1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
---|
2 | * |
---|
3 | * This file is a part of LEMON, a generic C++ optimization library. |
---|
4 | * |
---|
5 | * Copyright (C) 2003-2010 |
---|
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 | namespace lemon { |
---|
20 | /** |
---|
21 | [PAGE]sec_graph_adaptors[PAGE] Graph Adaptors |
---|
22 | |
---|
23 | In typical algorithms and applications related to graphs and networks, |
---|
24 | we usually encounter situations in which a specific alteration of a graph |
---|
25 | has to be considered. |
---|
26 | If some nodes or arcs have to be hidden (maybe temporarily) or the reverse |
---|
27 | oriented graph has to be used, then this is the case. |
---|
28 | However, actually modifing physical storage of the graph or |
---|
29 | making a copy of the graph structure along with the required maps |
---|
30 | could be rather expensive (in time or in memory usage) compared to the |
---|
31 | operations that should be performed on the altered graph. |
---|
32 | In such cases, the LEMON \e graph \e adaptor \e classes could be used. |
---|
33 | |
---|
34 | [SEC]sec_reverse_digraph[SEC] Reverse Oriented Digraph |
---|
35 | |
---|
36 | Let us suppose that we have an instance \c g of a directed graph type, say |
---|
37 | \ref ListDigraph and an algorithm |
---|
38 | \code |
---|
39 | template <typename Digraph> |
---|
40 | int algorithm(const Digraph&); |
---|
41 | \endcode |
---|
42 | is needed to run on the reverse oriented digraph. |
---|
43 | In this situation, a certain adaptor class |
---|
44 | \code |
---|
45 | template <typename Digraph> |
---|
46 | class ReverseDigraph; |
---|
47 | \endcode |
---|
48 | can be used. |
---|
49 | |
---|
50 | The graph adaptors are special classes that serve for considering other graph |
---|
51 | structures in different ways. They can be used exactly the same as "real" |
---|
52 | graphs, i.e. they conform to the \ref graph_concepts "graph concepts", thus all |
---|
53 | generic algorithms can be performed on them. However, the adaptor classes |
---|
54 | cannot be used alone but only in conjunction with actual graph representations. |
---|
55 | They do not alter the physical graph storage, they just give another view of it. |
---|
56 | When the methods of the adaptors are called, they use the underlying |
---|
57 | graph structures and their operations, thus these classes have only negligible |
---|
58 | memory usage and do not perform sophisticated algorithmic actions. |
---|
59 | |
---|
60 | This technique yields convenient tools that help writing compact and elegant |
---|
61 | code, and makes it possible to easily implement complex algorithms based on |
---|
62 | well tested standard components. |
---|
63 | |
---|
64 | For solving the problem introduced above, we could use the follwing code. |
---|
65 | |
---|
66 | \code |
---|
67 | ListDigraph g; |
---|
68 | ReverseDigraph<ListDigraph> rg(g); |
---|
69 | int result = algorithm(rg); |
---|
70 | \endcode |
---|
71 | |
---|
72 | Note that the original digraph \c g remains untouched during the whole |
---|
73 | procedure. |
---|
74 | |
---|
75 | LEMON also provides simple "creator functions" for the adaptor |
---|
76 | classes to make their usage even simpler. |
---|
77 | For example, \ref reverseDigraph() returns an instance of \ref ReverseDigraph, |
---|
78 | thus the above code can be written like this. |
---|
79 | |
---|
80 | \code |
---|
81 | ListDigraph g; |
---|
82 | int result = algorithm(reverseDigraph(g)); |
---|
83 | \endcode |
---|
84 | |
---|
85 | Another essential feature of the adaptors is that their \c Node and \c Arc |
---|
86 | types convert to the original item types. |
---|
87 | Therefore, the maps of the original graph can be used in connection with |
---|
88 | the adaptor. |
---|
89 | |
---|
90 | In the following code, Dijksta's algorithm is run on the reverse oriented |
---|
91 | graph but using the original node and arc maps. |
---|
92 | |
---|
93 | \code |
---|
94 | ListDigraph g; |
---|
95 | ListDigraph::ArcMap length(g); |
---|
96 | ListDigraph::NodeMap dist(g); |
---|
97 | |
---|
98 | ListDigraph::Node s = g.addNode(); |
---|
99 | // add more nodes and arcs |
---|
100 | |
---|
101 | dijkstra(reverseDigraph(g), length).distMap(dist).run(s); |
---|
102 | \endcode |
---|
103 | |
---|
104 | In the above examples, we used \ref ReverseDigraph in such a way that the |
---|
105 | underlying digraph was not changed. However, the adaptor class can even be |
---|
106 | used for modifying the original graph structure. |
---|
107 | It allows adding and deleting arcs or nodes, and these operations are carried |
---|
108 | out by calling suitable functions of the underlying digraph (if it supports |
---|
109 | them). |
---|
110 | |
---|
111 | For this, \ref ReverseDigraph "ReverseDigraph<GR>" has a constructor of the |
---|
112 | following form. |
---|
113 | \code |
---|
114 | ReverseDigraph(GR& gr); |
---|
115 | \endcode |
---|
116 | |
---|
117 | This means that in a situation, when the modification of the original graph |
---|
118 | has to be avoided (e.g. it is given as a const reference), then the adaptor |
---|
119 | class has to be instantiated with \c GR set to be \c const type |
---|
120 | (e.g. <tt>GR = const %ListDigraph</tt>), as in the following example. |
---|
121 | |
---|
122 | \code |
---|
123 | int algorithm1(const ListDigraph& g) { |
---|
124 | ReverseDigraph<const ListDigraph> rg(g); |
---|
125 | return algorithm2(rg); |
---|
126 | } |
---|
127 | \endcode |
---|
128 | |
---|
129 | \note Modification capabilities are not supported for all adaptors. |
---|
130 | E.g. for \ref ResidualDigraph (see \ref sec_other_adaptors "later"), |
---|
131 | this makes no sense. |
---|
132 | |
---|
133 | As a more complex example, let us see how \ref ReverseDigraph can be used |
---|
134 | together with a graph search algorithm to decide whether a directed graph is |
---|
135 | strongly connected or not. |
---|
136 | We exploit the fact the a digraph is strongly connected if and only if |
---|
137 | for an arbitrarily selected node \c u, each other node is reachable from |
---|
138 | \c u (along a directed path) and \c u is reachable from each node. |
---|
139 | The latter condition is the same that each node is reachable from \c u |
---|
140 | in the reversed digraph. |
---|
141 | |
---|
142 | \code |
---|
143 | template <typename Digraph> |
---|
144 | bool stronglyConnected(const Digraph& g) { |
---|
145 | typedef typename Digraph::NodeIt NodeIt; |
---|
146 | NodeIt u(g); |
---|
147 | if (u == INVALID) return true; |
---|
148 | |
---|
149 | // Run BFS on the original digraph |
---|
150 | Bfs<Digraph> bfs(g); |
---|
151 | bfs.run(u); |
---|
152 | for (NodeIt n(g); n != INVALID; ++n) { |
---|
153 | if (!bfs.reached(n)) return false; |
---|
154 | } |
---|
155 | |
---|
156 | // Run BFS on the reverse oriented digraph |
---|
157 | typedef ReverseDigraph<const Digraph> RDigraph; |
---|
158 | RDigraph rg(g); |
---|
159 | Bfs<RDigraph> rbfs(rg); |
---|
160 | rbfs.run(u); |
---|
161 | for (NodeIt n(g); n != INVALID; ++n) { |
---|
162 | if (!rbfs.reached(n)) return false; |
---|
163 | } |
---|
164 | |
---|
165 | return true; |
---|
166 | } |
---|
167 | \endcode |
---|
168 | |
---|
169 | Note that we have to use the adaptor with '<tt>const Digraph</tt>' type, since |
---|
170 | \c g is a \c const reference to the original graph structure. |
---|
171 | The \ref stronglyConnected() function provided in LEMON has a quite |
---|
172 | similar implementation. |
---|
173 | |
---|
174 | |
---|
175 | [SEC]sec_subgraphs[SEC] Subgraph Adaptorts |
---|
176 | |
---|
177 | Another typical requirement is the use of certain subgraphs of a graph, |
---|
178 | or in other words, hiding nodes and/or arcs from a graph. |
---|
179 | LEMON provides several convenient adaptors for these purposes. |
---|
180 | In the following image, a \ref SubDigraph adaptor is applied to an |
---|
181 | underlying digraph structure to obtain a suitable subgraph. |
---|
182 | |
---|
183 | \image html adaptors1.png |
---|
184 | \image latex adaptors1.eps "SubDigraph adaptor" width=\textwidth |
---|
185 | |
---|
186 | \ref FilterArcs can be used when some arcs have to be hidden from a digraph. |
---|
187 | A \e filter \e map has to be given to the constructor, which assign \c bool |
---|
188 | values to the arcs specifying whether they have to be shown or not in the |
---|
189 | subgraph structure. |
---|
190 | Suppose we have a \ref ListDigraph structure \c g. |
---|
191 | Then we can construct a subgraph in which some arcs (\c a1, \c a2 etc.) |
---|
192 | are hidden as follows. |
---|
193 | |
---|
194 | \code |
---|
195 | ListDigraph::ArcMap filter(g, true); |
---|
196 | filter[a1] = false; |
---|
197 | filter[a2] = false; |
---|
198 | // ... |
---|
199 | FilterArcs<ListDigraph> subgraph(g, filter); |
---|
200 | \endcode |
---|
201 | |
---|
202 | The following more complex code runs Dijkstra's algorithm on a digraph |
---|
203 | that is obtained from another digraph by hiding all arcs having negative |
---|
204 | lengths. |
---|
205 | |
---|
206 | \code |
---|
207 | ListDigraph::ArcMap<int> length(g); |
---|
208 | ListDigraph::NodeMap<int> dist(g); |
---|
209 | |
---|
210 | dijkstra(filterArcs( g, lessMap(length, constMap<ListDigraph::Arc>(0)) ), |
---|
211 | length).distMap(dist).run(s); |
---|
212 | \endcode |
---|
213 | |
---|
214 | Note the extensive use of map adaptors and creator functions, which makes |
---|
215 | the code really compact and elegant. |
---|
216 | |
---|
217 | \note Implicit maps and graphs (e.g. created using functions) can only be |
---|
218 | used with the function-type interfaces of the algorithms, since they store |
---|
219 | only references for the used structures. |
---|
220 | |
---|
221 | \ref FilterEdges can be used for hiding edges from an undirected graph (like |
---|
222 | \ref FilterArcs is used for digraphs). \ref FilterNodes serves for filtering |
---|
223 | nodes along with the incident arcs or edges in a directed or undirected graph. |
---|
224 | If both arcs/edges and nodes have to be hidden, then you could use |
---|
225 | \ref SubDigraph or \ref SubGraph adaptors. |
---|
226 | |
---|
227 | \code |
---|
228 | ListGraph ug; |
---|
229 | ListGraph::NodeMap<bool> node_filter(ug); |
---|
230 | ListGraph::EdgeMap<bool> edge_filter(ug); |
---|
231 | |
---|
232 | SubGraph<ListGraph> sg(ug, node_filter, edge_filter); |
---|
233 | \endcode |
---|
234 | |
---|
235 | As you see, we needed two filter maps in this case: one for the nodes and |
---|
236 | another for the edges. If a node is hidden, then all of its incident edges |
---|
237 | are also considered to be hidden independently of their own filter values. |
---|
238 | |
---|
239 | The subgraph adaptors also make it possible to modify the filter values |
---|
240 | even after the construction of the adaptor class, thus the corresponding |
---|
241 | graph items can be hidden or shown on the fly. |
---|
242 | The adaptors store references to the filter maps, thus the map values can be |
---|
243 | set directly and even by using the \c enable(), \c disable() and \c status() |
---|
244 | functions. |
---|
245 | |
---|
246 | \code |
---|
247 | ListDigraph g; |
---|
248 | ListDigraph::Node x = g.addNode(); |
---|
249 | ListDigraph::Node y = g.addNode(); |
---|
250 | ListDigraph::Node z = g.addNode(); |
---|
251 | |
---|
252 | ListDigraph::NodeMap<bool> filter(g, true); |
---|
253 | FilterNodes<ListDigraph> subgraph(g, filter); |
---|
254 | std::cout << countNodes(subgraph) << ", "; |
---|
255 | |
---|
256 | filter[x] = false; |
---|
257 | std::cout << countNodes(subgraph) << ", "; |
---|
258 | |
---|
259 | subgraph.enable(x); |
---|
260 | subgraph.disable(y); |
---|
261 | subgraph.status(z, !subgraph.status(z)); |
---|
262 | std::cout << countNodes(subgraph) << std::endl; |
---|
263 | \endcode |
---|
264 | |
---|
265 | The above example prints out this line. |
---|
266 | \code |
---|
267 | 3, 2, 1 |
---|
268 | \endcode |
---|
269 | |
---|
270 | Similarly to \ref ReverseDigraph, the subgraph adaptors also allow the |
---|
271 | modification of the underlying graph structures unless the graph template |
---|
272 | parameter is set to be \c const type. |
---|
273 | Moreover the item types of the original graphs and the subgraphs are |
---|
274 | convertible to each other. |
---|
275 | |
---|
276 | The iterators of the subgraph adaptors use the iterators of the original |
---|
277 | graph structures in such a way that each item with \c false filter value |
---|
278 | is skipped. If both the node and arc sets are filtered, then the arc iterators |
---|
279 | check for each arc the status of its end nodes in addition to its own assigned |
---|
280 | filter value. If the arc or one of its end nodes is hidden, then the arc |
---|
281 | is left out and the next arc is considered. |
---|
282 | (It is the same for edges in undirected graphs.) |
---|
283 | Therefore, the iterators of these adaptors are significantly slower than the |
---|
284 | original iterators. |
---|
285 | |
---|
286 | Using adaptors, these efficiency aspects should be kept in mind. |
---|
287 | For example, if rather complex algorithms have to be performed on a |
---|
288 | subgraph (e.g. the nodes and arcs need to be traversed several times), |
---|
289 | then it could worth copying the altered graph into an efficient |
---|
290 | structure (e.g. \ref StaticDigraph) and run the algorithm on it. |
---|
291 | Note that the adaptor classes can also be used for doing this easily, |
---|
292 | without having to copy the graph manually, as shown in the following |
---|
293 | example. |
---|
294 | |
---|
295 | \code |
---|
296 | ListDigraph g; |
---|
297 | ListDigraph::NodeMap<bool> filter_map(g); |
---|
298 | // construct the graph and fill the filter map |
---|
299 | |
---|
300 | { |
---|
301 | StaticDigraph tmp_graph; |
---|
302 | ListDigraph::NodeMap<StaticDigraph::Node> node_ref(g); |
---|
303 | digraphCopy(filterNodes(g, filter_map), tmp_graph) |
---|
304 | .nodeRef(node_ref).run(); |
---|
305 | |
---|
306 | // use tmp_graph |
---|
307 | } |
---|
308 | \endcode |
---|
309 | |
---|
310 | \note Using \ref ReverseDigraph could be as efficient as working with the |
---|
311 | original graph, but most of the adaptors cannot be so fast, of course. |
---|
312 | |
---|
313 | |
---|
314 | [SEC]sec_other_adaptors[SEC] Other Graph Adaptors |
---|
315 | |
---|
316 | Two other practical adaptors are \ref Undirector and \ref Orienter. |
---|
317 | \ref Undirector makes an undirected graph from a digraph disregarding the |
---|
318 | orientations of the arcs. More precisely, an arc of the original digraph |
---|
319 | is considered as an edge (and two arcs, as well) in the adaptor. |
---|
320 | \ref Orienter can be used for the reverse alteration, it assigns a certain |
---|
321 | orientation to each edge of an undirected graph to form a directed graph. |
---|
322 | A \c bool edge map of the underlying graph must be given to the constructor |
---|
323 | of the class, which define the direction of the arcs in the created adaptor |
---|
324 | (with respect to the inherent orientation of the original edges). |
---|
325 | |
---|
326 | \code |
---|
327 | ListGraph graph; |
---|
328 | ListGraph::EdgeMap<bool> dir_map(graph, true); |
---|
329 | Orienter<ListGraph> directed_graph(graph, dir_map); |
---|
330 | \endcode |
---|
331 | |
---|
332 | Sine the adaptor classes conform to the \ref graph_concepts "graph concepts", |
---|
333 | we can even apply an adaptor to another one. |
---|
334 | The following image illustrates a situation when a \ref SubDigraph and an |
---|
335 | \ref Undirector adaptor is applied to a digraph. |
---|
336 | |
---|
337 | \image html adaptors2.png |
---|
338 | \image latex adaptors2.eps "Arc disjoint paths" width=\textwidth |
---|
339 | |
---|
340 | LEMON also provides some more complex adaptors, for |
---|
341 | instance, \ref SplitNodes, which can be used for splitting each node of a |
---|
342 | directed graph into an in-node and an out-node. |
---|
343 | Formally, the adaptor replaces each node u in the graph with two nodes, |
---|
344 | namely u<sub>in</sub> and u<sub>out</sub>. Each arc (u,v) of the original |
---|
345 | graph will correspond to an arc (u<sub>out</sub>,v<sub>in</sub>). |
---|
346 | The adaptor also adds an additional bind arc (u<sub>in</sub>,u<sub>out</sub>) |
---|
347 | for each node u of the original digraph. |
---|
348 | |
---|
349 | The aim of this class is to assign costs or capacities to the nodes when using |
---|
350 | algorithms which would otherwise consider arc costs or capacities only. |
---|
351 | For example, let us suppose that we have a digraph \c g with costs assigned to |
---|
352 | both the nodes and the arcs. Then Dijkstra's algorithm can be used in |
---|
353 | connection with \ref SplitNodes as follows. |
---|
354 | |
---|
355 | \code |
---|
356 | typedef SplitNodes<ListDigraph> SplitGraph; |
---|
357 | SplitGraph sg(g); |
---|
358 | SplitGraph::CombinedArcMap<NodeCostMap, ArcCostMap> |
---|
359 | combined_cost(node_cost, arc_cost); |
---|
360 | SplitGraph::NodeMap<double> dist(sg); |
---|
361 | dijkstra(sg, combined_cost).distMap(dist).run(sg.outNode(u)); |
---|
362 | \endcode |
---|
363 | |
---|
364 | \note This problem can also be solved using map adaptors to create |
---|
365 | an implicit arc map that assigns for each arc the sum of its cost |
---|
366 | and the cost of its target node. This map can be used with the original |
---|
367 | graph more efficiently than using the above solution. |
---|
368 | |
---|
369 | Another nice application is the problem of finding disjoint paths in |
---|
370 | a digraph. |
---|
371 | The maximum number of \e edge \e disjoint paths from a source node to |
---|
372 | a sink node in a digraph can be easily computed using a maximum flow |
---|
373 | algorithm with all arc capacities set to 1. |
---|
374 | For example, in the following digraph, four arc disjoint paths can be found |
---|
375 | from the node on the left to the node on the right. |
---|
376 | |
---|
377 | \image html splitnodes1.png |
---|
378 | \image latex splitnodes1.eps "Arc disjoint paths" width=\textwidth |
---|
379 | |
---|
380 | On the other hand, \e node \e disjoint paths cannot be found directly |
---|
381 | using a standard algorithm. |
---|
382 | However, \ref SplitNodes adaptor makes it really simple. |
---|
383 | If a maximum flow computation is performed on this adaptor, then the |
---|
384 | bottleneck of the flow (i.e. the minimum cut) will be formed by bind arcs, |
---|
385 | thus the found flow will correspond to the union of some node disjoint |
---|
386 | paths in terms of the original digraph. |
---|
387 | For example, in the above digraph, there are only three node disjoint paths. |
---|
388 | |
---|
389 | \image html splitnodes2.png |
---|
390 | \image latex splitnodes2.eps "Node disjoint paths" width=\textwidth |
---|
391 | |
---|
392 | In flow, circulation and matching problems, the residual network is of |
---|
393 | particular importance, which is implemented in \ref ResidualDigraph. |
---|
394 | Combining this adaptor with various algorithms, a range of weighted and |
---|
395 | cardinality optimization methods can be implemented easily. |
---|
396 | |
---|
397 | To construct a residual network, a digraph structure, a flow map and a |
---|
398 | capacity map have to be given to the constructor of the adaptor as shown |
---|
399 | in the following code. |
---|
400 | |
---|
401 | \code |
---|
402 | ListDigraph g; |
---|
403 | ListDigraph::ArcMap<int> flow(g); |
---|
404 | ListDigraph::ArcMap<int> capacity(g); |
---|
405 | |
---|
406 | ResidualDigraph<ListDigraph> res_graph(g, capacity, flow); |
---|
407 | \endcode |
---|
408 | |
---|
409 | \note In fact, this class is implemented using two other adaptors: |
---|
410 | \ref Undirector and \ref FilterArcs. |
---|
411 | |
---|
412 | [TRAILER] |
---|
413 | */ |
---|
414 | } |
---|