1 | /* -*- C++ -*- |
---|
2 | * |
---|
3 | * This file is a part of LEMON, a generic C++ optimization library |
---|
4 | * |
---|
5 | * Copyright (C) 2003-2008 |
---|
6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
---|
7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
---|
8 | * |
---|
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. |
---|
12 | * |
---|
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 |
---|
15 | * purpose. |
---|
16 | * |
---|
17 | */ |
---|
18 | |
---|
19 | #ifndef LEMON_GOMORY_HU_TREE_H |
---|
20 | #define LEMON_GOMORY_HU_TREE_H |
---|
21 | |
---|
22 | #include <limits> |
---|
23 | |
---|
24 | #include <lemon/core.h> |
---|
25 | #include <lemon/preflow.h> |
---|
26 | #include <lemon/concept_check.h> |
---|
27 | #include <lemon/concepts/maps.h> |
---|
28 | |
---|
29 | /// \ingroup min_cut |
---|
30 | /// \file |
---|
31 | /// \brief Gomory-Hu cut tree in graphs. |
---|
32 | |
---|
33 | namespace lemon { |
---|
34 | |
---|
35 | /// \ingroup min_cut |
---|
36 | /// |
---|
37 | /// \brief Gomory-Hu cut tree algorithm |
---|
38 | /// |
---|
39 | /// The Gomory-Hu tree is a tree on the node set of a given graph, but it |
---|
40 | /// may contain edges which are not in the original graph. It has the |
---|
41 | /// property that the minimum capacity edge of the path between two nodes |
---|
42 | /// in this tree has the same weight as the minimum cut in the graph |
---|
43 | /// between these nodes. Moreover the components obtained by removing |
---|
44 | /// this edge from the tree determine the corresponding minimum cut. |
---|
45 | /// |
---|
46 | /// Therefore once this tree is computed, the minimum cut between any pair |
---|
47 | /// of nodes can easily be obtained. |
---|
48 | /// |
---|
49 | /// The algorithm calculates \e n-1 distinct minimum cuts (currently with |
---|
50 | /// the \ref Preflow algorithm), therefore the algorithm has |
---|
51 | /// \f$(O(n^3\sqrt{e})\f$ overall time complexity. It calculates a |
---|
52 | /// rooted Gomory-Hu tree, its structure and the weights can be obtained |
---|
53 | /// by \c predNode(), \c predValue() and \c rootDist(). |
---|
54 | /// |
---|
55 | /// The members \c minCutMap() and \c minCutValue() calculate |
---|
56 | /// the minimum cut and the minimum cut value between any two nodes |
---|
57 | /// in the graph. You can also list (iterate on) the nodes and the |
---|
58 | /// edges of the cuts using \c MinCutNodeIt and \c MinCutEdgeIt. |
---|
59 | /// |
---|
60 | /// \tparam GR The type of the undirected graph the algorithm runs on. |
---|
61 | /// \tparam CAP The type of the edge map describing the edge capacities. |
---|
62 | /// It is \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>" by default. |
---|
63 | #ifdef DOXYGEN |
---|
64 | template <typename GR, |
---|
65 | typename CAP> |
---|
66 | #else |
---|
67 | template <typename GR, |
---|
68 | typename CAP = typename GR::template EdgeMap<int> > |
---|
69 | #endif |
---|
70 | class GomoryHu { |
---|
71 | public: |
---|
72 | |
---|
73 | /// The graph type |
---|
74 | typedef GR Graph; |
---|
75 | /// The type of the edge capacity map |
---|
76 | typedef CAP Capacity; |
---|
77 | /// The value type of capacities |
---|
78 | typedef typename Capacity::Value Value; |
---|
79 | |
---|
80 | private: |
---|
81 | |
---|
82 | TEMPLATE_GRAPH_TYPEDEFS(Graph); |
---|
83 | |
---|
84 | const Graph& _graph; |
---|
85 | const Capacity& _capacity; |
---|
86 | |
---|
87 | Node _root; |
---|
88 | typename Graph::template NodeMap<Node>* _pred; |
---|
89 | typename Graph::template NodeMap<Value>* _weight; |
---|
90 | typename Graph::template NodeMap<int>* _order; |
---|
91 | |
---|
92 | void createStructures() { |
---|
93 | if (!_pred) { |
---|
94 | _pred = new typename Graph::template NodeMap<Node>(_graph); |
---|
95 | } |
---|
96 | if (!_weight) { |
---|
97 | _weight = new typename Graph::template NodeMap<Value>(_graph); |
---|
98 | } |
---|
99 | if (!_order) { |
---|
100 | _order = new typename Graph::template NodeMap<int>(_graph); |
---|
101 | } |
---|
102 | } |
---|
103 | |
---|
104 | void destroyStructures() { |
---|
105 | if (_pred) { |
---|
106 | delete _pred; |
---|
107 | } |
---|
108 | if (_weight) { |
---|
109 | delete _weight; |
---|
110 | } |
---|
111 | if (_order) { |
---|
112 | delete _order; |
---|
113 | } |
---|
114 | } |
---|
115 | |
---|
116 | public: |
---|
117 | |
---|
118 | /// \brief Constructor |
---|
119 | /// |
---|
120 | /// Constructor |
---|
121 | /// \param graph The undirected graph the algorithm runs on. |
---|
122 | /// \param capacity The edge capacity map. |
---|
123 | GomoryHu(const Graph& graph, const Capacity& capacity) |
---|
124 | : _graph(graph), _capacity(capacity), |
---|
125 | _pred(0), _weight(0), _order(0) |
---|
126 | { |
---|
127 | checkConcept<concepts::ReadMap<Edge, Value>, Capacity>(); |
---|
128 | } |
---|
129 | |
---|
130 | |
---|
131 | /// \brief Destructor |
---|
132 | /// |
---|
133 | /// Destructor |
---|
134 | ~GomoryHu() { |
---|
135 | destroyStructures(); |
---|
136 | } |
---|
137 | |
---|
138 | private: |
---|
139 | |
---|
140 | // Initialize the internal data structures |
---|
141 | void init() { |
---|
142 | createStructures(); |
---|
143 | |
---|
144 | _root = NodeIt(_graph); |
---|
145 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
146 | (*_pred)[n] = _root; |
---|
147 | (*_order)[n] = -1; |
---|
148 | } |
---|
149 | (*_pred)[_root] = INVALID; |
---|
150 | (*_weight)[_root] = std::numeric_limits<Value>::max(); |
---|
151 | } |
---|
152 | |
---|
153 | |
---|
154 | // Start the algorithm |
---|
155 | void start() { |
---|
156 | Preflow<Graph, Capacity> fa(_graph, _capacity, _root, INVALID); |
---|
157 | |
---|
158 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
159 | if (n == _root) continue; |
---|
160 | |
---|
161 | Node pn = (*_pred)[n]; |
---|
162 | fa.source(n); |
---|
163 | fa.target(pn); |
---|
164 | |
---|
165 | fa.runMinCut(); |
---|
166 | |
---|
167 | (*_weight)[n] = fa.flowValue(); |
---|
168 | |
---|
169 | for (NodeIt nn(_graph); nn != INVALID; ++nn) { |
---|
170 | if (nn != n && fa.minCut(nn) && (*_pred)[nn] == pn) { |
---|
171 | (*_pred)[nn] = n; |
---|
172 | } |
---|
173 | } |
---|
174 | if ((*_pred)[pn] != INVALID && fa.minCut((*_pred)[pn])) { |
---|
175 | (*_pred)[n] = (*_pred)[pn]; |
---|
176 | (*_pred)[pn] = n; |
---|
177 | (*_weight)[n] = (*_weight)[pn]; |
---|
178 | (*_weight)[pn] = fa.flowValue(); |
---|
179 | } |
---|
180 | } |
---|
181 | |
---|
182 | (*_order)[_root] = 0; |
---|
183 | int index = 1; |
---|
184 | |
---|
185 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
186 | std::vector<Node> st; |
---|
187 | Node nn = n; |
---|
188 | while ((*_order)[nn] == -1) { |
---|
189 | st.push_back(nn); |
---|
190 | nn = (*_pred)[nn]; |
---|
191 | } |
---|
192 | while (!st.empty()) { |
---|
193 | (*_order)[st.back()] = index++; |
---|
194 | st.pop_back(); |
---|
195 | } |
---|
196 | } |
---|
197 | } |
---|
198 | |
---|
199 | public: |
---|
200 | |
---|
201 | ///\name Execution Control |
---|
202 | |
---|
203 | ///@{ |
---|
204 | |
---|
205 | /// \brief Run the Gomory-Hu algorithm. |
---|
206 | /// |
---|
207 | /// This function runs the Gomory-Hu algorithm. |
---|
208 | void run() { |
---|
209 | init(); |
---|
210 | start(); |
---|
211 | } |
---|
212 | |
---|
213 | /// @} |
---|
214 | |
---|
215 | ///\name Query Functions |
---|
216 | ///The results of the algorithm can be obtained using these |
---|
217 | ///functions.\n |
---|
218 | ///\ref run() "run()" should be called before using them.\n |
---|
219 | ///See also \ref MinCutNodeIt and \ref MinCutEdgeIt. |
---|
220 | |
---|
221 | ///@{ |
---|
222 | |
---|
223 | /// \brief Return the predecessor node in the Gomory-Hu tree. |
---|
224 | /// |
---|
225 | /// This function returns the predecessor node in the Gomory-Hu tree. |
---|
226 | /// If the node is |
---|
227 | /// the root of the Gomory-Hu tree, then it returns \c INVALID. |
---|
228 | Node predNode(const Node& node) { |
---|
229 | return (*_pred)[node]; |
---|
230 | } |
---|
231 | |
---|
232 | /// \brief Return the distance from the root node in the Gomory-Hu tree. |
---|
233 | /// |
---|
234 | /// This function returns the distance of \c node from the root node |
---|
235 | /// in the Gomory-Hu tree. |
---|
236 | int rootDist(const Node& node) { |
---|
237 | return (*_order)[node]; |
---|
238 | } |
---|
239 | |
---|
240 | /// \brief Return the weight of the predecessor edge in the |
---|
241 | /// Gomory-Hu tree. |
---|
242 | /// |
---|
243 | /// This function returns the weight of the predecessor edge in the |
---|
244 | /// Gomory-Hu tree. If the node is the root, the result is undefined. |
---|
245 | Value predValue(const Node& node) { |
---|
246 | return (*_weight)[node]; |
---|
247 | } |
---|
248 | |
---|
249 | /// \brief Return the minimum cut value between two nodes |
---|
250 | /// |
---|
251 | /// This function returns the minimum cut value between two nodes. The |
---|
252 | /// algorithm finds the nearest common ancestor in the Gomory-Hu |
---|
253 | /// tree and calculates the minimum weight edge on the paths to |
---|
254 | /// the ancestor. |
---|
255 | Value minCutValue(const Node& s, const Node& t) const { |
---|
256 | Node sn = s, tn = t; |
---|
257 | Value value = std::numeric_limits<Value>::max(); |
---|
258 | |
---|
259 | while (sn != tn) { |
---|
260 | if ((*_order)[sn] < (*_order)[tn]) { |
---|
261 | if ((*_weight)[tn] <= value) value = (*_weight)[tn]; |
---|
262 | tn = (*_pred)[tn]; |
---|
263 | } else { |
---|
264 | if ((*_weight)[sn] <= value) value = (*_weight)[sn]; |
---|
265 | sn = (*_pred)[sn]; |
---|
266 | } |
---|
267 | } |
---|
268 | return value; |
---|
269 | } |
---|
270 | |
---|
271 | /// \brief Return the minimum cut between two nodes |
---|
272 | /// |
---|
273 | /// This function returns the minimum cut between the nodes \c s and \c t |
---|
274 | /// in the \c cutMap parameter by setting the nodes in the component of |
---|
275 | /// \c s to \c true and the other nodes to \c false. |
---|
276 | /// |
---|
277 | /// For higher level interfaces, see MinCutNodeIt and MinCutEdgeIt. |
---|
278 | template <typename CutMap> |
---|
279 | Value minCutMap(const Node& s, ///< The base node. |
---|
280 | const Node& t, |
---|
281 | ///< The node you want to separate from node \c s. |
---|
282 | CutMap& cutMap |
---|
283 | ///< The cut will be returned in this map. |
---|
284 | /// It must be a \c bool (or convertible) |
---|
285 | /// \ref concepts::ReadWriteMap "ReadWriteMap" |
---|
286 | /// on the graph nodes. |
---|
287 | ) const { |
---|
288 | Node sn = s, tn = t; |
---|
289 | bool s_root=false; |
---|
290 | Node rn = INVALID; |
---|
291 | Value value = std::numeric_limits<Value>::max(); |
---|
292 | |
---|
293 | while (sn != tn) { |
---|
294 | if ((*_order)[sn] < (*_order)[tn]) { |
---|
295 | if ((*_weight)[tn] <= value) { |
---|
296 | rn = tn; |
---|
297 | s_root = false; |
---|
298 | value = (*_weight)[tn]; |
---|
299 | } |
---|
300 | tn = (*_pred)[tn]; |
---|
301 | } else { |
---|
302 | if ((*_weight)[sn] <= value) { |
---|
303 | rn = sn; |
---|
304 | s_root = true; |
---|
305 | value = (*_weight)[sn]; |
---|
306 | } |
---|
307 | sn = (*_pred)[sn]; |
---|
308 | } |
---|
309 | } |
---|
310 | |
---|
311 | typename Graph::template NodeMap<bool> reached(_graph, false); |
---|
312 | reached[_root] = true; |
---|
313 | cutMap.set(_root, !s_root); |
---|
314 | reached[rn] = true; |
---|
315 | cutMap.set(rn, s_root); |
---|
316 | |
---|
317 | std::vector<Node> st; |
---|
318 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
319 | st.clear(); |
---|
320 | Node nn = n; |
---|
321 | while (!reached[nn]) { |
---|
322 | st.push_back(nn); |
---|
323 | nn = (*_pred)[nn]; |
---|
324 | } |
---|
325 | while (!st.empty()) { |
---|
326 | cutMap.set(st.back(), cutMap[nn]); |
---|
327 | st.pop_back(); |
---|
328 | } |
---|
329 | } |
---|
330 | |
---|
331 | return value; |
---|
332 | } |
---|
333 | |
---|
334 | ///@} |
---|
335 | |
---|
336 | friend class MinCutNodeIt; |
---|
337 | |
---|
338 | /// Iterate on the nodes of a minimum cut |
---|
339 | |
---|
340 | /// This iterator class lists the nodes of a minimum cut found by |
---|
341 | /// GomoryHu. Before using it, you must allocate a GomoryHu class, |
---|
342 | /// and call its \ref GomoryHu::run() "run()" method. |
---|
343 | /// |
---|
344 | /// This example counts the nodes in the minimum cut separating \c s from |
---|
345 | /// \c t. |
---|
346 | /// \code |
---|
347 | /// GomoruHu<Graph> gom(g, capacities); |
---|
348 | /// gom.run(); |
---|
349 | /// int cnt=0; |
---|
350 | /// for(GomoruHu<Graph>::MinCutNodeIt n(gom,s,t); n!=INVALID; ++n) ++cnt; |
---|
351 | /// \endcode |
---|
352 | class MinCutNodeIt |
---|
353 | { |
---|
354 | bool _side; |
---|
355 | typename Graph::NodeIt _node_it; |
---|
356 | typename Graph::template NodeMap<bool> _cut; |
---|
357 | public: |
---|
358 | /// Constructor |
---|
359 | |
---|
360 | /// Constructor. |
---|
361 | /// |
---|
362 | MinCutNodeIt(GomoryHu const &gomory, |
---|
363 | ///< The GomoryHu class. You must call its |
---|
364 | /// run() method |
---|
365 | /// before initializing this iterator. |
---|
366 | const Node& s, ///< The base node. |
---|
367 | const Node& t, |
---|
368 | ///< The node you want to separate from node \c s. |
---|
369 | bool side=true |
---|
370 | ///< If it is \c true (default) then the iterator lists |
---|
371 | /// the nodes of the component containing \c s, |
---|
372 | /// otherwise it lists the other component. |
---|
373 | /// \note As the minimum cut is not always unique, |
---|
374 | /// \code |
---|
375 | /// MinCutNodeIt(gomory, s, t, true); |
---|
376 | /// \endcode |
---|
377 | /// and |
---|
378 | /// \code |
---|
379 | /// MinCutNodeIt(gomory, t, s, false); |
---|
380 | /// \endcode |
---|
381 | /// does not necessarily give the same set of nodes. |
---|
382 | /// However it is ensured that |
---|
383 | /// \code |
---|
384 | /// MinCutNodeIt(gomory, s, t, true); |
---|
385 | /// \endcode |
---|
386 | /// and |
---|
387 | /// \code |
---|
388 | /// MinCutNodeIt(gomory, s, t, false); |
---|
389 | /// \endcode |
---|
390 | /// together list each node exactly once. |
---|
391 | ) |
---|
392 | : _side(side), _cut(gomory._graph) |
---|
393 | { |
---|
394 | gomory.minCutMap(s,t,_cut); |
---|
395 | for(_node_it=typename Graph::NodeIt(gomory._graph); |
---|
396 | _node_it!=INVALID && _cut[_node_it]!=_side; |
---|
397 | ++_node_it) {} |
---|
398 | } |
---|
399 | /// Conversion to \c Node |
---|
400 | |
---|
401 | /// Conversion to \c Node. |
---|
402 | /// |
---|
403 | operator typename Graph::Node() const |
---|
404 | { |
---|
405 | return _node_it; |
---|
406 | } |
---|
407 | bool operator==(Invalid) { return _node_it==INVALID; } |
---|
408 | bool operator!=(Invalid) { return _node_it!=INVALID; } |
---|
409 | /// Next node |
---|
410 | |
---|
411 | /// Next node. |
---|
412 | /// |
---|
413 | MinCutNodeIt &operator++() |
---|
414 | { |
---|
415 | for(++_node_it;_node_it!=INVALID&&_cut[_node_it]!=_side;++_node_it) {} |
---|
416 | return *this; |
---|
417 | } |
---|
418 | /// Postfix incrementation |
---|
419 | |
---|
420 | /// Postfix incrementation. |
---|
421 | /// |
---|
422 | /// \warning This incrementation |
---|
423 | /// returns a \c Node, not a \c MinCutNodeIt, as one may |
---|
424 | /// expect. |
---|
425 | typename Graph::Node operator++(int) |
---|
426 | { |
---|
427 | typename Graph::Node n=*this; |
---|
428 | ++(*this); |
---|
429 | return n; |
---|
430 | } |
---|
431 | }; |
---|
432 | |
---|
433 | friend class MinCutEdgeIt; |
---|
434 | |
---|
435 | /// Iterate on the edges of a minimum cut |
---|
436 | |
---|
437 | /// This iterator class lists the edges of a minimum cut found by |
---|
438 | /// GomoryHu. Before using it, you must allocate a GomoryHu class, |
---|
439 | /// and call its \ref GomoryHu::run() "run()" method. |
---|
440 | /// |
---|
441 | /// This example computes the value of the minimum cut separating \c s from |
---|
442 | /// \c t. |
---|
443 | /// \code |
---|
444 | /// GomoruHu<Graph> gom(g, capacities); |
---|
445 | /// gom.run(); |
---|
446 | /// int value=0; |
---|
447 | /// for(GomoruHu<Graph>::MinCutEdgeIt e(gom,s,t); e!=INVALID; ++e) |
---|
448 | /// value+=capacities[e]; |
---|
449 | /// \endcode |
---|
450 | /// the result will be the same as it is returned by |
---|
451 | /// \ref GomoryHu::minCutValue() "gom.minCutValue(s,t)" |
---|
452 | class MinCutEdgeIt |
---|
453 | { |
---|
454 | bool _side; |
---|
455 | const Graph &_graph; |
---|
456 | typename Graph::NodeIt _node_it; |
---|
457 | typename Graph::OutArcIt _arc_it; |
---|
458 | typename Graph::template NodeMap<bool> _cut; |
---|
459 | void step() |
---|
460 | { |
---|
461 | ++_arc_it; |
---|
462 | while(_node_it!=INVALID && _arc_it==INVALID) |
---|
463 | { |
---|
464 | for(++_node_it;_node_it!=INVALID&&!_cut[_node_it];++_node_it) {} |
---|
465 | if(_node_it!=INVALID) |
---|
466 | _arc_it=typename Graph::OutArcIt(_graph,_node_it); |
---|
467 | } |
---|
468 | } |
---|
469 | |
---|
470 | public: |
---|
471 | MinCutEdgeIt(GomoryHu const &gomory, |
---|
472 | ///< The GomoryHu class. You must call its |
---|
473 | /// run() method |
---|
474 | /// before initializing this iterator. |
---|
475 | const Node& s, ///< The base node. |
---|
476 | const Node& t, |
---|
477 | ///< The node you want to separate from node \c s. |
---|
478 | bool side=true |
---|
479 | ///< If it is \c true (default) then the listed arcs |
---|
480 | /// will be oriented from the |
---|
481 | /// the nodes of the component containing \c s, |
---|
482 | /// otherwise they will be oriented in the opposite |
---|
483 | /// direction. |
---|
484 | ) |
---|
485 | : _graph(gomory._graph), _cut(_graph) |
---|
486 | { |
---|
487 | gomory.minCutMap(s,t,_cut); |
---|
488 | if(!side) |
---|
489 | for(typename Graph::NodeIt n(_graph);n!=INVALID;++n) |
---|
490 | _cut[n]=!_cut[n]; |
---|
491 | |
---|
492 | for(_node_it=typename Graph::NodeIt(_graph); |
---|
493 | _node_it!=INVALID && !_cut[_node_it]; |
---|
494 | ++_node_it) {} |
---|
495 | _arc_it = _node_it!=INVALID ? |
---|
496 | typename Graph::OutArcIt(_graph,_node_it) : INVALID; |
---|
497 | while(_node_it!=INVALID && _arc_it == INVALID) |
---|
498 | { |
---|
499 | for(++_node_it; _node_it!=INVALID&&!_cut[_node_it]; ++_node_it) {} |
---|
500 | if(_node_it!=INVALID) |
---|
501 | _arc_it= typename Graph::OutArcIt(_graph,_node_it); |
---|
502 | } |
---|
503 | while(_arc_it!=INVALID && _cut[_graph.target(_arc_it)]) step(); |
---|
504 | } |
---|
505 | /// Conversion to \c Arc |
---|
506 | |
---|
507 | /// Conversion to \c Arc. |
---|
508 | /// |
---|
509 | operator typename Graph::Arc() const |
---|
510 | { |
---|
511 | return _arc_it; |
---|
512 | } |
---|
513 | /// Conversion to \c Edge |
---|
514 | |
---|
515 | /// Conversion to \c Edge. |
---|
516 | /// |
---|
517 | operator typename Graph::Edge() const |
---|
518 | { |
---|
519 | return _arc_it; |
---|
520 | } |
---|
521 | bool operator==(Invalid) { return _node_it==INVALID; } |
---|
522 | bool operator!=(Invalid) { return _node_it!=INVALID; } |
---|
523 | /// Next edge |
---|
524 | |
---|
525 | /// Next edge. |
---|
526 | /// |
---|
527 | MinCutEdgeIt &operator++() |
---|
528 | { |
---|
529 | step(); |
---|
530 | while(_arc_it!=INVALID && _cut[_graph.target(_arc_it)]) step(); |
---|
531 | return *this; |
---|
532 | } |
---|
533 | /// Postfix incrementation |
---|
534 | |
---|
535 | /// Postfix incrementation. |
---|
536 | /// |
---|
537 | /// \warning This incrementation |
---|
538 | /// returns an \c Arc, not a \c MinCutEdgeIt, as one may expect. |
---|
539 | typename Graph::Arc operator++(int) |
---|
540 | { |
---|
541 | typename Graph::Arc e=*this; |
---|
542 | ++(*this); |
---|
543 | return e; |
---|
544 | } |
---|
545 | }; |
---|
546 | |
---|
547 | }; |
---|
548 | |
---|
549 | } |
---|
550 | |
---|
551 | #endif |
---|