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1 | .. Copyright (C) 2004-2008 The Trustees of Indiana University. |
2 | Use, modification and distribution is subject to the Boost Software | |
3 | License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at | |
4 | http://www.boost.org/LICENSE_1_0.txt) | |
5 | ||
6 | ============================ | |
7 | |Logo| Minimum Spanning Tree | |
8 | ============================ | |
9 | ||
10 | The Parallel BGL contains four `minimum spanning tree`_ (MST) | |
11 | algorithms [DG98]_ for use on undirected, weighted, distributed | |
12 | graphs. The graphs need not be connected: each algorithm will compute | |
13 | a minimum spanning forest (MSF) when provided with a disconnected | |
14 | graph. | |
15 | ||
16 | The interface to each of the four algorithms is similar to the | |
17 | implementation of 'Kruskal's algorithm'_ in the sequential BGL. Each | |
18 | accepts, at a minimum, a graph, a weight map, and an output | |
19 | iterator. The edges of the MST (or MSF) will be output via the output | |
20 | iterator on process 0: other processes may receive edges on their | |
21 | output iterators, but the set may not be complete, depending on the | |
22 | algorithm. The algorithm parameters are documented together, because | |
23 | they do not vary greatly. See the section `Selecting an MST | |
24 | algorithm`_ for advice on algorithm selection. | |
25 | ||
26 | The graph itself must model the `Vertex List Graph`_ concept and the | |
27 | Distributed Edge List Graph concept. Since the most common | |
28 | distributed graph structure, the `distributed adjacency list`_, only | |
29 | models the Distributed Vertex List Graph concept, it may only be used | |
30 | with these algorithms when wrapped in a suitable adaptor, such as the | |
31 | `vertex_list_adaptor`_. | |
32 | ||
33 | .. contents:: | |
34 | ||
35 | Where Defined | |
36 | ------------- | |
37 | <``boost/graph/distributed/dehne_gotz_min_spanning_tree.hpp``> | |
38 | ||
39 | Parameters | |
40 | ---------- | |
41 | ||
42 | IN: ``Graph& g`` | |
43 | The graph type must be a model of `Vertex List Graph`_ and | |
44 | `Distributed Edge List Graph`_. | |
45 | ||
46 | ||
47 | ||
48 | IN/OUT: ``WeightMap weight`` | |
49 | The weight map must be a `Distributed Property Map`_ and a `Readable | |
50 | Property Map`_ whose key type is the edge descriptor of the graph | |
51 | and whose value type is numerical. | |
52 | ||
53 | ||
54 | ||
55 | IN/OUT: ``OutputIterator out`` | |
56 | The output iterator through which the edges of the MSF will be | |
57 | written. Must be capable of accepting edge descriptors for output. | |
58 | ||
59 | ||
60 | ||
61 | ||
62 | IN: ``VertexIndexMap index`` | |
63 | A mapping from vertex descriptors to indices in the range *[0, | |
64 | num_vertices(g))*. This must be a `Readable Property Map`_ whose | |
65 | key type is a vertex descriptor and whose value type is an integral | |
66 | type, typically the ``vertices_size_type`` of the graph. | |
67 | ||
68 | **Default:** ``get(vertex_index, g)`` | |
69 | ||
70 | ||
71 | IN/UTIL: ``RankMap rank_map`` | |
72 | Stores the rank of each vertex, which is used for maintaining | |
73 | union-find data structures. This must be a `Read/Write Property Map`_ | |
74 | whose key type is a vertex descriptor and whose value type is an | |
75 | integral type. | |
76 | ||
77 | **Default:** An ``iterator_property_map`` built from an STL vector | |
78 | of the ``vertices_size_type`` of the graph and the vertex index map. | |
79 | ||
80 | ||
81 | IN/UTIL: ``ParentMap parent_map`` | |
82 | Stores the parent (representative) of each vertex, which is used for | |
83 | maintaining union-find data structures. This must be a `Read/Write | |
84 | Property Map`_ whose key type is a vertex descriptor and whose value | |
85 | type is also a vertex descriptor. | |
86 | ||
87 | **Default:** An ``iterator_property_map`` built from an STL vector | |
88 | of the ``vertex_descriptor`` of the graph and the vertex index map. | |
89 | ||
90 | ||
91 | IN/UTIL: ``SupervertexMap supervertex_map`` | |
92 | Stores the supervertex index of each vertex, which is used for | |
93 | maintaining the supervertex list data structures. This must be a | |
94 | `Read/Write Property Map`_ whose key type is a vertex descriptor and | |
95 | whose value type is an integral type. | |
96 | ||
97 | **Default:** An ``iterator_property_map`` built from an STL vector | |
98 | of the ``vertices_size_type`` of the graph and the vertex index map. | |
99 | ||
100 | ||
101 | ||
102 | ``dense_boruvka_minimum_spanning_tree`` | |
103 | --------------------------------------- | |
104 | ||
105 | :: | |
106 | ||
107 | namespace graph { | |
108 | template<typename Graph, typename WeightMap, typename OutputIterator, | |
109 | typename VertexIndexMap, typename RankMap, typename ParentMap, | |
110 | typename SupervertexMap> | |
111 | OutputIterator | |
112 | dense_boruvka_minimum_spanning_tree(const Graph& g, WeightMap weight_map, | |
113 | OutputIterator out, | |
114 | VertexIndexMap index, | |
115 | RankMap rank_map, ParentMap parent_map, | |
116 | SupervertexMap supervertex_map); | |
117 | ||
118 | template<typename Graph, typename WeightMap, typename OutputIterator, | |
119 | typename VertexIndex> | |
120 | OutputIterator | |
121 | dense_boruvka_minimum_spanning_tree(const Graph& g, WeightMap weight_map, | |
122 | OutputIterator out, VertexIndex index); | |
123 | ||
124 | template<typename Graph, typename WeightMap, typename OutputIterator> | |
125 | OutputIterator | |
126 | dense_boruvka_minimum_spanning_tree(const Graph& g, WeightMap weight_map, | |
127 | OutputIterator out); | |
128 | } | |
129 | ||
130 | Description | |
131 | ~~~~~~~~~~~ | |
132 | ||
133 | The dense Boruvka distributed minimum spanning tree algorithm is a | |
134 | direct parallelization of the sequential MST algorithm by | |
135 | Boruvka. The algorithm first creates a *supervertex* out of each | |
136 | vertex. Then, in each iteration, it finds the smallest-weight edge | |
137 | incident to each vertex, collapses supervertices along these edges, | |
138 | and removals all self loops. The only difference between the | |
139 | sequential and parallel algorithms is that the parallel algorithm | |
140 | performs an all-reduce operation so that all processes have the | |
141 | global minimum set of edges. | |
142 | ||
143 | Unlike the other three algorithms, this algorithm emits the complete | |
144 | list of edges in the minimum spanning forest via the output iterator | |
145 | on all processes. It may therefore be more useful than the others | |
146 | when parallelizing sequential BGL programs. | |
147 | ||
148 | Complexity | |
149 | ~~~~~~~~~~ | |
150 | ||
151 | The distributed algorithm requires *O(log n)* BSP supersteps, each of | |
152 | which requires *O(m/p + n)* time and *O(n)* communication per | |
153 | process. | |
154 | ||
155 | Performance | |
156 | ~~~~~~~~~~~ | |
157 | ||
158 | The following charts illustrate the performance of this algorithm on | |
159 | various random graphs. We see that the algorithm scales well up to 64 | |
160 | or 128 processors, depending on the type of graph, for dense | |
161 | graphs. However, for sparse graphs performance tapers off as the | |
162 | number of processors surpases *m/n*, i.e., the average degree (which | |
163 | is 30 for this graph). This behavior is expected from the algorithm. | |
164 | ||
165 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=5 | |
166 | :align: left | |
167 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=5&speedup=1 | |
168 | ||
169 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=5 | |
170 | :align: left | |
171 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=5&speedup=1 | |
172 | ||
173 | ``merge_local_minimum_spanning_trees`` | |
174 | -------------------------------------- | |
175 | ||
176 | :: | |
177 | ||
178 | namespace graph { | |
179 | template<typename Graph, typename WeightMap, typename OutputIterator, | |
180 | typename VertexIndexMap> | |
181 | OutputIterator | |
182 | merge_local_minimum_spanning_trees(const Graph& g, WeightMap weight, | |
183 | OutputIterator out, | |
184 | VertexIndexMap index); | |
185 | ||
186 | template<typename Graph, typename WeightMap, typename OutputIterator> | |
187 | inline OutputIterator | |
188 | merge_local_minimum_spanning_trees(const Graph& g, WeightMap weight, | |
189 | OutputIterator out); | |
190 | } | |
191 | ||
192 | Description | |
193 | ~~~~~~~~~~~ | |
194 | ||
195 | The merging local MSTs algorithm operates by computing minimum | |
196 | spanning forests from the edges stored on each process. Then the | |
197 | processes merge their edge lists along a tree. The child nodes cease | |
198 | participating in the computation, but the parent nodes recompute MSFs | |
199 | from the newly acquired edges. In the final round, the root of the | |
200 | tree computes the global MSFs, having received candidate edges from | |
201 | every other process via the tree. | |
202 | ||
203 | Complexity | |
204 | ~~~~~~~~~~ | |
205 | ||
206 | This algorithm requires *O(log_D p)* BSP supersteps (where *D* is the | |
207 | number of children in the tree, and is currently fixed at 3). Each | |
208 | superstep requires *O((m/p) log (m/p) + n)* time and *O(m/p)* | |
209 | communication per process. | |
210 | ||
211 | Performance | |
212 | ~~~~~~~~~~~ | |
213 | ||
214 | The following charts illustrate the performance of this algorithm on | |
215 | various random graphs. The algorithm only scales well for very dense | |
216 | graphs, where most of the work is performed in the initial stage and | |
217 | there is very little work in the later stages. | |
218 | ||
219 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=6 | |
220 | :align: left | |
221 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=6&speedup=1 | |
222 | ||
223 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=6 | |
224 | :align: left | |
225 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=6&speedup=1 | |
226 | ||
227 | ||
228 | ``boruvka_then_merge`` | |
229 | ---------------------- | |
230 | ||
231 | :: | |
232 | ||
233 | namespace graph { | |
234 | template<typename Graph, typename WeightMap, typename OutputIterator, | |
235 | typename VertexIndexMap, typename RankMap, typename ParentMap, | |
236 | typename SupervertexMap> | |
237 | OutputIterator | |
238 | boruvka_then_merge(const Graph& g, WeightMap weight, OutputIterator out, | |
239 | VertexIndexMap index, RankMap rank_map, | |
240 | ParentMap parent_map, SupervertexMap | |
241 | supervertex_map); | |
242 | ||
243 | template<typename Graph, typename WeightMap, typename OutputIterator, | |
244 | typename VertexIndexMap> | |
245 | inline OutputIterator | |
246 | boruvka_then_merge(const Graph& g, WeightMap weight, OutputIterator out, | |
247 | VertexIndexMap index); | |
248 | ||
249 | template<typename Graph, typename WeightMap, typename OutputIterator> | |
250 | inline OutputIterator | |
251 | boruvka_then_merge(const Graph& g, WeightMap weight, OutputIterator out); | |
252 | } | |
253 | ||
254 | Description | |
255 | ~~~~~~~~~~~ | |
256 | ||
257 | This algorithm applies both Boruvka steps and local MSF merging steps | |
258 | together to achieve better asymptotic performance than either | |
259 | algorithm alone. It first executes Boruvka steps until only *n/(log_d | |
260 | p)^2* supervertices remain, then completes the MSF computation by | |
261 | performing local MSF merging on the remaining edges and | |
262 | supervertices. | |
263 | ||
264 | Complexity | |
265 | ~~~~~~~~~~ | |
266 | ||
267 | This algorithm requires *log_D p* + *log log_D p* BSP supersteps. The | |
268 | time required by each superstep depends on the type of superstep | |
269 | being performed; see the distributed Boruvka or merging local MSFS | |
270 | algorithms for details. | |
271 | ||
272 | Performance | |
273 | ~~~~~~~~~~~ | |
274 | ||
275 | The following charts illustrate the performance of this algorithm on | |
276 | various random graphs. We see that the algorithm scales well up to 64 | |
277 | or 128 processors, depending on the type of graph, for dense | |
278 | graphs. However, for sparse graphs performance tapers off as the | |
279 | number of processors surpases *m/n*, i.e., the average degree (which | |
280 | is 30 for this graph). This behavior is expected from the algorithm. | |
281 | ||
282 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=7 | |
283 | :align: left | |
284 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=7&speedup=1 | |
285 | ||
286 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=7 | |
287 | :align: left | |
288 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=7&speedup=1 | |
289 | ||
290 | ``boruvka_mixed_merge`` | |
291 | ----------------------- | |
292 | ||
293 | :: | |
294 | ||
295 | namespace { | |
296 | template<typename Graph, typename WeightMap, typename OutputIterator, | |
297 | typename VertexIndexMap, typename RankMap, typename ParentMap, | |
298 | typename SupervertexMap> | |
299 | OutputIterator | |
300 | boruvka_mixed_merge(const Graph& g, WeightMap weight, OutputIterator out, | |
301 | VertexIndexMap index, RankMap rank_map, | |
302 | ParentMap parent_map, SupervertexMap | |
303 | supervertex_map); | |
304 | ||
305 | template<typename Graph, typename WeightMap, typename OutputIterator, | |
306 | typename VertexIndexMap> | |
307 | inline OutputIterator | |
308 | boruvka_mixed_merge(const Graph& g, WeightMap weight, OutputIterator out, | |
309 | VertexIndexMap index); | |
310 | ||
311 | template<typename Graph, typename WeightMap, typename OutputIterator> | |
312 | inline OutputIterator | |
313 | boruvka_mixed_merge(const Graph& g, WeightMap weight, OutputIterator out); | |
314 | } | |
315 | ||
316 | Description | |
317 | ~~~~~~~~~~~ | |
318 | ||
319 | This algorithm applies both Boruvka steps and local MSF merging steps | |
320 | together to achieve better asymptotic performance than either method | |
321 | alone. In each iteration, the algorithm first performs a Boruvka step | |
322 | and then merges the local MSFs computed based on the supervertex | |
323 | graph. | |
324 | ||
325 | Complexity | |
326 | ~~~~~~~~~~ | |
327 | ||
328 | This algorithm requires *log_D p* BSP supersteps. The | |
329 | time required by each superstep depends on the type of superstep | |
330 | being performed; see the distributed Boruvka or merging local MSFS | |
331 | algorithms for details. However, the algorithm is | |
332 | communication-optional (requiring *O(n)* communication overall) when | |
333 | the graph is sufficiently dense, i.e., *m/n >= p*. | |
334 | ||
335 | Performance | |
336 | ~~~~~~~~~~~ | |
337 | ||
338 | The following charts illustrate the performance of this algorithm on | |
339 | various random graphs. We see that the algorithm scales well up to 64 | |
340 | or 128 processors, depending on the type of graph, for dense | |
341 | graphs. However, for sparse graphs performance tapers off as the | |
342 | number of processors surpases *m/n*, i.e., the average degree (which | |
343 | is 30 for this graph). This behavior is expected from the algorithm. | |
344 | ||
345 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=8 | |
346 | :align: left | |
347 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=8&speedup=1 | |
348 | ||
349 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=8 | |
350 | :align: left | |
351 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=8&speedup=1 | |
352 | ||
353 | ||
354 | Selecting an MST algorithm | |
355 | -------------------------- | |
356 | ||
357 | Dehne and Gotz reported [DG98]_ mixed results when evaluating these | |
358 | four algorithms. No particular algorithm was clearly better than the | |
359 | others in all cases. However, the asymptotically best algorithm | |
360 | (``boruvka_mixed_merge``) seemed to perform more poorly in their tests | |
361 | than the other merging-based algorithms. The following performance | |
362 | charts illustrate the performance of these four minimum spanning tree | |
363 | implementations. | |
364 | ||
365 | Overall, ``dense_boruvka_minimum_spanning_tree`` gives the most | |
366 | consistent performance and scalability for the graphs we | |
367 | tested. Additionally, it may be more suitable for sequential programs | |
368 | that are being parallelized, because it emits complete MSF edge lists | |
369 | via the output iterators in every process. | |
370 | ||
371 | Performance on Sparse Graphs | |
372 | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | |
373 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeSparse&columns=5,6,7,8 | |
374 | :align: left | |
375 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeSparse&columns=5,6,7,8&speedup=1 | |
376 | ||
377 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeSparse&columns=5,6,7,8 | |
378 | :align: left | |
379 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeSparse&columns=5,6,7,8&speedup=1 | |
380 | ||
381 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeSparse&columns=5,6,7,8 | |
382 | :align: left | |
383 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeSparse&columns=5,6,7,8&speedup=1 | |
384 | ||
385 | Performance on Dense Graphs | |
386 | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ | |
387 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeDense&columns=5,6,7,8 | |
388 | :align: left | |
389 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeDense&columns=5,6,7,8&speedup=1 | |
390 | ||
391 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeDense&columns=5,6,7,8 | |
392 | :align: left | |
393 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeDense&columns=5,6,7,8&speedup=1 | |
394 | ||
395 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeDense&columns=5,6,7,8 | |
396 | :align: left | |
397 | .. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeDense&columns=5,6,7,8&speedup=1 | |
398 | ||
399 | ----------------------------------------------------------------------------- | |
400 | ||
401 | Copyright (C) 2004 The Trustees of Indiana University. | |
402 | ||
403 | Authors: Douglas Gregor and Andrew Lumsdaine | |
404 | ||
405 | .. |Logo| image:: pbgl-logo.png | |
406 | :align: middle | |
407 | :alt: Parallel BGL | |
408 | :target: http://www.osl.iu.edu/research/pbgl | |
409 | ||
410 | .. _minimum spanning tree: http://www.boost.org/libs/graph/doc/graph_theory_review.html#sec:minimum-spanning-tree | |
411 | .. _Kruskal's algorithm: http://www.boost.org/libs/graph/doc/kruskal_min_spanning_tree.html | |
412 | .. _Vertex list graph: http://www.boost.org/libs/graph/doc/VertexListGraph.html | |
413 | .. _distributed adjacency list: distributed_adjacency_list.html | |
414 | .. _vertex_list_adaptor: vertex_list_adaptor.html | |
415 | .. _Distributed Edge List Graph: DistributedEdgeListGraph.html | |
416 | .. _Distributed property map: distributed_property_map.html | |
417 | .. _Readable Property Map: http://www.boost.org/libs/property_map/ReadablePropertyMap.html | |
418 | .. _Read/Write Property Map: http://www.boost.org/libs/property_map/ReadWritePropertyMap.html | |
419 | ||
420 | .. [DG98] Frank Dehne and Silvia Gotz. *Practical Parallel Algorithms | |
421 | for Minimum Spanning Trees*. In Symposium on Reliable Distributed Systems, | |
422 | pages 366--371, 1998. | |
423 |