[906] | 1 | /* -*- C++ -*- |
---|
| 2 | * |
---|
[1956] | 3 | * This file is a part of LEMON, a generic C++ optimization library |
---|
| 4 | * |
---|
| 5 | * Copyright (C) 2003-2006 |
---|
| 6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
---|
[1359] | 7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
---|
[906] | 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 | |
---|
[921] | 19 | #ifndef LEMON_KRUSKAL_H |
---|
| 20 | #define LEMON_KRUSKAL_H |
---|
[810] | 21 | |
---|
| 22 | #include <algorithm> |
---|
[1942] | 23 | #include <vector> |
---|
[921] | 24 | #include <lemon/unionfind.h> |
---|
[1942] | 25 | #include <lemon/utility.h> |
---|
[1979] | 26 | #include <lemon/traits.h> |
---|
[810] | 27 | |
---|
| 28 | /** |
---|
| 29 | @defgroup spantree Minimum Cost Spanning Tree Algorithms |
---|
| 30 | @ingroup galgs |
---|
| 31 | \brief This group containes the algorithms for finding a minimum cost spanning |
---|
| 32 | tree in a graph |
---|
| 33 | |
---|
| 34 | This group containes the algorithms for finding a minimum cost spanning |
---|
| 35 | tree in a graph |
---|
| 36 | */ |
---|
| 37 | |
---|
| 38 | ///\ingroup spantree |
---|
| 39 | ///\file |
---|
| 40 | ///\brief Kruskal's algorithm to compute a minimum cost tree |
---|
| 41 | /// |
---|
| 42 | ///Kruskal's algorithm to compute a minimum cost tree. |
---|
[1557] | 43 | /// |
---|
| 44 | ///\todo The file still needs some clean-up. |
---|
[810] | 45 | |
---|
[921] | 46 | namespace lemon { |
---|
[810] | 47 | |
---|
| 48 | /// \addtogroup spantree |
---|
| 49 | /// @{ |
---|
| 50 | |
---|
| 51 | /// Kruskal's algorithm to find a minimum cost tree of a graph. |
---|
| 52 | |
---|
| 53 | /// This function runs Kruskal's algorithm to find a minimum cost tree. |
---|
[1557] | 54 | /// Due to hard C++ hacking, it accepts various input and output types. |
---|
| 55 | /// |
---|
[1555] | 56 | /// \param g The graph the algorithm runs on. |
---|
| 57 | /// It can be either \ref concept::StaticGraph "directed" or |
---|
[1909] | 58 | /// \ref concept::UGraph "undirected". |
---|
[1555] | 59 | /// If the graph is directed, the algorithm consider it to be |
---|
| 60 | /// undirected by disregarding the direction of the edges. |
---|
[810] | 61 | /// |
---|
[1557] | 62 | /// \param in This object is used to describe the edge costs. It can be one |
---|
| 63 | /// of the following choices. |
---|
| 64 | /// - An STL compatible 'Forward Container' |
---|
[824] | 65 | /// with <tt>std::pair<GR::Edge,X></tt> as its <tt>value_type</tt>, |
---|
[1557] | 66 | /// where \c X is the type of the costs. The pairs indicates the edges along |
---|
| 67 | /// with the assigned cost. <em>They must be in a |
---|
| 68 | /// cost-ascending order.</em> |
---|
| 69 | /// - Any readable Edge map. The values of the map indicate the edge costs. |
---|
[810] | 70 | /// |
---|
[1557] | 71 | /// \retval out Here we also have a choise. |
---|
| 72 | /// - Is can be a writable \c bool edge map. |
---|
[810] | 73 | /// After running the algorithm |
---|
| 74 | /// this will contain the found minimum cost spanning tree: the value of an |
---|
| 75 | /// edge will be set to \c true if it belongs to the tree, otherwise it will |
---|
| 76 | /// be set to \c false. The value of each edge will be set exactly once. |
---|
[1557] | 77 | /// - It can also be an iteraror of an STL Container with |
---|
| 78 | /// <tt>GR::Edge</tt> as its <tt>value_type</tt>. |
---|
| 79 | /// The algorithm copies the elements of the found tree into this sequence. |
---|
| 80 | /// For example, if we know that the spanning tree of the graph \c g has |
---|
[1603] | 81 | /// say 53 edges, then |
---|
[1557] | 82 | /// we can put its edges into a STL vector \c tree with a code like this. |
---|
[1946] | 83 | ///\code |
---|
[1557] | 84 | /// std::vector<Edge> tree(53); |
---|
| 85 | /// kruskal(g,cost,tree.begin()); |
---|
[1946] | 86 | ///\endcode |
---|
[1557] | 87 | /// Or if we don't know in advance the size of the tree, we can write this. |
---|
[1946] | 88 | ///\code |
---|
[1557] | 89 | /// std::vector<Edge> tree; |
---|
| 90 | /// kruskal(g,cost,std::back_inserter(tree)); |
---|
[1946] | 91 | ///\endcode |
---|
[810] | 92 | /// |
---|
| 93 | /// \return The cost of the found tree. |
---|
[1449] | 94 | /// |
---|
[1631] | 95 | /// \warning If kruskal is run on an |
---|
[1909] | 96 | /// \ref lemon::concept::UGraph "undirected graph", be sure that the |
---|
[1603] | 97 | /// map storing the tree is also undirected |
---|
[1909] | 98 | /// (e.g. ListUGraph::UEdgeMap<bool>, otherwise the values of the |
---|
[1603] | 99 | /// half of the edges will not be set. |
---|
| 100 | /// |
---|
[1449] | 101 | /// \todo Discuss the case of undirected graphs: In this case the algorithm |
---|
[1909] | 102 | /// also require <tt>Edge</tt>s instead of <tt>UEdge</tt>s, as some |
---|
[1449] | 103 | /// people would expect. So, one should be careful not to add both of the |
---|
[1909] | 104 | /// <tt>Edge</tt>s belonging to a certain <tt>UEdge</tt>. |
---|
[1570] | 105 | /// (\ref kruskal() and \ref KruskalMapInput are kind enough to do so.) |
---|
[810] | 106 | |
---|
[1557] | 107 | #ifdef DOXYGEN |
---|
[824] | 108 | template <class GR, class IN, class OUT> |
---|
| 109 | typename IN::value_type::second_type |
---|
[1547] | 110 | kruskal(GR const& g, IN const& in, |
---|
[1557] | 111 | OUT& out) |
---|
| 112 | #else |
---|
| 113 | template <class GR, class IN, class OUT> |
---|
| 114 | typename IN::value_type::second_type |
---|
| 115 | kruskal(GR const& g, IN const& in, |
---|
| 116 | OUT& out, |
---|
| 117 | // typename IN::value_type::first_type = typename GR::Edge() |
---|
| 118 | // ,typename OUT::Key = OUT::Key() |
---|
| 119 | // //,typename OUT::Key = typename GR::Edge() |
---|
| 120 | const typename IN::value_type::first_type * = |
---|
| 121 | (const typename IN::value_type::first_type *)(0), |
---|
| 122 | const typename OUT::Key * = (const typename OUT::Key *)(0) |
---|
| 123 | ) |
---|
| 124 | #endif |
---|
[810] | 125 | { |
---|
[824] | 126 | typedef typename IN::value_type::second_type EdgeCost; |
---|
| 127 | typedef typename GR::template NodeMap<int> NodeIntMap; |
---|
| 128 | typedef typename GR::Node Node; |
---|
[810] | 129 | |
---|
[1547] | 130 | NodeIntMap comp(g, -1); |
---|
[810] | 131 | UnionFind<Node,NodeIntMap> uf(comp); |
---|
| 132 | |
---|
| 133 | EdgeCost tot_cost = 0; |
---|
[824] | 134 | for (typename IN::const_iterator p = in.begin(); |
---|
[810] | 135 | p!=in.end(); ++p ) { |
---|
[1547] | 136 | if ( uf.join(g.target((*p).first), |
---|
| 137 | g.source((*p).first)) ) { |
---|
[810] | 138 | out.set((*p).first, true); |
---|
| 139 | tot_cost += (*p).second; |
---|
| 140 | } |
---|
| 141 | else { |
---|
| 142 | out.set((*p).first, false); |
---|
| 143 | } |
---|
| 144 | } |
---|
| 145 | return tot_cost; |
---|
| 146 | } |
---|
| 147 | |
---|
[1557] | 148 | |
---|
| 149 | /// @} |
---|
| 150 | |
---|
| 151 | |
---|
[810] | 152 | /* A work-around for running Kruskal with const-reference bool maps... */ |
---|
| 153 | |
---|
[885] | 154 | /// Helper class for calling kruskal with "constant" output map. |
---|
| 155 | |
---|
| 156 | /// Helper class for calling kruskal with output maps constructed |
---|
| 157 | /// on-the-fly. |
---|
[810] | 158 | /// |
---|
[885] | 159 | /// A typical examle is the following call: |
---|
[1547] | 160 | /// <tt>kruskal(g, some_input, makeSequenceOutput(iterator))</tt>. |
---|
[885] | 161 | /// Here, the third argument is a temporary object (which wraps around an |
---|
| 162 | /// iterator with a writable bool map interface), and thus by rules of C++ |
---|
| 163 | /// is a \c const object. To enable call like this exist this class and |
---|
| 164 | /// the prototype of the \ref kruskal() function with <tt>const& OUT</tt> |
---|
| 165 | /// third argument. |
---|
[824] | 166 | template<class Map> |
---|
[810] | 167 | class NonConstMapWr { |
---|
| 168 | const Map &m; |
---|
| 169 | public: |
---|
[1557] | 170 | typedef typename Map::Key Key; |
---|
[987] | 171 | typedef typename Map::Value Value; |
---|
[810] | 172 | |
---|
| 173 | NonConstMapWr(const Map &_m) : m(_m) {} |
---|
| 174 | |
---|
[987] | 175 | template<class Key> |
---|
| 176 | void set(Key const& k, Value const &v) const { m.set(k,v); } |
---|
[810] | 177 | }; |
---|
| 178 | |
---|
[824] | 179 | template <class GR, class IN, class OUT> |
---|
[810] | 180 | inline |
---|
[885] | 181 | typename IN::value_type::second_type |
---|
[1557] | 182 | kruskal(GR const& g, IN const& edges, OUT const& out_map, |
---|
| 183 | // typename IN::value_type::first_type = typename GR::Edge(), |
---|
| 184 | // typename OUT::Key = GR::Edge() |
---|
| 185 | const typename IN::value_type::first_type * = |
---|
| 186 | (const typename IN::value_type::first_type *)(0), |
---|
| 187 | const typename OUT::Key * = (const typename OUT::Key *)(0) |
---|
| 188 | ) |
---|
[810] | 189 | { |
---|
[824] | 190 | NonConstMapWr<OUT> map_wr(out_map); |
---|
[1547] | 191 | return kruskal(g, edges, map_wr); |
---|
[810] | 192 | } |
---|
| 193 | |
---|
| 194 | /* ** ** Input-objects ** ** */ |
---|
| 195 | |
---|
[1274] | 196 | /// Kruskal's input source. |
---|
[1557] | 197 | |
---|
[1274] | 198 | /// Kruskal's input source. |
---|
[810] | 199 | /// |
---|
[1570] | 200 | /// In most cases you possibly want to use the \ref kruskal() instead. |
---|
[810] | 201 | /// |
---|
| 202 | /// \sa makeKruskalMapInput() |
---|
| 203 | /// |
---|
[824] | 204 | ///\param GR The type of the graph the algorithm runs on. |
---|
[810] | 205 | ///\param Map An edge map containing the cost of the edges. |
---|
| 206 | ///\par |
---|
| 207 | ///The cost type can be any type satisfying |
---|
| 208 | ///the STL 'LessThan comparable' |
---|
| 209 | ///concept if it also has an operator+() implemented. (It is necessary for |
---|
| 210 | ///computing the total cost of the tree). |
---|
| 211 | /// |
---|
[824] | 212 | template<class GR, class Map> |
---|
[810] | 213 | class KruskalMapInput |
---|
[824] | 214 | : public std::vector< std::pair<typename GR::Edge, |
---|
[987] | 215 | typename Map::Value> > { |
---|
[810] | 216 | |
---|
| 217 | public: |
---|
[824] | 218 | typedef std::vector< std::pair<typename GR::Edge, |
---|
[987] | 219 | typename Map::Value> > Parent; |
---|
[810] | 220 | typedef typename Parent::value_type value_type; |
---|
| 221 | |
---|
| 222 | private: |
---|
| 223 | class comparePair { |
---|
| 224 | public: |
---|
| 225 | bool operator()(const value_type& a, |
---|
| 226 | const value_type& b) { |
---|
| 227 | return a.second < b.second; |
---|
| 228 | } |
---|
| 229 | }; |
---|
| 230 | |
---|
[1449] | 231 | template<class _GR> |
---|
[1979] | 232 | typename enable_if<UndirectedTagIndicator<_GR>,void>::type |
---|
[1547] | 233 | fillWithEdges(const _GR& g, const Map& m,dummy<0> = 0) |
---|
[1449] | 234 | { |
---|
[1909] | 235 | for(typename GR::UEdgeIt e(g);e!=INVALID;++e) |
---|
[1679] | 236 | push_back(value_type(g.direct(e, true), m[e])); |
---|
[1449] | 237 | } |
---|
| 238 | |
---|
| 239 | template<class _GR> |
---|
[1979] | 240 | typename disable_if<UndirectedTagIndicator<_GR>,void>::type |
---|
[1547] | 241 | fillWithEdges(const _GR& g, const Map& m,dummy<1> = 1) |
---|
[1449] | 242 | { |
---|
[1547] | 243 | for(typename GR::EdgeIt e(g);e!=INVALID;++e) |
---|
[1449] | 244 | push_back(value_type(e, m[e])); |
---|
| 245 | } |
---|
| 246 | |
---|
| 247 | |
---|
[810] | 248 | public: |
---|
| 249 | |
---|
| 250 | void sort() { |
---|
| 251 | std::sort(this->begin(), this->end(), comparePair()); |
---|
| 252 | } |
---|
| 253 | |
---|
[1547] | 254 | KruskalMapInput(GR const& g, Map const& m) { |
---|
| 255 | fillWithEdges(g,m); |
---|
[810] | 256 | sort(); |
---|
| 257 | } |
---|
| 258 | }; |
---|
| 259 | |
---|
| 260 | /// Creates a KruskalMapInput object for \ref kruskal() |
---|
| 261 | |
---|
[1274] | 262 | /// It makes easier to use |
---|
[810] | 263 | /// \ref KruskalMapInput by making it unnecessary |
---|
| 264 | /// to explicitly give the type of the parameters. |
---|
| 265 | /// |
---|
| 266 | /// In most cases you possibly |
---|
[1570] | 267 | /// want to use \ref kruskal() instead. |
---|
[810] | 268 | /// |
---|
[1547] | 269 | ///\param g The type of the graph the algorithm runs on. |
---|
[810] | 270 | ///\param m An edge map containing the cost of the edges. |
---|
| 271 | ///\par |
---|
| 272 | ///The cost type can be any type satisfying the |
---|
| 273 | ///STL 'LessThan Comparable' |
---|
| 274 | ///concept if it also has an operator+() implemented. (It is necessary for |
---|
| 275 | ///computing the total cost of the tree). |
---|
| 276 | /// |
---|
| 277 | ///\return An appropriate input source for \ref kruskal(). |
---|
| 278 | /// |
---|
[824] | 279 | template<class GR, class Map> |
---|
[810] | 280 | inline |
---|
[1547] | 281 | KruskalMapInput<GR,Map> makeKruskalMapInput(const GR &g,const Map &m) |
---|
[810] | 282 | { |
---|
[1547] | 283 | return KruskalMapInput<GR,Map>(g,m); |
---|
[810] | 284 | } |
---|
| 285 | |
---|
| 286 | |
---|
[885] | 287 | |
---|
| 288 | /* ** ** Output-objects: simple writable bool maps ** ** */ |
---|
[810] | 289 | |
---|
[885] | 290 | |
---|
| 291 | |
---|
[810] | 292 | /// A writable bool-map that makes a sequence of "true" keys |
---|
| 293 | |
---|
| 294 | /// A writable bool-map that creates a sequence out of keys that receives |
---|
| 295 | /// the value "true". |
---|
[885] | 296 | /// |
---|
| 297 | /// \sa makeKruskalSequenceOutput() |
---|
| 298 | /// |
---|
| 299 | /// Very often, when looking for a min cost spanning tree, we want as |
---|
| 300 | /// output a container containing the edges of the found tree. For this |
---|
| 301 | /// purpose exist this class that wraps around an STL iterator with a |
---|
| 302 | /// writable bool map interface. When a key gets value "true" this key |
---|
| 303 | /// is added to sequence pointed by the iterator. |
---|
| 304 | /// |
---|
| 305 | /// A typical usage: |
---|
[1946] | 306 | ///\code |
---|
[885] | 307 | /// std::vector<Graph::Edge> v; |
---|
| 308 | /// kruskal(g, input, makeKruskalSequenceOutput(back_inserter(v))); |
---|
[1946] | 309 | ///\endcode |
---|
[885] | 310 | /// |
---|
| 311 | /// For the most common case, when the input is given by a simple edge |
---|
| 312 | /// map and the output is a sequence of the tree edges, a special |
---|
| 313 | /// wrapper function exists: \ref kruskalEdgeMap_IteratorOut(). |
---|
| 314 | /// |
---|
[987] | 315 | /// \warning Not a regular property map, as it doesn't know its Key |
---|
[885] | 316 | |
---|
[824] | 317 | template<class Iterator> |
---|
[885] | 318 | class KruskalSequenceOutput { |
---|
[810] | 319 | mutable Iterator it; |
---|
| 320 | |
---|
| 321 | public: |
---|
[1942] | 322 | typedef typename std::iterator_traits<Iterator>::value_type Key; |
---|
[987] | 323 | typedef bool Value; |
---|
[810] | 324 | |
---|
[885] | 325 | KruskalSequenceOutput(Iterator const &_it) : it(_it) {} |
---|
[810] | 326 | |
---|
[987] | 327 | template<typename Key> |
---|
| 328 | void set(Key const& k, bool v) const { if(v) {*it=k; ++it;} } |
---|
[810] | 329 | }; |
---|
| 330 | |
---|
[824] | 331 | template<class Iterator> |
---|
[810] | 332 | inline |
---|
[885] | 333 | KruskalSequenceOutput<Iterator> |
---|
| 334 | makeKruskalSequenceOutput(Iterator it) { |
---|
| 335 | return KruskalSequenceOutput<Iterator>(it); |
---|
[810] | 336 | } |
---|
| 337 | |
---|
[885] | 338 | |
---|
| 339 | |
---|
[810] | 340 | /* ** ** Wrapper funtions ** ** */ |
---|
| 341 | |
---|
[1557] | 342 | // \brief Wrapper function to kruskal(). |
---|
| 343 | // Input is from an edge map, output is a plain bool map. |
---|
| 344 | // |
---|
| 345 | // Wrapper function to kruskal(). |
---|
| 346 | // Input is from an edge map, output is a plain bool map. |
---|
| 347 | // |
---|
| 348 | // \param g The type of the graph the algorithm runs on. |
---|
| 349 | // \param in An edge map containing the cost of the edges. |
---|
| 350 | // \par |
---|
| 351 | // The cost type can be any type satisfying the |
---|
| 352 | // STL 'LessThan Comparable' |
---|
| 353 | // concept if it also has an operator+() implemented. (It is necessary for |
---|
| 354 | // computing the total cost of the tree). |
---|
| 355 | // |
---|
| 356 | // \retval out This must be a writable \c bool edge map. |
---|
| 357 | // After running the algorithm |
---|
| 358 | // this will contain the found minimum cost spanning tree: the value of an |
---|
| 359 | // edge will be set to \c true if it belongs to the tree, otherwise it will |
---|
| 360 | // be set to \c false. The value of each edge will be set exactly once. |
---|
| 361 | // |
---|
| 362 | // \return The cost of the found tree. |
---|
[810] | 363 | |
---|
[824] | 364 | template <class GR, class IN, class RET> |
---|
[810] | 365 | inline |
---|
[987] | 366 | typename IN::Value |
---|
[1557] | 367 | kruskal(GR const& g, |
---|
| 368 | IN const& in, |
---|
| 369 | RET &out, |
---|
| 370 | // typename IN::Key = typename GR::Edge(), |
---|
| 371 | //typename IN::Key = typename IN::Key (), |
---|
| 372 | // typename RET::Key = typename GR::Edge() |
---|
| 373 | const typename IN::Key * = (const typename IN::Key *)(0), |
---|
| 374 | const typename RET::Key * = (const typename RET::Key *)(0) |
---|
| 375 | ) |
---|
| 376 | { |
---|
[1547] | 377 | return kruskal(g, |
---|
| 378 | KruskalMapInput<GR,IN>(g,in), |
---|
[810] | 379 | out); |
---|
| 380 | } |
---|
| 381 | |
---|
[1557] | 382 | // \brief Wrapper function to kruskal(). |
---|
| 383 | // Input is from an edge map, output is an STL Sequence. |
---|
| 384 | // |
---|
| 385 | // Wrapper function to kruskal(). |
---|
| 386 | // Input is from an edge map, output is an STL Sequence. |
---|
| 387 | // |
---|
| 388 | // \param g The type of the graph the algorithm runs on. |
---|
| 389 | // \param in An edge map containing the cost of the edges. |
---|
| 390 | // \par |
---|
| 391 | // The cost type can be any type satisfying the |
---|
| 392 | // STL 'LessThan Comparable' |
---|
| 393 | // concept if it also has an operator+() implemented. (It is necessary for |
---|
| 394 | // computing the total cost of the tree). |
---|
| 395 | // |
---|
| 396 | // \retval out This must be an iteraror of an STL Container with |
---|
| 397 | // <tt>GR::Edge</tt> as its <tt>value_type</tt>. |
---|
| 398 | // The algorithm copies the elements of the found tree into this sequence. |
---|
| 399 | // For example, if we know that the spanning tree of the graph \c g has |
---|
[1603] | 400 | // say 53 edges, then |
---|
[1557] | 401 | // we can put its edges into a STL vector \c tree with a code like this. |
---|
[1946] | 402 | //\code |
---|
[1557] | 403 | // std::vector<Edge> tree(53); |
---|
[1570] | 404 | // kruskal(g,cost,tree.begin()); |
---|
[1946] | 405 | //\endcode |
---|
[1557] | 406 | // Or if we don't know in advance the size of the tree, we can write this. |
---|
[1946] | 407 | //\code |
---|
[1557] | 408 | // std::vector<Edge> tree; |
---|
[1570] | 409 | // kruskal(g,cost,std::back_inserter(tree)); |
---|
[1946] | 410 | //\endcode |
---|
[1557] | 411 | // |
---|
| 412 | // \return The cost of the found tree. |
---|
| 413 | // |
---|
| 414 | // \bug its name does not follow the coding style. |
---|
[885] | 415 | |
---|
[824] | 416 | template <class GR, class IN, class RET> |
---|
[810] | 417 | inline |
---|
[987] | 418 | typename IN::Value |
---|
[1557] | 419 | kruskal(const GR& g, |
---|
| 420 | const IN& in, |
---|
| 421 | RET out, |
---|
| 422 | const typename RET::value_type * = |
---|
| 423 | (const typename RET::value_type *)(0) |
---|
| 424 | ) |
---|
[810] | 425 | { |
---|
[885] | 426 | KruskalSequenceOutput<RET> _out(out); |
---|
[1547] | 427 | return kruskal(g, KruskalMapInput<GR,IN>(g, in), _out); |
---|
[810] | 428 | } |
---|
[1557] | 429 | |
---|
[1942] | 430 | template <class GR, class IN, class RET> |
---|
| 431 | inline |
---|
| 432 | typename IN::Value |
---|
| 433 | kruskal(const GR& g, |
---|
| 434 | const IN& in, |
---|
| 435 | RET *out |
---|
| 436 | ) |
---|
| 437 | { |
---|
| 438 | KruskalSequenceOutput<RET*> _out(out); |
---|
| 439 | return kruskal(g, KruskalMapInput<GR,IN>(g, in), _out); |
---|
| 440 | } |
---|
| 441 | |
---|
[810] | 442 | /// @} |
---|
| 443 | |
---|
[921] | 444 | } //namespace lemon |
---|
[810] | 445 | |
---|
[921] | 446 | #endif //LEMON_KRUSKAL_H |
---|