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_HOWARD_H
20 #define LEMON_HOWARD_H
22 /// \ingroup min_mean_cycle
25 /// \brief Howard's algorithm for finding a minimum mean cycle.
29 #include <lemon/core.h>
30 #include <lemon/path.h>
31 #include <lemon/tolerance.h>
32 #include <lemon/connectivity.h>
36 /// \brief Default traits class of Howard class.
38 /// Default traits class of Howard class.
39 /// \tparam GR The type of the digraph.
40 /// \tparam LEN The type of the length map.
41 /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
43 template <typename GR, typename LEN>
45 template <typename GR, typename LEN,
46 bool integer = std::numeric_limits<typename LEN::Value>::is_integer>
48 struct HowardDefaultTraits
50 /// The type of the digraph
52 /// The type of the length map
53 typedef LEN LengthMap;
54 /// The type of the arc lengths
55 typedef typename LengthMap::Value Value;
57 /// \brief The large value type used for internal computations
59 /// The large value type used for internal computations.
60 /// It is \c long \c long if the \c Value type is integer,
61 /// otherwise it is \c double.
62 /// \c Value must be convertible to \c LargeValue.
63 typedef double LargeValue;
65 /// The tolerance type used for internal computations
66 typedef lemon::Tolerance<LargeValue> Tolerance;
68 /// \brief The path type of the found cycles
70 /// The path type of the found cycles.
71 /// It must conform to the \ref lemon::concepts::Path "Path" concept
72 /// and it must have an \c addBack() function.
73 typedef lemon::Path<Digraph> Path;
76 // Default traits class for integer value types
77 template <typename GR, typename LEN>
78 struct HowardDefaultTraits<GR, LEN, true>
81 typedef LEN LengthMap;
82 typedef typename LengthMap::Value Value;
83 #ifdef LEMON_HAVE_LONG_LONG
84 typedef long long LargeValue;
86 typedef long LargeValue;
88 typedef lemon::Tolerance<LargeValue> Tolerance;
89 typedef lemon::Path<Digraph> Path;
93 /// \addtogroup min_mean_cycle
96 /// \brief Implementation of Howard's algorithm for finding a minimum
99 /// This class implements Howard's policy iteration algorithm for finding
100 /// a directed cycle of minimum mean length (cost) in a digraph.
101 /// This class provides the most efficient algorithm for the
102 /// minimum mean cycle problem, though the best known theoretical
103 /// bound on its running time is exponential.
105 /// \tparam GR The type of the digraph the algorithm runs on.
106 /// \tparam LEN The type of the length map. The default
107 /// map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".
109 template <typename GR, typename LEN, typename TR>
111 template < typename GR,
112 typename LEN = typename GR::template ArcMap<int>,
113 typename TR = HowardDefaultTraits<GR, LEN> >
119 /// The type of the digraph
120 typedef typename TR::Digraph Digraph;
121 /// The type of the length map
122 typedef typename TR::LengthMap LengthMap;
123 /// The type of the arc lengths
124 typedef typename TR::Value Value;
126 /// \brief The large value type
128 /// The large value type used for internal computations.
129 /// Using the \ref HowardDefaultTraits "default traits class",
130 /// it is \c long \c long if the \c Value type is integer,
131 /// otherwise it is \c double.
132 typedef typename TR::LargeValue LargeValue;
134 /// The tolerance type
135 typedef typename TR::Tolerance Tolerance;
137 /// \brief The path type of the found cycles
139 /// The path type of the found cycles.
140 /// Using the \ref HowardDefaultTraits "default traits class",
141 /// it is \ref lemon::Path "Path<Digraph>".
142 typedef typename TR::Path Path;
144 /// The \ref HowardDefaultTraits "traits class" of the algorithm
149 TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
151 // The digraph the algorithm runs on
153 // The length of the arcs
154 const LengthMap &_length;
156 // Data for the found cycles
157 bool _curr_found, _best_found;
158 LargeValue _curr_length, _best_length;
159 int _curr_size, _best_size;
160 Node _curr_node, _best_node;
165 // Internal data used by the algorithm
166 typename Digraph::template NodeMap<Arc> _policy;
167 typename Digraph::template NodeMap<bool> _reached;
168 typename Digraph::template NodeMap<int> _level;
169 typename Digraph::template NodeMap<LargeValue> _dist;
171 // Data for storing the strongly connected components
173 typename Digraph::template NodeMap<int> _comp;
174 std::vector<std::vector<Node> > _comp_nodes;
175 std::vector<Node>* _nodes;
176 typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs;
178 // Queue used for BFS search
179 std::vector<Node> _queue;
182 Tolerance _tolerance;
185 const LargeValue INF;
189 /// \name Named Template Parameters
192 template <typename T>
193 struct SetLargeValueTraits : public Traits {
194 typedef T LargeValue;
195 typedef lemon::Tolerance<T> Tolerance;
198 /// \brief \ref named-templ-param "Named parameter" for setting
199 /// \c LargeValue type.
201 /// \ref named-templ-param "Named parameter" for setting \c LargeValue
202 /// type. It is used for internal computations in the algorithm.
203 template <typename T>
205 : public Howard<GR, LEN, SetLargeValueTraits<T> > {
206 typedef Howard<GR, LEN, SetLargeValueTraits<T> > Create;
209 template <typename T>
210 struct SetPathTraits : public Traits {
214 /// \brief \ref named-templ-param "Named parameter" for setting
217 /// \ref named-templ-param "Named parameter" for setting the \c %Path
218 /// type of the found cycles.
219 /// It must conform to the \ref lemon::concepts::Path "Path" concept
220 /// and it must have an \c addBack() function.
221 template <typename T>
223 : public Howard<GR, LEN, SetPathTraits<T> > {
224 typedef Howard<GR, LEN, SetPathTraits<T> > Create;
231 /// \brief Constructor.
233 /// The constructor of the class.
235 /// \param digraph The digraph the algorithm runs on.
236 /// \param length The lengths (costs) of the arcs.
237 Howard( const Digraph &digraph,
238 const LengthMap &length ) :
239 _gr(digraph), _length(length), _best_found(false),
240 _best_length(0), _best_size(1), _cycle_path(NULL), _local_path(false),
241 _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),
242 _comp(digraph), _in_arcs(digraph),
243 INF(std::numeric_limits<LargeValue>::has_infinity ?
244 std::numeric_limits<LargeValue>::infinity() :
245 std::numeric_limits<LargeValue>::max())
250 if (_local_path) delete _cycle_path;
253 /// \brief Set the path structure for storing the found cycle.
255 /// This function sets an external path structure for storing the
258 /// If you don't call this function before calling \ref run() or
259 /// \ref findMinMean(), it will allocate a local \ref Path "path"
260 /// structure. The destuctor deallocates this automatically
261 /// allocated object, of course.
263 /// \note The algorithm calls only the \ref lemon::Path::addBack()
264 /// "addBack()" function of the given path structure.
266 /// \return <tt>(*this)</tt>
267 Howard& cycle(Path &path) {
276 /// \brief Set the tolerance used by the algorithm.
278 /// This function sets the tolerance object used by the algorithm.
280 /// \return <tt>(*this)</tt>
281 Howard& tolerance(const Tolerance& tolerance) {
282 _tolerance = tolerance;
286 /// \brief Return a const reference to the tolerance.
288 /// This function returns a const reference to the tolerance object
289 /// used by the algorithm.
290 const Tolerance& tolerance() const {
294 /// \name Execution control
295 /// The simplest way to execute the algorithm is to call the \ref run()
297 /// If you only need the minimum mean length, you may call
298 /// \ref findMinMean().
302 /// \brief Run the algorithm.
304 /// This function runs the algorithm.
305 /// It can be called more than once (e.g. if the underlying digraph
306 /// and/or the arc lengths have been modified).
308 /// \return \c true if a directed cycle exists in the digraph.
310 /// \note <tt>mmc.run()</tt> is just a shortcut of the following code.
312 /// return mmc.findMinMean() && mmc.findCycle();
315 return findMinMean() && findCycle();
318 /// \brief Find the minimum cycle mean.
320 /// This function finds the minimum mean length of the directed
321 /// cycles in the digraph.
323 /// \return \c true if a directed cycle exists in the digraph.
325 // Initialize and find strongly connected components
329 // Find the minimum cycle mean in the components
330 for (int comp = 0; comp < _comp_num; ++comp) {
331 // Find the minimum mean cycle in the current component
332 if (!buildPolicyGraph(comp)) continue;
335 if (!computeNodeDistances()) break;
337 // Update the best cycle (global minimum mean cycle)
338 if ( _curr_found && (!_best_found ||
339 _curr_length * _best_size < _best_length * _curr_size) ) {
341 _best_length = _curr_length;
342 _best_size = _curr_size;
343 _best_node = _curr_node;
349 /// \brief Find a minimum mean directed cycle.
351 /// This function finds a directed cycle of minimum mean length
352 /// in the digraph using the data computed by findMinMean().
354 /// \return \c true if a directed cycle exists in the digraph.
356 /// \pre \ref findMinMean() must be called before using this function.
358 if (!_best_found) return false;
359 _cycle_path->addBack(_policy[_best_node]);
360 for ( Node v = _best_node;
361 (v = _gr.target(_policy[v])) != _best_node; ) {
362 _cycle_path->addBack(_policy[v]);
369 /// \name Query Functions
370 /// The results of the algorithm can be obtained using these
372 /// The algorithm should be executed before using them.
376 /// \brief Return the total length of the found cycle.
378 /// This function returns the total length of the found cycle.
380 /// \pre \ref run() or \ref findMinMean() must be called before
381 /// using this function.
382 LargeValue cycleLength() const {
386 /// \brief Return the number of arcs on the found cycle.
388 /// This function returns the number of arcs on the found cycle.
390 /// \pre \ref run() or \ref findMinMean() must be called before
391 /// using this function.
392 int cycleArcNum() const {
396 /// \brief Return the mean length of the found cycle.
398 /// This function returns the mean length of the found cycle.
400 /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
403 /// return static_cast<double>(alg.cycleLength()) / alg.cycleArcNum();
406 /// \pre \ref run() or \ref findMinMean() must be called before
407 /// using this function.
408 double cycleMean() const {
409 return static_cast<double>(_best_length) / _best_size;
412 /// \brief Return the found cycle.
414 /// This function returns a const reference to the path structure
415 /// storing the found cycle.
417 /// \pre \ref run() or \ref findCycle() must be called before using
419 const Path& cycle() const {
431 _cycle_path = new Path;
433 _queue.resize(countNodes(_gr));
437 _cycle_path->clear();
440 // Find strongly connected components and initialize _comp_nodes
442 void findComponents() {
443 _comp_num = stronglyConnectedComponents(_gr, _comp);
444 _comp_nodes.resize(_comp_num);
445 if (_comp_num == 1) {
446 _comp_nodes[0].clear();
447 for (NodeIt n(_gr); n != INVALID; ++n) {
448 _comp_nodes[0].push_back(n);
450 for (InArcIt a(_gr, n); a != INVALID; ++a) {
451 _in_arcs[n].push_back(a);
455 for (int i = 0; i < _comp_num; ++i)
456 _comp_nodes[i].clear();
457 for (NodeIt n(_gr); n != INVALID; ++n) {
459 _comp_nodes[k].push_back(n);
461 for (InArcIt a(_gr, n); a != INVALID; ++a) {
462 if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);
468 // Build the policy graph in the given strongly connected component
469 // (the out-degree of every node is 1)
470 bool buildPolicyGraph(int comp) {
471 _nodes = &(_comp_nodes[comp]);
472 if (_nodes->size() < 1 ||
473 (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {
476 for (int i = 0; i < int(_nodes->size()); ++i) {
477 _dist[(*_nodes)[i]] = INF;
481 for (int i = 0; i < int(_nodes->size()); ++i) {
483 for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
486 if (_length[e] < _dist[u]) {
487 _dist[u] = _length[e];
495 // Find the minimum mean cycle in the policy graph
496 void findPolicyCycle() {
497 for (int i = 0; i < int(_nodes->size()); ++i) {
498 _level[(*_nodes)[i]] = -1;
504 for (int i = 0; i < int(_nodes->size()); ++i) {
506 if (_level[u] >= 0) continue;
507 for (; _level[u] < 0; u = _gr.target(_policy[u])) {
510 if (_level[u] == i) {
512 clength = _length[_policy[u]];
514 for (v = u; (v = _gr.target(_policy[v])) != u; ) {
515 clength += _length[_policy[v]];
519 (clength * _curr_size < _curr_length * csize) ) {
521 _curr_length = clength;
529 // Contract the policy graph and compute node distances
530 bool computeNodeDistances() {
531 // Find the component of the main cycle and compute node distances
533 for (int i = 0; i < int(_nodes->size()); ++i) {
534 _reached[(*_nodes)[i]] = false;
536 _qfront = _qback = 0;
537 _queue[0] = _curr_node;
538 _reached[_curr_node] = true;
539 _dist[_curr_node] = 0;
542 while (_qfront <= _qback) {
543 v = _queue[_qfront++];
544 for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
547 if (_policy[u] == e && !_reached[u]) {
549 _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
550 _queue[++_qback] = u;
555 // Connect all other nodes to this component and compute node
556 // distances using reverse BFS
558 while (_qback < int(_nodes->size())-1) {
559 v = _queue[_qfront++];
560 for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
566 _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
567 _queue[++_qback] = u;
572 // Improve node distances
573 bool improved = false;
574 for (int i = 0; i < int(_nodes->size()); ++i) {
576 for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
579 LargeValue delta = _dist[v] + _length[e] * _curr_size - _curr_length;
580 if (_tolerance.less(delta, _dist[u])) {
596 #endif //LEMON_HOWARD_H