL), _pupper(NULL), _pcost(NULL),
582 _psupply(NULL), _pstsup(false),
583 _flow_map(NULL), _potential_map(NULL),
584 _local_flow(false), _local_potential(false),
587 LEMON_ASSERT(std::numeric_limits<Value>::is_integer &&
588 std::numeric_limits<Value>::is_signed,
589 "The value type of NetworkSimplex must be a signed integer");
594 if (_local_flow) delete _flow_map;
595 if (_local_potential) delete _potential_map;
598 /// \brief Set the lower bounds on the arcs.
600 /// This function sets the lower bounds on the arcs.
601 /// If neither this function nor \ref boundMaps() is used before
602 /// calling \ref run(), the lower bounds will be set to zero
605 /// \param map An arc map storing the lower bounds.
606 /// Its \c Value type must be convertible to the \c Value type
607 /// of the algorithm.
609 /// \return <tt>(*this)</tt>
610 template <typename LOWER>
611 NetworkSimplex& lowerMap(const LOWER& map) {
613 _plower = new ValueArcMap(_graph);
614 for (ArcIt a(_graph); a != INVALID; ++a) {
615 (*_plower)[a] = map[a];
620 /// \brief Set the upper bounds (capacities) on the arcs.
622 /// This function sets the upper bounds (capacities) on the arcs.
623 /// If none of the functions \ref upperMap(), \ref capacityMap()
624 /// and \ref boundMaps() is used before calling \ref run(),
625 /// the upper bounds (capacities) will be set to
626 /// \c std::numeric_limits<Value>::max() on all arcs.
628 /// \param map An arc map storing the upper bounds.
629 /// Its \c Value type must be convertible to the \c Value type
630 /// of the algorithm.
632 /// \return <tt>(*this)</tt>
633 template<typename UPPER>
634 NetworkSimplex& upperMap(const UPPER& map) {
636 _pupper = new ValueArcMap(_graph);
637 for (ArcIt a(_graph); a != INVALID; ++a) {
638 (*_pupper)[a] = map[a];
643 /// \brief Set the upper bounds (capacities) on the arcs.
645 /// This function sets the upper bounds (capacities) on the arcs.
646 /// It is just an alias for \ref upperMap().
648 /// \return <tt>(*this)</tt>
649 template<typename CAP>
650 NetworkSimplex& capacityMap(const CAP& map) {
651 return upperMap(map);
654 /// \brief Set the lower and upper bounds on the arcs.
656 /// This function sets the lower and upper bounds on the arcs.
657 /// If neither this function nor \ref lowerMap() is used before
658 /// calling \ref run(), the lower bounds will be set to zero
660 /// If none of the functions \ref upperMap(), \ref capacityMap()
661 /// and \ref boundMaps() is used before calling \ref run(),
662 /// the upper bounds (capacities) will be set to
663 /// \c std::numeric_limits<Value>::max() on all arcs.
665 /// \param lower An arc map storing the lower bounds.
666 /// \param upper An arc map storing the upper bounds.
668 /// The \c Value type of the maps must be convertible to the
669 /// \c Value type of the algorithm.
671 /// \note This function is just a shortcut of calling \ref lowerMap()
672 /// and \ref upperMap() separately.
674 /// \return <tt>(*this)</tt>
675 template <typename LOWER, typename UPPER>
676 NetworkSimplex& boundMaps(const LOWER& lower, const UPPER& upper) {
677 return lowerMap(lower).upperMap(upper);
680 /// \brief Set the costs of the arcs.
682 /// This function sets the costs of the arcs.
683 /// If it is not used before calling \ref run(), the costs
684 /// will be set to \c 1 on all arcs.
686 /// \param map An arc map storing the costs.
687 /// Its \c Value type must be convertible to the \c Value type
688 /// of the algorithm.
690 /// \return <tt>(*this)</tt>
691 template<typename COST>
692 NetworkSimplex& costMap(const COST& map) {
694 _pcost = new ValueArcMap(_graph);
695 for (ArcIt a(_graph); a != INVALID; ++a) {
696 (*_pcost)[a] = map[a];
701 /// \brief Set the supply values of the nodes.
703 /// This function sets the supply values of the nodes.
704 /// If neither this function nor \ref stSupply() is used before
705 /// calling \ref run(), the supply of each node will be set to zero.
706 /// (It makes sense only if non-zero lower bounds are given.)
708 /// \param map A node map storing the supply values.
709 /// Its \c Value type must be convertible to the \c Value type
710 /// of the algorithm.
712 /// \return <tt>(*this)</tt>
713 template<typename SUP>
714 NetworkSimplex& supplyMap(const SUP& map) {
717 _psupply = new ValueNodeMap(_graph);
718 for (NodeIt n(_graph); n != INVALID; ++n) {
719 (*_psupply)[n] = map[n];
724 /// \brief Set single source and target nodes and a supply value.
726 /// This function sets a single source node and a single target node
727 /// and the required flow value.
728 /// If neither this function nor \ref supplyMap() is used before
729 /// calling \ref run(), the supply of each node will be set to zero.
730 /// (It makes sense only if non-zero lower bounds are given.)
732 /// \param s The source node.
733 /// \param t The target node.
734 /// \param k The required amount of flow from node \c s to node \c t
735 /// (i.e. the supply of \c s and the demand of \c t).
737 /// \return <tt>(*this)</tt>
738 NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
748 /// \brief Set the flow map.
750 /// This function sets the flow map.
751 /// If it is not used before calling \ref run(), an instance will
752 /// be allocated automatically. The destructor deallocates this
753 /// automatically allocated map, of course.
755 /// \return <tt>(*this)</tt>
756 NetworkSimplex& flowMap(FlowMap& map) {
765 /// \brief Set the potential map.
767 /// This function sets the potential map, which is used for storing
768 /// the dual solution.
769 /// If it is not used before calling \ref run(), an instance will
770 /// be allocated automatically. The destructor deallocates this
771 /// automatically allocated map, of course.
773 /// \return <tt>(*this)</tt>
774 NetworkSimplex& potentialMap(PotentialMap& map) {
775 if (_local_potential) {
776 delete _potential_map;
777 _local_potential = false;
779 _potential_map = ↦
783 /// \name Execution Control
784 /// The algorithm can be executed using \ref run().
788 /// \brief Run the algorithm.
790 /// This function runs the algorithm.
791 /// The paramters can be specified using \ref lowerMap(),
792 /// \ref upperMap(), \ref capacityMap(), \ref boundMaps(),
793 /// \ref costMap(), \ref supplyMap() and \ref stSupply()
794 /// functions. For example,
796 /// NetworkSimplex<ListDigraph> ns(graph);
797 /// ns.boundMaps(lower, upper).costMap(cost)
798 /// .supplyMap(sup).run();
801 /// \param pivot_rule The pivot rule that will be used during the
802 /// algorithm. For more information see \ref PivotRule.
804 /// \return \c true if a feasible flow can be found.
805 bool run(PivotRule pivot_rule = BLOCK_SEARCH) {
806 return init() && start(pivot_rule);
811 /// \name Query Functions
812 /// The results of the algorithm can be obtained using these
814 /// The \ref run() function must be called before using them.
818 /// \brief Return the total cost of the found flow.
820 /// This function returns the total cost of the found flow.
821 /// The complexity of the function is \f$ O(e) \f$.
823 /// \note The return type of the function can be specified as a
824 /// template parameter. For example,
826 /// ns.totalCost<double>();
828 /// It is useful if the total cost cannot be stored in the \c Value
829 /// type of the algorithm, which is the default return type of the
832 /// \pre \ref run() must be called before using this function.
833 template <typename Num>
834 Num totalCost() const {
837 for (ArcIt e(_graph); e != INVALID; ++e)
838 c += (*_flow_map)[e] * (*_pcost)[e];
840 for (ArcIt e(_graph); e != INVALID; ++e)
841 c += (*_flow_map)[e];
847 Value totalCost() const {
848 return totalCost<Value>();
852 /// \brief Return the flow on the given arc.
854 /// This function returns the flow on the given arc.
856 /// \pre \ref run() must be called before using this function.
857 Value flow(const Arc& a) const {
858 return (*_flow_map)[a];
861 /// \brief Return a const reference to the flow map.
863 /// This function returns a const reference to an arc map storing
866 /// \pre \ref run() must be called before using this function.
867 const FlowMap& flowMap() const {
871 /// \brief Return the potential (dual value) of the given node.
873 /// This function returns the potential (dual value) of the
876 /// \pre \ref run() must be called before using this function.
877 Value potential(const Node& n) const {
878 return (*_potential_map)[n];
881 /// \brief Return a const reference to the potential map
882 /// (the dual solution).
884 /// This function returns a const reference to a node map storing
885 /// the found potentials, which form the dual solution of the
886 /// \ref min_cost_flow "minimum cost flow" problem.
888 /// \pre \ref run() must be called before using this function.
889 const PotentialMap& potentialMap() const {
890 return *_potential_map;
897 // Initialize internal data structures
899 // Initialize result maps
901 _flow_map = new FlowMap(_graph);
904 if (!_potential_map) {
905 _potential_map = new PotentialMap(_graph);
906 _local_potential = true;
909 // Initialize vectors
910 _node_num = countNodes(_graph);
911 _arc_num = countArcs(_graph);
912 int all_node_num = _node_num + 1;
913 int all_arc_num = _arc_num + _node_num;
914 if (_node_num == 0) return false;
916 _arc_ref.resize(_arc_num);
917 _source.resize(all_arc_num);
918 _target.resize(all_arc_num);
920 _cap.resize(all_arc_num);
921 _cost.resize(all_arc_num);
922 _supply.resize(all_node_num);
923 _flow.resize(all_arc_num, 0);
924 _pi.resize(all_node_num, 0);
926 _parent.resize(all_node_num);
927 _pred.resize(all_node_num);
928 _forward.resize(all_node_num);
929 _thread.resize(all_node_num);
930 _rev_thread.resize(all_node_num);
931 _succ_num.resize(all_node_num);
932 _last_succ.resize(all_node_num);
933 _state.resize(all_arc_num, STATE_LOWER);
935 // Initialize node related data
936 bool valid_supply = true;
937 if (!_pstsup && !_psupply) {
939 _psource = _ptarget = NodeIt(_graph);
945 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
947 _supply[i] = (*_psupply)[n];
950 valid_supply = (sum == 0);
953 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
957 _supply[_node_id[_psource]] = _pstflow;
958 _supply[_node_id[_ptarget]] = -_pstflow;
960 if (!valid_supply) return false;
962 // Set data for the artificial root node
967 _rev_thread[0] = _root;
968 _succ_num[_root] = all_node_num;
969 _last_succ[_root] = _root - 1;
973 // Store the arcs in a mixed order
974 int k = std::max(int(sqrt(_arc_num)), 10);
976 for (ArcIt e(_graph); e != INVALID; ++e) {
978 if ((i += k) >= _arc_num) i = (i % k) + 1;
981 // Initialize arc maps
982 if (_pupper && _pcost) {
983 for (int i = 0; i != _arc_num; ++i) {
985 _source[i] = _node_id[_graph.source(e)];
986 _target[i] = _node_id[_graph.target(e)];
987 _cap[i] = (*_pupper)[e];
988 _cost[i] = (*_pcost)[e];
991 for (int i = 0; i != _arc_num; ++i) {
993 _source[i] = _node_id[_graph.source(e)];
994 _target[i] = _node_id[_graph.target(e)];
997 for (int i = 0; i != _arc_num; ++i)
998 _cap[i] = (*_pupper)[_arc_ref[i]];
1000 Value val = std::numeric_limits<Value>::max();
1001 for (int i = 0; i != _arc_num; ++i)
1005 for (int i = 0; i != _arc_num; ++i)
1006 _cost[i] = (*_pcost)[_arc_ref[i]];
1008 for (int i = 0; i != _arc_num; ++i)
1013 // Remove non-zero lower bounds
1015 for (int i = 0; i != _arc_num; ++i) {
1016 Value c = (*_plower)[_arc_ref[i]];
1019 _supply[_source[i]] -= c;
1020 _supply[_target[i]] += c;
1025 // Add artificial arcs and initialize the spanning tree data structure
1026 Value max_cap = std::numeric_limits<Value>::max();
1027 Value max_cost = std::numeric_limits<Value>::max() / 4;
1028 for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1030 _rev_thread[u + 1] = u;
1035 if (_supply[u] >= 0) {
1036 _flow[e] = _supply[u];
1040 _flow[e] = -_supply[u];
1041 _forward[u] = false;
1044 _cost[e] = max_cost;
1046 _state[e] = STATE_TREE;
1052 // Find the join node
1053 void findJoinNode() {
1054 int u = _source[in_arc];
1055 int v = _target[in_arc];
1057 if (_succ_num[u] < _succ_num[v]) {
1066 // Find the leaving arc of the cycle and returns true if the
1067 // leaving arc is not the same as the entering arc
1068 bool findLeavingArc() {
1069 // Initialize first and second nodes according to the direction
1071 if (_state[in_arc] == STATE_LOWER) {
1072 first = _source[in_arc];
1073 second = _target[in_arc];
1075 first = _target[in_arc];
1076 second = _source[in_arc];
1078 delta = _cap[in_arc];
1083 // Search the cycle along the path form the first node to the root
1084 for (int u = first; u != join; u = _parent[u]) {
1086 d = _forward[u] ? _flow[e] : _cap[e] - _flow[e];
1093 // Search the cycle along the path form the second node to the root
1094 for (int u = second; u != join; u = _parent[u]) {
1096 d = _forward[u] ? _cap[e] - _flow[e] : _flow[e];
1114 // Change _flow and _state vectors
1115 void changeFlow(bool change) {
1116 // Augment along the cycle
1118 Value val = _state[in_arc] * delta;
1119 _flow[in_arc] += val;
1120 for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1121 _flow[_pred[u]] += _forward[u] ? -val : val;
1123 for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1124 _flow[_pred[u]] += _forward[u] ? val : -val;
1127 // Update the state of the entering and leaving arcs
1129 _state[in_arc] = STATE_TREE;
1130 _state[_pred[u_out]] =
1131 (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1133 _state[in_arc] = -_state[in_arc];
1137 // Update the tree structure
1138 void updateTreeStructure() {
1140 int old_rev_thread = _rev_thread[u_out];
1141 int old_succ_num = _succ_num[u_out];
1142 int old_last_succ = _last_succ[u_out];
1143 v_out = _parent[u_out];
1145 u = _last_succ[u_in]; // the last successor of u_in
1146 right = _thread[u]; // the node after it
1148 // Handle the case when old_rev_thread equals to v_in
1149 // (it also means that join and v_out coincide)
1150 if (old_rev_thread == v_in) {
1151 last = _thread[_last_succ[u_out]];
1153 last = _thread[v_in];
1156 // Update _thread and _parent along the stem nodes (i.e. the nodes
1157 // between u_in and u_out, whose parent have to be changed)
1158 _thread[v_in] = stem = u_in;
1159 _dirty_revs.clear();
1160 _dirty_revs.push_back(v_in);
1162 while (stem != u_out) {
1163 // Insert the next stem node into the thread list
1164 new_stem = _parent[stem];
1165 _thread[u] = new_stem;
1166 _dirty_revs.push_back(u);
1168 // Remove the subtree of stem from the thread list
1169 w = _rev_thread[stem];
1171 _rev_thread[right] = w;
1173 // Change the parent node and shift stem nodes
1174 _parent[stem] = par_stem;
1178 // Update u and right
1179 u = _last_succ[stem] == _last_succ[par_stem] ?
1180 _rev_thread[par_stem] : _last_succ[stem];
1183 _parent[u_out] = par_stem;
1185 _rev_thread[last] = u;
1186 _last_succ[u_out] = u;
1188 // Remove the subtree of u_out from the thread list except for
1189 // the case when old_rev_thread equals to v_in
1190 // (it also means that join and v_out coincide)
1191 if (old_rev_thread != v_in) {
1192 _thread[old_rev_thread] = right;
1193 _rev_thread[right] = old_rev_thread;
1196 // Update _rev_thread using the new _thread values
1197 for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1199 _rev_thread[_thread[u]] = u;
1202 // Update _pred, _forward, _last_succ and _succ_num for the
1203 // stem nodes from u_out to u_in
1204 int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1208 _pred[u] = _pred[w];
1209 _forward[u] = !_forward[w];
1210 tmp_sc += _succ_num[u] - _succ_num[w];
1211 _succ_num[u] = tmp_sc;
1212 _last_succ[w] = tmp_ls;
1215 _pred[u_in] = in_arc;
1216 _forward[u_in] = (u_in == _source[in_arc]);
1217 _succ_num[u_in] = old_succ_num;
1219 // Set limits for updating _last_succ form v_in and v_out
1221 int up_limit_in = -1;
1222 int up_limit_out = -1;
1223 if (_last_succ[join] == v_in) {
1224 up_limit_out = join;
1229 // Update _last_succ from v_in towards the root
1230 for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1232 _last_succ[u] = _last_succ[u_out];
1234 // Update _last_succ from v_out towards the root
1235 if (join != old_rev_thread && v_in != old_rev_thread) {
1236 for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1238 _last_succ[u] = old_rev_thread;
1241 for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1243 _last_succ[u] = _last_succ[u_out];
1247 // Update _succ_num from v_in to join
1248 for (u = v_in; u != join; u = _parent[u]) {
1249 _succ_num[u] += old_succ_num;
1251 // Update _succ_num from v_out to join
1252 for (u = v_out; u != join; u = _parent[u]) {
1253 _succ_num[u] -= old_succ_num;
1257 // Update potentials
1258 void updatePotential() {
1259 Value sigma = _forward[u_in] ?
1260 _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1261 _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1262 if (_succ_num[u_in] > _node_num / 2) {
1263 // Update in the upper subtree (which contains the root)
1264 int before = _rev_thread[u_in];
1265 int after = _thread[_last_succ[u_in]];
1266 _thread[before] = after;
1267 _pi[_root] -= sigma;
1268 for (int u = _thread[_root]; u != _root; u = _thread[u]) {
1271 _thread[before] = u_in;
1273 // Update in the lower subtree (which has been moved)
1274 int end = _thread[_last_succ[u_in]];
1275 for (int u = u_in; u != end; u = _thread[u]) {
1281 // Execute the algorithm
1282 bool start(PivotRule pivot_rule) {
1283 // Select the pivot rule implementation
1284 switch (pivot_rule) {
1285 case FIRST_ELIGIBLE:
1286 return start<FirstEligiblePivotRule>();
1288 return start<BestEligiblePivotRule>();
1290 return start<BlockSearchPivotRule>();
1291 case CANDIDATE_LIST:
1292 return start<CandidateListPivotRule>();
1294 return start<AlteringListPivotRule>();
1299 template <typename PivotRuleImpl>
1301 PivotRuleImpl pivot(*this);
1303 // Execute the Network Simplex algorithm
1304 while (pivot.findEnteringArc()) {
1306 bool change = findLeavingArc();
1309 updateTreeStructure();
1314 // Check if the flow amount equals zero on all the artificial arcs
1315 for (int e = _arc_num; e != _arc_num + _node_num; ++e) {
1316 if (_flow[e] > 0) return false;
1319 // Copy flow values to _flow_map
1321 for (int i = 0; i != _arc_num; ++i) {
1322 Arc e = _arc_ref[i];
1323 _flow_map->set(e, (*_plower)[e] + _flow[i]);
1326 for (int i = 0; i != _arc_num; ++i) {
1327 _flow_map->set(_arc_ref[i], _flow[i]);
1330 // Copy potential values to _potential_map
1331 for (NodeIt n(_graph); n != INVALID; ++n) {
1332 _potential_map->set(n, _pi[_node_id[n]]);
1338 }; //class NetworkSimplex
1344 #endif //LEMON_NETWORK_SIMPLEX_H