3 * This file is a part of LEMON, a generic C++ optimization library
5 * Copyright (C) 2003-2008
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_CAPACITY_SCALING_H
20 #define LEMON_CAPACITY_SCALING_H
22 /// \ingroup min_cost_flow_algs
25 /// \brief Capacity Scaling algorithm for finding a minimum cost flow.
29 #include <lemon/core.h>
30 #include <lemon/bin_heap.h>
34 /// \brief Default traits class of CapacityScaling algorithm.
36 /// Default traits class of CapacityScaling algorithm.
37 /// \tparam GR Digraph type.
38 /// \tparam V The number type used for flow amounts, capacity bounds
39 /// and supply values. By default it is \c int.
40 /// \tparam C The number type used for costs and potentials.
41 /// By default it is the same as \c V.
42 template <typename GR, typename V = int, typename C = V>
43 struct CapacityScalingDefaultTraits
45 /// The type of the digraph
47 /// The type of the flow amounts, capacity bounds and supply values
49 /// The type of the arc costs
52 /// \brief The type of the heap used for internal Dijkstra computations.
54 /// The type of the heap used for internal Dijkstra computations.
55 /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
56 /// its priority type must be \c Cost and its cross reference type
57 /// must be \ref RangeMap "RangeMap<int>".
58 typedef BinHeap<Cost, RangeMap<int> > Heap;
61 /// \addtogroup min_cost_flow_algs
64 /// \brief Implementation of the Capacity Scaling algorithm for
65 /// finding a \ref min_cost_flow "minimum cost flow".
67 /// \ref CapacityScaling implements the capacity scaling version
68 /// of the successive shortest path algorithm for finding a
69 /// \ref min_cost_flow "minimum cost flow" \ref amo93networkflows,
70 /// \ref edmondskarp72theoretical. It is an efficient dual
73 /// Most of the parameters of the problem (except for the digraph)
74 /// can be given using separate functions, and the algorithm can be
75 /// executed using the \ref run() function. If some parameters are not
76 /// specified, then default values will be used.
78 /// \tparam GR The digraph type the algorithm runs on.
79 /// \tparam V The number type used for flow amounts, capacity bounds
80 /// and supply values in the algorithm. By default it is \c int.
81 /// \tparam C The number type used for costs and potentials in the
82 /// algorithm. By default it is the same as \c V.
84 /// \warning Both number types must be signed and all input data must
86 /// \warning This algorithm does not support negative costs for such
87 /// arcs that have infinite upper bound.
89 template <typename GR, typename V, typename C, typename TR>
91 template < typename GR, typename V = int, typename C = V,
92 typename TR = CapacityScalingDefaultTraits<GR, V, C> >
98 /// The type of the digraph
99 typedef typename TR::Digraph Digraph;
100 /// The type of the flow amounts, capacity bounds and supply values
101 typedef typename TR::Value Value;
102 /// The type of the arc costs
103 typedef typename TR::Cost Cost;
105 /// The type of the heap used for internal Dijkstra computations
106 typedef typename TR::Heap Heap;
108 /// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm
113 /// \brief Problem type constants for the \c run() function.
115 /// Enum type containing the problem type constants that can be
116 /// returned by the \ref run() function of the algorithm.
118 /// The problem has no feasible solution (flow).
120 /// The problem has optimal solution (i.e. it is feasible and
121 /// bounded), and the algorithm has found optimal flow and node
122 /// potentials (primal and dual solutions).
124 /// The digraph contains an arc of negative cost and infinite
125 /// upper bound. It means that the objective function is unbounded
126 /// on that arc, however, note that it could actually be bounded
127 /// over the feasible flows, but this algroithm cannot handle
134 TEMPLATE_DIGRAPH_TYPEDEFS(GR);
136 typedef std::vector<int> IntVector;
137 typedef std::vector<Value> ValueVector;
138 typedef std::vector<Cost> CostVector;
139 typedef std::vector<char> BoolVector;
140 // Note: vector<char> is used instead of vector<bool> for efficiency reasons
144 // Data related to the underlying digraph
151 // Parameters of the problem
155 // Data structures for storing the digraph
159 IntVector _first_out;
171 ValueVector _res_cap;
174 IntVector _excess_nodes;
175 IntVector _deficit_nodes;
183 /// \brief Constant for infinite upper bounds (capacities).
185 /// Constant for infinite upper bounds (capacities).
186 /// It is \c std::numeric_limits<Value>::infinity() if available,
187 /// \c std::numeric_limits<Value>::max() otherwise.
192 // Special implementation of the Dijkstra algorithm for finding
193 // shortest paths in the residual network of the digraph with
194 // respect to the reduced arc costs and modifying the node
195 // potentials according to the found distance labels.
196 class ResidualDijkstra
202 const IntVector &_first_out;
203 const IntVector &_target;
204 const CostVector &_cost;
205 const ValueVector &_res_cap;
206 const ValueVector &_excess;
210 IntVector _proc_nodes;
215 ResidualDijkstra(CapacityScaling& cs) :
216 _node_num(cs._node_num), _geq(cs._sum_supply < 0),
217 _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
218 _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
219 _pred(cs._pred), _dist(cs._node_num)
222 int run(int s, Value delta = 1) {
223 RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
224 Heap heap(heap_cross_ref);
230 while (!heap.empty() && _excess[heap.top()] > -delta) {
231 int u = heap.top(), v;
232 Cost d = heap.prio() + _pi[u], dn;
233 _dist[u] = heap.prio();
234 _proc_nodes.push_back(u);
237 // Traverse outgoing residual arcs
238 int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
239 for (int a = _first_out[u]; a != last_out; ++a) {
240 if (_res_cap[a] < delta) continue;
242 switch (heap.state(v)) {
244 heap.push(v, d + _cost[a] - _pi[v]);
248 dn = d + _cost[a] - _pi[v];
250 heap.decrease(v, dn);
254 case Heap::POST_HEAP:
259 if (heap.empty()) return -1;
261 // Update potentials of processed nodes
263 Cost dt = heap.prio();
264 for (int i = 0; i < int(_proc_nodes.size()); ++i) {
265 _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
271 }; //class ResidualDijkstra
275 /// \name Named Template Parameters
278 template <typename T>
279 struct SetHeapTraits : public Traits {
283 /// \brief \ref named-templ-param "Named parameter" for setting
286 /// \ref named-templ-param "Named parameter" for setting \c Heap
287 /// type, which is used for internal Dijkstra computations.
288 /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
289 /// its priority type must be \c Cost and its cross reference type
290 /// must be \ref RangeMap "RangeMap<int>".
291 template <typename T>
293 : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
294 typedef CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
301 /// \brief Constructor.
303 /// The constructor of the class.
305 /// \param graph The digraph the algorithm runs on.
306 CapacityScaling(const GR& graph) :
307 _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
308 INF(std::numeric_limits<Value>::has_infinity ?
309 std::numeric_limits<Value>::infinity() :
310 std::numeric_limits<Value>::max())
312 // Check the number types
313 LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
314 "The flow type of CapacityScaling must be signed");
315 LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
316 "The cost type of CapacityScaling must be signed");
319 _node_num = countNodes(_graph);
320 _arc_num = countArcs(_graph);
321 _res_arc_num = 2 * (_arc_num + _node_num);
325 _first_out.resize(_node_num + 1);
326 _forward.resize(_res_arc_num);
327 _source.resize(_res_arc_num);
328 _target.resize(_res_arc_num);
329 _reverse.resize(_res_arc_num);
331 _lower.resize(_res_arc_num);
332 _upper.resize(_res_arc_num);
333 _cost.resize(_res_arc_num);
334 _supply.resize(_node_num);
336 _res_cap.resize(_res_arc_num);
337 _pi.resize(_node_num);
338 _excess.resize(_node_num);
339 _pred.resize(_node_num);
342 int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
343 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
347 for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
349 for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
353 _target[j] = _node_id[_graph.runningNode(a)];
355 for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
359 _target[j] = _node_id[_graph.runningNode(a)];
372 _first_out[_node_num] = k;
373 for (ArcIt a(_graph); a != INVALID; ++a) {
374 int fi = _arc_idf[a];
375 int bi = _arc_idb[a];
385 /// The parameters of the algorithm can be specified using these
390 /// \brief Set the lower bounds on the arcs.
392 /// This function sets the lower bounds on the arcs.
393 /// If it is not used before calling \ref run(), the lower bounds
394 /// will be set to zero on all arcs.
396 /// \param map An arc map storing the lower bounds.
397 /// Its \c Value type must be convertible to the \c Value type
398 /// of the algorithm.
400 /// \return <tt>(*this)</tt>
401 template <typename LowerMap>
402 CapacityScaling& lowerMap(const LowerMap& map) {
404 for (ArcIt a(_graph); a != INVALID; ++a) {
405 _lower[_arc_idf[a]] = map[a];
406 _lower[_arc_idb[a]] = map[a];
411 /// \brief Set the upper bounds (capacities) on the arcs.
413 /// This function sets the upper bounds (capacities) on the arcs.
414 /// If it is not used before calling \ref run(), the upper bounds
415 /// will be set to \ref INF on all arcs (i.e. the flow value will be
416 /// unbounded from above).
418 /// \param map An arc map storing the upper bounds.
419 /// Its \c Value type must be convertible to the \c Value type
420 /// of the algorithm.
422 /// \return <tt>(*this)</tt>
423 template<typename UpperMap>
424 CapacityScaling& upperMap(const UpperMap& map) {
425 for (ArcIt a(_graph); a != INVALID; ++a) {
426 _upper[_arc_idf[a]] = map[a];
431 /// \brief Set the costs of the arcs.
433 /// This function sets the costs of the arcs.
434 /// If it is not used before calling \ref run(), the costs
435 /// will be set to \c 1 on all arcs.
437 /// \param map An arc map storing the costs.
438 /// Its \c Value type must be convertible to the \c Cost type
439 /// of the algorithm.
441 /// \return <tt>(*this)</tt>
442 template<typename CostMap>
443 CapacityScaling& costMap(const CostMap& map) {
444 for (ArcIt a(_graph); a != INVALID; ++a) {
445 _cost[_arc_idf[a]] = map[a];
446 _cost[_arc_idb[a]] = -map[a];
451 /// \brief Set the supply values of the nodes.
453 /// This function sets the supply values of the nodes.
454 /// If neither this function nor \ref stSupply() is used before
455 /// calling \ref run(), the supply of each node will be set to zero.
457 /// \param map A node map storing the supply values.
458 /// Its \c Value type must be convertible to the \c Value type
459 /// of the algorithm.
461 /// \return <tt>(*this)</tt>
462 template<typename SupplyMap>
463 CapacityScaling& supplyMap(const SupplyMap& map) {
464 for (NodeIt n(_graph); n != INVALID; ++n) {
465 _supply[_node_id[n]] = map[n];
470 /// \brief Set single source and target nodes and a supply value.
472 /// This function sets a single source node and a single target node
473 /// and the required flow value.
474 /// If neither this function nor \ref supplyMap() is used before
475 /// calling \ref run(), the supply of each node will be set to zero.
477 /// Using this function has the same effect as using \ref supplyMap()
478 /// with such a map in which \c k is assigned to \c s, \c -k is
479 /// assigned to \c t and all other nodes have zero supply value.
481 /// \param s The source node.
482 /// \param t The target node.
483 /// \param k The required amount of flow from node \c s to node \c t
484 /// (i.e. the supply of \c s and the demand of \c t).
486 /// \return <tt>(*this)</tt>
487 CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
488 for (int i = 0; i != _node_num; ++i) {
491 _supply[_node_id[s]] = k;
492 _supply[_node_id[t]] = -k;
498 /// \name Execution control
499 /// The algorithm can be executed using \ref run().
503 /// \brief Run the algorithm.
505 /// This function runs the algorithm.
506 /// The paramters can be specified using functions \ref lowerMap(),
507 /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
510 /// CapacityScaling<ListDigraph> cs(graph);
511 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
512 /// .supplyMap(sup).run();
515 /// This function can be called more than once. All the parameters
516 /// that have been given are kept for the next call, unless
517 /// \ref reset() is called, thus only the modified parameters
518 /// have to be set again. See \ref reset() for examples.
519 /// However, the underlying digraph must not be modified after this
520 /// class have been constructed, since it copies and extends the graph.
522 /// \param factor The capacity scaling factor. It must be larger than
523 /// one to use scaling. If it is less or equal to one, then scaling
524 /// will be disabled.
526 /// \return \c INFEASIBLE if no feasible flow exists,
527 /// \n \c OPTIMAL if the problem has optimal solution
528 /// (i.e. it is feasible and bounded), and the algorithm has found
529 /// optimal flow and node potentials (primal and dual solutions),
530 /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
531 /// and infinite upper bound. It means that the objective function
532 /// is unbounded on that arc, however, note that it could actually be
533 /// bounded over the feasible flows, but this algroithm cannot handle
537 ProblemType run(int factor = 4) {
539 ProblemType pt = init();
540 if (pt != OPTIMAL) return pt;
544 /// \brief Reset all the parameters that have been given before.
546 /// This function resets all the paramaters that have been given
547 /// before using functions \ref lowerMap(), \ref upperMap(),
548 /// \ref costMap(), \ref supplyMap(), \ref stSupply().
550 /// It is useful for multiple run() calls. If this function is not
551 /// used, all the parameters given before are kept for the next
553 /// However, the underlying digraph must not be modified after this
554 /// class have been constructed, since it copies and extends the graph.
558 /// CapacityScaling<ListDigraph> cs(graph);
561 /// cs.lowerMap(lower).upperMap(upper).costMap(cost)
562 /// .supplyMap(sup).run();
564 /// // Run again with modified cost map (reset() is not called,
565 /// // so only the cost map have to be set again)
567 /// cs.costMap(cost).run();
569 /// // Run again from scratch using reset()
570 /// // (the lower bounds will be set to zero on all arcs)
572 /// cs.upperMap(capacity).costMap(cost)
573 /// .supplyMap(sup).run();
576 /// \return <tt>(*this)</tt>
577 CapacityScaling& reset() {
578 for (int i = 0; i != _node_num; ++i) {
581 for (int j = 0; j != _res_arc_num; ++j) {
584 _cost[j] = _forward[j] ? 1 : -1;
592 /// \name Query Functions
593 /// The results of the algorithm can be obtained using these
595 /// The \ref run() function must be called before using them.
599 /// \brief Return the total cost of the found flow.
601 /// This function returns the total cost of the found flow.
602 /// Its complexity is O(e).
604 /// \note The return type of the function can be specified as a
605 /// template parameter. For example,
607 /// cs.totalCost<double>();
609 /// It is useful if the total cost cannot be stored in the \c Cost
610 /// type of the algorithm, which is the default return type of the
613 /// \pre \ref run() must be called before using this function.
614 template <typename Number>
615 Number totalCost() const {
617 for (ArcIt a(_graph); a != INVALID; ++a) {
619 c += static_cast<Number>(_res_cap[i]) *
620 (-static_cast<Number>(_cost[i]));
626 Cost totalCost() const {
627 return totalCost<Cost>();
631 /// \brief Return the flow on the given arc.
633 /// This function returns the flow on the given arc.
635 /// \pre \ref run() must be called before using this function.
636 Value flow(const Arc& a) const {
637 return _res_cap[_arc_idb[a]];
640 /// \brief Return the flow map (the primal solution).
642 /// This function copies the flow value on each arc into the given
643 /// map. The \c Value type of the algorithm must be convertible to
644 /// the \c Value type of the map.
646 /// \pre \ref run() must be called before using this function.
647 template <typename FlowMap>
648 void flowMap(FlowMap &map) const {
649 for (ArcIt a(_graph); a != INVALID; ++a) {
650 map.set(a, _res_cap[_arc_idb[a]]);
654 /// \brief Return the potential (dual value) of the given node.
656 /// This function returns the potential (dual value) of the
659 /// \pre \ref run() must be called before using this function.
660 Cost potential(const Node& n) const {
661 return _pi[_node_id[n]];
664 /// \brief Return the potential map (the dual solution).
666 /// This function copies the potential (dual value) of each node
667 /// into the given map.
668 /// The \c Cost type of the algorithm must be convertible to the
669 /// \c Value type of the map.
671 /// \pre \ref run() must be called before using this function.
672 template <typename PotentialMap>
673 void potentialMap(PotentialMap &map) const {
674 for (NodeIt n(_graph); n != INVALID; ++n) {
675 map.set(n, _pi[_node_id[n]]);
683 // Initialize the algorithm
685 if (_node_num <= 1) return INFEASIBLE;
687 // Check the sum of supply values
689 for (int i = 0; i != _root; ++i) {
690 _sum_supply += _supply[i];
692 if (_sum_supply > 0) return INFEASIBLE;
694 // Initialize vectors
695 for (int i = 0; i != _root; ++i) {
697 _excess[i] = _supply[i];
700 // Remove non-zero lower bounds
701 const Value MAX = std::numeric_limits<Value>::max();
704 for (int i = 0; i != _root; ++i) {
705 last_out = _first_out[i+1];
706 for (int j = _first_out[i]; j != last_out; ++j) {
710 _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
712 _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
715 _excess[_target[j]] += c;
722 for (int j = 0; j != _res_arc_num; ++j) {
723 _res_cap[j] = _forward[j] ? _upper[j] : 0;
727 // Handle negative costs
728 for (int i = 0; i != _root; ++i) {
729 last_out = _first_out[i+1] - 1;
730 for (int j = _first_out[i]; j != last_out; ++j) {
731 Value rc = _res_cap[j];
732 if (_cost[j] < 0 && rc > 0) {
733 if (rc >= MAX) return UNBOUNDED;
735 _excess[_target[j]] += rc;
737 _res_cap[_reverse[j]] += rc;
742 // Handle GEQ supply type
743 if (_sum_supply < 0) {
745 _excess[_root] = -_sum_supply;
746 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
747 int ra = _reverse[a];
748 _res_cap[a] = -_sum_supply + 1;
756 for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
757 int ra = _reverse[a];
765 // Initialize delta value
768 Value max_sup = 0, max_dem = 0, max_cap = 0;
769 for (int i = 0; i != _root; ++i) {
770 Value ex = _excess[i];
771 if ( ex > max_sup) max_sup = ex;
772 if (-ex > max_dem) max_dem = -ex;
773 int last_out = _first_out[i+1] - 1;
774 for (int j = _first_out[i]; j != last_out; ++j) {
775 if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
778 max_sup = std::min(std::min(max_sup, max_dem), max_cap);
779 for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
788 ProblemType start() {
789 // Execute the algorithm
792 pt = startWithScaling();
794 pt = startWithoutScaling();
796 // Handle non-zero lower bounds
798 int limit = _first_out[_root];
799 for (int j = 0; j != limit; ++j) {
800 if (!_forward[j]) _res_cap[j] += _lower[j];
804 // Shift potentials if necessary
805 Cost pr = _pi[_root];
806 if (_sum_supply < 0 || pr > 0) {
807 for (int i = 0; i != _node_num; ++i) {
815 // Execute the capacity scaling algorithm
816 ProblemType startWithScaling() {
817 // Perform capacity scaling phases
819 ResidualDijkstra _dijkstra(*this);
821 // Saturate all arcs not satisfying the optimality condition
823 for (int u = 0; u != _node_num; ++u) {
824 last_out = _sum_supply < 0 ?
825 _first_out[u+1] : _first_out[u+1] - 1;
826 for (int a = _first_out[u]; a != last_out; ++a) {
828 Cost c = _cost[a] + _pi[u] - _pi[v];
829 Value rc = _res_cap[a];
830 if (c < 0 && rc >= _delta) {
834 _res_cap[_reverse[a]] += rc;
839 // Find excess nodes and deficit nodes
840 _excess_nodes.clear();
841 _deficit_nodes.clear();
842 for (int u = 0; u != _node_num; ++u) {
843 Value ex = _excess[u];
844 if (ex >= _delta) _excess_nodes.push_back(u);
845 if (ex <= -_delta) _deficit_nodes.push_back(u);
847 int next_node = 0, next_def_node = 0;
849 // Find augmenting shortest paths
850 while (next_node < int(_excess_nodes.size())) {
851 // Check deficit nodes
853 bool delta_deficit = false;
854 for ( ; next_def_node < int(_deficit_nodes.size());
856 if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
857 delta_deficit = true;
861 if (!delta_deficit) break;
864 // Run Dijkstra in the residual network
865 s = _excess_nodes[next_node];
866 if ((t = _dijkstra.run(s, _delta)) == -1) {
874 // Augment along a shortest path from s to t
875 Value d = std::min(_excess[s], -_excess[t]);
879 while ((a = _pred[u]) != -1) {
880 if (_res_cap[a] < d) d = _res_cap[a];
885 while ((a = _pred[u]) != -1) {
887 _res_cap[_reverse[a]] += d;
893 if (_excess[s] < _delta) ++next_node;
896 if (_delta == 1) break;
897 _delta = _delta <= _factor ? 1 : _delta / _factor;
903 // Execute the successive shortest path algorithm
904 ProblemType startWithoutScaling() {
906 _excess_nodes.clear();
907 for (int i = 0; i != _node_num; ++i) {
908 if (_excess[i] > 0) _excess_nodes.push_back(i);
910 if (_excess_nodes.size() == 0) return OPTIMAL;
913 // Find shortest paths
915 ResidualDijkstra _dijkstra(*this);
916 while ( _excess[_excess_nodes[next_node]] > 0 ||
917 ++next_node < int(_excess_nodes.size()) )
919 // Run Dijkstra in the residual network
920 s = _excess_nodes[next_node];
921 if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
923 // Augment along a shortest path from s to t
924 Value d = std::min(_excess[s], -_excess[t]);
928 while ((a = _pred[u]) != -1) {
929 if (_res_cap[a] < d) d = _res_cap[a];
934 while ((a = _pred[u]) != -1) {
936 _res_cap[_reverse[a]] += d;
946 }; //class CapacityScaling
952 #endif //LEMON_CAPACITY_SCALING_H