| 1 | r""" |
| 2 | Linear orderings of graphs and digraphs |
| 3 | |
| 4 | This module implements several algorithms for computing and evaluating linear |
| 5 | vertex ordering of graphs and digraphs. The <width> of a linear ordering is one |
| 6 | of the following values. |
| 7 | |
| 8 | +--------------+----------------------------+ |
| 9 | | short name | long name | |
| 10 | +==============+============================+ |
| 11 | | vs | vertex_separation | |
| 12 | +--------------+----------------------------+ |
| 13 | | pw | pathwidth | |
| 14 | +--------------+----------------------------+ |
| 15 | | tw | treewidth | |
| 16 | +--------------+----------------------------+ |
| 17 | | bw | bandwidth | |
| 18 | +--------------+----------------------------+ |
| 19 | | pn | process_number | |
| 20 | +--------------+----------------------------+ |
| 21 | | cw | cutwidth | |
| 22 | +--------------+----------------------------+ |
| 23 | | mcw | modified_cutwidth | |
| 24 | +--------------+----------------------------+ |
| 25 | | ola | optimal_linear_arrangement | |
| 26 | +--------------+----------------------------+ |
| 27 | | sc | sum_cut | |
| 28 | +--------------+----------------------------+ |
| 29 | | mfi | minimun_fill_in | |
| 30 | +--------------+----------------------------+ |
| 31 | |
| 32 | Some of these widths are defined for both graphs and digraphs, and others only |
| 33 | for graphs or for digraphs. |
| 34 | |
| 35 | We follow the formulations proposed in [Bod98]_ and [BFK+11]_ for the evaluation |
| 36 | of the widths induced by a linear vertex ordering `L=[v_1, ..., v_n]` of the |
| 37 | vertices of a (di)graph `G=(V,E)`. Thus, we consider that we have a function |
| 38 | `f` associating an integer to each 3-tuple, consisting of a (di)graph `G=(V,E)`, |
| 39 | a vertex set `S\subseteq V`, and a vertex `v\in V`. Let `L[:i]=[v_1, ..., v_i]` |
| 40 | be a prefix of the ordering and so a subset of `V`. We consider the problems |
| 41 | that can be formulated as the computation of: |
| 42 | |
| 43 | .. MATH:: |
| 44 | |
| 45 | \text{cost}(G, L) = \max_{0\leq i<|V(G)|} f(G, L[:i]\setminus\{v\}, v) |
| 46 | |
| 47 | or |
| 48 | |
| 49 | .. MATH:: |
| 50 | |
| 51 | \text{cost}(G, L) = \sum_{0\leq i<|V(G)|} f(G, L[:i]\setminus\{v\}, v) |
| 52 | |
| 53 | |
| 54 | |
| 55 | Given an ordering `v_1, ..., v_n` of the vertices of `V(G)`, its *cost* is |
| 56 | defined as: |
| 57 | |
| 58 | |
| 59 | **Vertex separation** |
| 60 | |
| 61 | The vertex separation of a linear ordering `L` of the vertices of a digraph `D` |
| 62 | is measured as the maximum number of out-neighbors with position `k>i` in the |
| 63 | ordering of the vertices with position `j<=i` in the ordering. That is, |
| 64 | |
| 65 | .. MATH:: |
| 66 | vs_L(D) = \max_{1\leq i\leq |V|-1} |\{N^+(L[:i])\setminus L[:i]\}| |
| 67 | |
| 68 | |
| 69 | **Path decomposition** |
| 70 | |
| 71 | The cost of a path decomposition induced by a linear ordering `L` of the |
| 72 | vertices of a graph `G` is measured as the maximum number of neighbors with |
| 73 | position `k>i` in the ordering of the vertices with position `j<=i` in the |
| 74 | ordering. That is, |
| 75 | |
| 76 | .. MATH:: |
| 77 | pw_L(G) = \max_{1\leq i\leq |V|-1} |\{N(L[:i])\setminus L[:i]\}| |
| 78 | |
| 79 | |
| 80 | |
| 81 | **Tree decomposition** |
| 82 | |
| 83 | The value ``tw_L(G)`` of the tree decomposition induced by the ordering `L` of |
| 84 | the vertices of `G`. |
| 85 | |
| 86 | The cost of a tree decomposition is mesured by the maximum number of vertices |
| 87 | with position `j > i` in the vertex ordering that can be reached from a vertex |
| 88 | `v` at position `i` through a path with internal vertices at positions `<= |
| 89 | i`. Let `Z(i)` be the set of vertices that can be reached from the vertex at |
| 90 | position `i`. We have |
| 91 | |
| 92 | .. MATH:: |
| 93 | |
| 94 | tw_L(G) = \max_{1\leq i\leq |V|-1} | Z( i ) | |
| 95 | |
| 96 | |
| 97 | |
| 98 | **Bandwidth** |
| 99 | |
| 100 | The bandwidth minimization problem is to find a linear vertex ordering that |
| 101 | minimizes the maximum dilation among all the edges, where the dilatation of an |
| 102 | edge is the distance between its endpoints in the vertex ordering. We can extend |
| 103 | the notion to digraphs where the dilatation of an arc `(u,v)` is the distance |
| 104 | between its endpoints if `v` has a higher index that `u` in the ordering, and 0 |
| 105 | otherwise. |
| 106 | |
| 107 | So, the bandwidth `bw_L(G)` of a graph `G=(V,E)`, or the bandwidth `bw_L(D)` for |
| 108 | a digraph `D=(V,A)`, induced by a given vertex ordering `L` are measured as |
| 109 | follows, |
| 110 | |
| 111 | .. MATH:: |
| 112 | |
| 113 | bw_L(G) = \max_{\{u,v\}\in E} | L[u] - L[v] | |
| 114 | |
| 115 | bw_L(D) = \max_{\{u,v\}\in A} max( 0, L[v] - L[u] ) |
| 116 | |
| 117 | |
| 118 | |
| 119 | **Process number** |
| 120 | |
| 121 | The process number of a linear ordering `L` of the vertices of a digraph `D` is |
| 122 | measured as the maximum number of out-neighbors with position `k>=i` (`i` |
| 123 | included) in the ordering of the vertices with position `j<=i` (`i` included) in |
| 124 | the ordering. Thus, loops on vertices of the digraph are also taken into |
| 125 | account. This notion can also be formulated for undirected graphs (see [CoSe11]_ |
| 126 | for more details on this notion). We have, |
| 127 | |
| 128 | .. MATH:: |
| 129 | |
| 130 | pn_L(D) = \max_{1\leq i\leq |V|} |\{N^+(L[:i])\setminus L[:i-1]\}| |
| 131 | |
| 132 | pn_L(G) = \max_{1\leq i\leq |V|} |\{N(L[:i])\setminus L[:i-1]\}| |
| 133 | |
| 134 | |
| 135 | |
| 136 | **Cutwidth and modified cutwidth** |
| 137 | |
| 138 | The cutwidth minimization problem is to find an ordering that minimizes the |
| 139 | maximum number of edges between vertices with index `j<=i` in the ordering and |
| 140 | vertices with index `k>i` in the ordering. When considering digraphs, only arcs |
| 141 | from `j<=i` to `k>i` are considered. For the modified cutwidth, only edges (or |
| 142 | arcs) from vertices with indexes `j<i` to vertices with indexes `k>i` are |
| 143 | considered. So we count the number of edges (or arcs) passing over the vertex |
| 144 | with index `i`. |
| 145 | |
| 146 | So, the cutwidth `cw_L(G)` of a graph `G=(V,E)`, or the cutwidth `cw_L(D)` for a |
| 147 | digraph `D=(V,A)`, or the modified cutwidth `mcw_L(G)` of a graph `G=(V,E)`, or |
| 148 | the modified cutwidth `mcw_L(D)` for a digraph `D=(V,A)`, induced by a given |
| 149 | vertex ordering `L` are measured as follows, |
| 150 | |
| 151 | .. MATH:: |
| 152 | |
| 153 | cw_L(D) = \max_{1\leq i\leq |V|-1}|\{(u,v)\in A,\ L[u]\leq i<L[v]\}| |
| 154 | |
| 155 | mcw_L(D) = \max_{1\leq i\leq |V|-1}|\{(u,v)\in A,\ L[u]< i<L[v]\}| |
| 156 | |
| 157 | |
| 158 | **Optimal linear arrangement** |
| 159 | |
| 160 | The optimal linear arrangement minimization problem is to find an ordering of |
| 161 | the vertices that minimizes the sum over all intervals `[i,i+1]` of the number |
| 162 | of edges between vertices with index `j<=i` in the ordering and vertices with |
| 163 | index `k>i` in the ordering. When considering digraphs, only arcs from `j<=i` to |
| 164 | `k>i` are considered. So, the cost of a linear arrangement `ola_L(G)` of a graph |
| 165 | `G=(V,E)`, or `ola_L(D)` for a digraph `D=(V,A)`, induced by a given vertex |
| 166 | ordering `L` are measured as follows, |
| 167 | |
| 168 | .. MATH:: |
| 169 | |
| 170 | ola_L(G) = \sum_{1\leq i\leq |V|-1} |\{(u,v)\in E,\ L[u]\leq i<L[v]\}| |
| 171 | |
| 172 | ola_L(D) = \sum_{1\leq i\leq |V|-1} |\{(u,v)\in A,\ L[u]\leq i<L[v]\}| |
| 173 | |
| 174 | However, realizing that in the sum edge `(u,v)` counts for the distance between |
| 175 | `u` and `v` in the ordering, we obtain a better formulation. Indeed, the optimal |
| 176 | linear arrangement minimization problem is to find an ordering that minimizes |
| 177 | the sum of the dilation of the edges, where the dilatation of an edge is the |
| 178 | distance between its endpoints in the vertex ordering. We can extend the notion |
| 179 | to digraphs where the dilatation of an arc `(u,v)` is the distance between its |
| 180 | endpoints if `v` has a higher index that `u` in the ordering, and 0 otherwise. |
| 181 | So, the cost of a linear arrangement `ola_L(G)` of a graph `G=(V,E)`, or |
| 182 | `ola_L(D)` for a digraph `D=(V,A)`, induced by a given vertex ordering `L` are |
| 183 | measured as follows, |
| 184 | |
| 185 | .. MATH:: |
| 186 | |
| 187 | ola_L(G) = \sum_{\{u,v\}\in E} | L[u] - L[v] | |
| 188 | |
| 189 | ola_L(D) = \sum_{\{u,v\}\in A} max( 0, L[v] - L[u] ) |
| 190 | |
| 191 | |
| 192 | **Sum Cut** |
| 193 | |
| 194 | The sum cut problem as a linear vertex ordering problem is to minimize the sum |
| 195 | of the vertex cuts induced by the ordering. A cut is the size of the |
| 196 | neighborhood in `L[i+1:]` of the vertices in `L[:i]`. This notion naturally |
| 197 | extend to digraphs. So the sum of the cost of the cuts induced by an ordering is |
| 198 | measured as, |
| 199 | |
| 200 | .. MATH:: |
| 201 | sc_L(D) = \sum_{1\leq i\leq |V|-1} |\{N^+(L[:i])\setminus L[:i]\}| |
| 202 | |
| 203 | sc_L(G) = \sum_{1\leq i\leq |V|-1} |\{N(L[:i])\setminus L[:i]\}| |
| 204 | |
| 205 | |
| 206 | |
| 207 | **Minimum fill-in** |
| 208 | |
| 209 | The cost of a fill-in is mesured by the sum of the number of vertices with |
| 210 | position `j > i` in the vertex ordering that can be reached from a vertex `v` at |
| 211 | position `i` through a path with internal vertices at positions `<= i` (see tree |
| 212 | decompositions). Let `Z(i)` be the set of vertices that can be reached from the |
| 213 | vertex at position `i`. We have |
| 214 | |
| 215 | .. MATH:: |
| 216 | |
| 217 | mfi_L(G) = \sum_{1\leq i\leq |V|-1} | Z( i ) | |
| 218 | |
| 219 | |
| 220 | |
| 221 | REFERENCES: |
| 222 | |
| 223 | .. [BFK+06] *On exact algorithms for treewidth*, Hans L. Bodlaender, Fedor |
| 224 | V. Fomin, Arie M.C.A. Koster, Dieter Kratsch, and Dimitrios M. Thilikos, |
| 225 | Proceedings 14th Annual European Symposium on Algorithms (ESA), volume 4168 of |
| 226 | Lecture Notes in Computer Science, pages 672-683. Springer, 2006. |
| 227 | |
| 228 | .. [BFK+11] *A note on exact algorithms for vertex ordering problems on graphs*, |
| 229 | Hans L. Bodlaender, Fedor V. Fomin, Arie M.C.A. Koster, Dieter Kratsch, and |
| 230 | Dimitrios M. Thilikos, Theory of Computing Systems, 2011 to appear. |
| 231 | http://dx.doi.org/10.1007/s00224-011-9312-0. |
| 232 | |
| 233 | .. [Bod98] *A partial k-arboretum of graphs with bounded treewidth*, Hans |
| 234 | L. Bodlaender, Theoretical Computer Science, Volume 209, Issues 1-2, Pages |
| 235 | 1-45, 6 December 1998 |
| 236 | |
| 237 | .. [CoSe11] *Characterization of graphs and digraphs with small process number*, |
| 238 | D. Coudert and J-S. Sereni, Discrete Applied Mathematics (DAM), |
| 239 | 159(11):1094-1109, July 2011. |
| 240 | |
| 241 | .. [Kin92] *The vertex separation number of a graph equals its path-width*, |
| 242 | Nancy G. Kinnersley, Information Processing Letters, Volume 42, Issue 6, Pages |
| 243 | 345-350, 24 July 1992. |
| 244 | |
| 245 | |
| 246 | |
| 247 | AUTHORS: |
| 248 | |
| 249 | - David Coudert (2012-03-20): initial version |
| 250 | |
| 251 | |
| 252 | |
| 253 | .. TODO:: |
| 254 | |
| 255 | * implement methods for computing linear orderings |
| 256 | |
| 257 | |
| 258 | |
| 259 | Methods |
| 260 | ------- |
| 261 | """ |
| 262 | |
| 263 | |
| 264 | #***************************************************************************** |
| 265 | # Copyright (C) 2012 David Coudert <david.coudert@inria.fr> |
| 266 | # |
| 267 | # Distributed under the terms of the GNU General Public License (GPL) |
| 268 | # http://www.gnu.org/licenses/ |
| 269 | #***************************************************************************** |
| 270 | |
| 271 | |
| 272 | class LinearOrdering(): |
| 273 | r""" |
| 274 | Class gathering methods for manipulating linear orderings of (di)graphs. |
| 275 | |
| 276 | This module implements several algorithms for computing and evaluating |
| 277 | linear vertex ordering of graphs and digraphs. The <width> of a linear |
| 278 | ordering is one of the following values. |
| 279 | |
| 280 | """ |
| 281 | |
| 282 | def _repr_(self): |
| 283 | r""" |
| 284 | Returns a string representation of the class LinearOrdering. |
| 285 | """ |
| 286 | return "Linear Ordering Class" |
| 287 | |
| 288 | def _latex_(self): |
| 289 | r""" |
| 290 | Returns the LaTeX representation of the class LinearOrdering. |
| 291 | """ |
| 292 | return 'Linear Ordering Class' |
| 293 | |
| 294 | |
| 295 | ########################################################## |
| 296 | # Function for testing the validity of a vertex ordering # |
| 297 | ########################################################## |
| 298 | |
| 299 | def is_valid(self, G, L): |
| 300 | r""" |
| 301 | Test if a given linear ordering `L` is valid for a given (di)graph `G`. |
| 302 | |
| 303 | Returns True if `L` is a valid vertex ordering for `G`, that is if all |
| 304 | vertices of `G` are in `L`, and `L` contains no other vertex and no |
| 305 | duplicated vertices. |
| 306 | |
| 307 | INPUT: |
| 308 | |
| 309 | - ``G`` -- a graph or a digraph |
| 310 | |
| 311 | - ``L`` -- an ordering of the vertices of ``G`` |
| 312 | |
| 313 | |
| 314 | OUTPUT: |
| 315 | |
| 316 | Returns True if `L` is a valid vertex ordering for `G`, and False |
| 317 | oterwise. |
| 318 | |
| 319 | |
| 320 | EXAMPLE: |
| 321 | |
| 322 | Path decomposition of a cycle:: |
| 323 | |
| 324 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 325 | sage: LO = LinearOrdering() |
| 326 | sage: G = graphs.CycleGraph(6) |
| 327 | sage: L = [u for u in G.vertices()] |
| 328 | sage: LO.is_valid(G, L) |
| 329 | True |
| 330 | sage: LO.is_valid(G, [1,2]) |
| 331 | False |
| 332 | |
| 333 | TEST: |
| 334 | |
| 335 | Giving anything else than a Graph or a DiGraph:: |
| 336 | |
| 337 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 338 | sage: LO = LinearOrdering() |
| 339 | sage: LO.is_valid(2, []) |
| 340 | Traceback (most recent call last): |
| 341 | ... |
| 342 | ValueError: The input parameter must be a Graph or a DiGraph. |
| 343 | """ |
| 344 | from sage.graphs.graph import Graph |
| 345 | from sage.graphs.digraph import DiGraph |
| 346 | if not isinstance(G, Graph) and not isinstance(G, DiGraph): |
| 347 | raise ValueError("The input parameter must be a Graph or a DiGraph.") |
| 348 | |
| 349 | A = set( G.vertices() ) |
| 350 | B = set( L ) |
| 351 | return len( A.symmetric_difference(B) ) == 0 |
| 352 | |
| 353 | |
| 354 | ############################################################################ |
| 355 | # Front end function for the evaluation of linear vertex orderings with # |
| 356 | # different widths # |
| 357 | ############################################################################ |
| 358 | |
| 359 | def width_of(self, G, L, width = 'vertex_separation'): |
| 360 | r""" |
| 361 | Returns the <width> of the linear ordering `L` for `G`. |
| 362 | |
| 363 | This function returns the <width> of the vertex ordering L, where |
| 364 | width is one of the following values |
| 365 | |
| 366 | +--------------+----------------------------+ |
| 367 | | short name | long name | |
| 368 | +==============+============================+ |
| 369 | | vs | vertex_separation | |
| 370 | +--------------+----------------------------+ |
| 371 | | pw | pathwidth | |
| 372 | +--------------+----------------------------+ |
| 373 | | tw | treewidth | |
| 374 | +--------------+----------------------------+ |
| 375 | | bw | bandwidth | |
| 376 | +--------------+----------------------------+ |
| 377 | | pn | process_number | |
| 378 | +--------------+----------------------------+ |
| 379 | | cw | cutwidth | |
| 380 | +--------------+----------------------------+ |
| 381 | | mcw | modified_cutwidth | |
| 382 | +--------------+----------------------------+ |
| 383 | | ola | optimal_linear_arrangement | |
| 384 | +--------------+----------------------------+ |
| 385 | | sc | sum_cut | |
| 386 | +--------------+----------------------------+ |
| 387 | | mfi | minimun_fill_in | |
| 388 | +--------------+----------------------------+ |
| 389 | |
| 390 | See the module's documentation for more details on these widths. |
| 391 | |
| 392 | INPUT: |
| 393 | |
| 394 | - ``G`` -- a graph or a digraph (possibly with multi-edges) |
| 395 | |
| 396 | - ``L`` -- an ordering of the vertices of ``G`` |
| 397 | |
| 398 | - ``width`` -- is the graph invariant to consider (vertex_separation by |
| 399 | default) |
| 400 | |
| 401 | |
| 402 | OUTPUT: |
| 403 | |
| 404 | The value of the measured graph invariants for this vertex ordering. |
| 405 | |
| 406 | |
| 407 | NOTES: |
| 408 | |
| 409 | All width are defined for undirected graphs. |
| 410 | |
| 411 | For directed graphs, only vertex_separation, bandwidth, |
| 412 | process_number, cutwidth, modified_cutwidth, |
| 413 | optimal_linear_arrangement, and sum_cut are defined. Measurements |
| 414 | are performed using arcs from u=L[i] to v=L[j], with i<=j. |
| 415 | |
| 416 | EXAMPLE: |
| 417 | |
| 418 | Path decomposition of a cycle:: |
| 419 | |
| 420 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 421 | sage: LO = LinearOrdering() |
| 422 | sage: G = graphs.CycleGraph(6) |
| 423 | sage: LO.width_of(G, [0, 1, 2, 3, 4, 5], 'pw') |
| 424 | 2 |
| 425 | sage: LO.width_of(G, [0, 2, 4, 1, 5, 3], 'pw') |
| 426 | 3 |
| 427 | |
| 428 | TEST: |
| 429 | |
| 430 | Giving anything else than a Graph or a DiGraph:: |
| 431 | |
| 432 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 433 | sage: LO = LinearOrdering() |
| 434 | sage: LO.width_of(2, [], 'pw') |
| 435 | Traceback (most recent call last): |
| 436 | ... |
| 437 | ValueError: The input parameter must be a Graph or a DiGraph. |
| 438 | |
| 439 | Giving a vertex ordering on a different set of vertices:: |
| 440 | |
| 441 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 442 | sage: LO = LinearOrdering() |
| 443 | sage: G = graphs.CycleGraph(6) |
| 444 | sage: LO.width_of(G, [1, 2, 3], 'pw') |
| 445 | Traceback (most recent call last): |
| 446 | ... |
| 447 | ValueError: The input parameter is not a valid vertex ordering. |
| 448 | |
| 449 | Giving a not implemented width parameter:: |
| 450 | |
| 451 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 452 | sage: LO = LinearOrdering() |
| 453 | sage: G = graphs.CycleGraph(6) |
| 454 | sage: L = [u for u in G.vertices()] |
| 455 | sage: LO.width_of(G, L, 'frite') |
| 456 | Traceback (most recent call last): |
| 457 | ... |
| 458 | ValueError: The desired width evaluation function has not been implemented so far. Feel free to add it. |
| 459 | """ |
| 460 | from sage.graphs.graph import Graph |
| 461 | from sage.graphs.digraph import DiGraph |
| 462 | if not isinstance(G, Graph) and not isinstance(G, DiGraph): |
| 463 | raise ValueError("The input parameter must be a Graph or a DiGraph.") |
| 464 | |
| 465 | if not self.is_valid(G, L): |
| 466 | raise ValueError("The input parameter is not a valid vertex ordering.") |
| 467 | |
| 468 | if width in ['pathwidth','pw']: |
| 469 | return self.width_of_path_decomposition(G, L) |
| 470 | elif width in ['vertex_separation','vs']: |
| 471 | return self.width_of_vertex_separation(G, L) |
| 472 | elif width in ['process_number','pn']: |
| 473 | return self.width_of_process_number(G, L) |
| 474 | elif width in ['treewidth','tw']: |
| 475 | return self.width_of_tree_decomposition(G, L) |
| 476 | elif width in ['cutwidth','cw']: |
| 477 | return self.width_of_cutwidth(G, L) |
| 478 | elif width in ['modified_cutwidth','mcw']: |
| 479 | return self.width_of_modified_cutwidth(G, L) |
| 480 | elif width in ['bandwidth','bw']: |
| 481 | return self.width_of_bandwidth(G, L) |
| 482 | elif width in ['sum_cut','sc']: |
| 483 | return self.width_of_sum_cut(G, L) |
| 484 | elif width in ['minimum_fill_in','mfi']: |
| 485 | return self.width_of_fill_in(G, L) |
| 486 | elif width in ['optimal_linear_arrangement','ola']: |
| 487 | return self.width_of_linear_arrangement(G, L) |
| 488 | else: |
| 489 | raise ValueError("The desired width evaluation function has not been implemented so far. Feel free to add it.") |
| 490 | |
| 491 | |
| 492 | |
| 493 | |
| 494 | ############################################### |
| 495 | # Methods for pathwidth and vertex separation # |
| 496 | ############################################### |
| 497 | |
| 498 | def width_of_path_decomposition(self, G, L): |
| 499 | r""" |
| 500 | Returns the value `pw_L(G)` of the path decomposition induced by the |
| 501 | vertex ordering `L` for `G`, where |
| 502 | |
| 503 | ..MATH:: |
| 504 | |
| 505 | pw_L(G) = \max_{0\leq i< |V|-1} | N(L[:i])\setminus L[:i] | |
| 506 | |
| 507 | INPUT: |
| 508 | |
| 509 | - ``G`` -- a Graph |
| 510 | |
| 511 | - ``L`` -- an ordering of the vertices of ``G`` |
| 512 | |
| 513 | |
| 514 | OUTPUT: |
| 515 | |
| 516 | The value ``pw_L(G)`` of the path decomposition induced by the ordering |
| 517 | `L` of the vertices of `G`. |
| 518 | |
| 519 | EXAMPLE: |
| 520 | |
| 521 | Path decomposition of a cycle:: |
| 522 | |
| 523 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 524 | sage: LO = LinearOrdering() |
| 525 | sage: G = graphs.CycleGraph(6) |
| 526 | sage: LO.width_of_path_decomposition(G, [0, 1, 2, 3, 4, 5]) |
| 527 | 2 |
| 528 | sage: LO.width_of_path_decomposition(G, [0, 2, 4, 1, 5, 3]) |
| 529 | 3 |
| 530 | |
| 531 | TEST: |
| 532 | |
| 533 | Giving anything else than a Graph:: |
| 534 | |
| 535 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 536 | sage: LO = LinearOrdering() |
| 537 | sage: G = digraphs.Circuit(4) |
| 538 | sage: LO.width_of_path_decomposition(G, []) |
| 539 | Traceback (most recent call last): |
| 540 | ... |
| 541 | ValueError: The input parameter must be a Graph. |
| 542 | |
| 543 | Giving a vertex ordering on a different set of vertices:: |
| 544 | |
| 545 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 546 | sage: LO = LinearOrdering() |
| 547 | sage: G = graphs.CycleGraph(6) |
| 548 | sage: LO.width_of_path_decomposition(G, [1, 2, 3]) |
| 549 | Traceback (most recent call last): |
| 550 | ... |
| 551 | ValueError: The input parameter is not a valid vertex ordering. |
| 552 | """ |
| 553 | # Path decompositions are defined only for graphs |
| 554 | from sage.graphs.graph import Graph |
| 555 | if not isinstance(G, Graph): |
| 556 | raise ValueError("The input parameter must be a Graph.") |
| 557 | |
| 558 | if not self.is_valid(G, L): |
| 559 | raise ValueError("The input parameter is not a valid vertex ordering.") |
| 560 | |
| 561 | pwL = 0 |
| 562 | S = [] |
| 563 | neighbors_of_S_in_V_minus_S = [] |
| 564 | |
| 565 | for u in L: |
| 566 | |
| 567 | # We remove u from the list of neighbors of S |
| 568 | if u in neighbors_of_S_in_V_minus_S: |
| 569 | neighbors_of_S_in_V_minus_S.remove(u) |
| 570 | |
| 571 | # We add vertex u to the set S |
| 572 | S += [u] |
| 573 | |
| 574 | # We add the neighbors of u to the list of neighbors of S |
| 575 | for v in G.neighbors(u): |
| 576 | if (not v in S) and (not v in neighbors_of_S_in_V_minus_S): |
| 577 | neighbors_of_S_in_V_minus_S += [v] |
| 578 | |
| 579 | # We update the cost of the path decomposition |
| 580 | pwL = max( pwL, len(neighbors_of_S_in_V_minus_S) ) |
| 581 | |
| 582 | return pwL |
| 583 | |
| 584 | |
| 585 | def width_of_vertex_separation(self, G, L): |
| 586 | r""" |
| 587 | Returns the value `vs_L(G)` of the vertex separation induced by the |
| 588 | vertex ordering `L` for `G`, where |
| 589 | |
| 590 | ..MATH:: |
| 591 | |
| 592 | vs_L(G) = \max_{0\leq i< |V|-1} | N^+(L[:i])\setminus L[:i] | |
| 593 | |
| 594 | INPUT: |
| 595 | |
| 596 | - ``G`` -- a graph or a digraph |
| 597 | |
| 598 | - ``L`` -- an ordering of the vertices of ``G`` |
| 599 | |
| 600 | |
| 601 | OUTPUT: |
| 602 | |
| 603 | The value ``vs_L(G)`` of the vertex separation induced by the ordering |
| 604 | `L` of the vertices of `G`. |
| 605 | |
| 606 | NOTES: |
| 607 | |
| 608 | The vertex separation is defined for both graphs and digraphs. The |
| 609 | vertex separation of a graph is also the cost of the path |
| 610 | decomposition induced by the vertex ordering (see the module's |
| 611 | documentation). |
| 612 | |
| 613 | EXAMPLE: |
| 614 | |
| 615 | Vertex separation of a cycle:: |
| 616 | |
| 617 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 618 | sage: LO = LinearOrdering() |
| 619 | sage: G = graphs.CycleGraph(6) |
| 620 | sage: L = [u for u in G.vertices()] |
| 621 | sage: LO.width_of_vertex_separation(G, L) |
| 622 | 2 |
| 623 | |
| 624 | Vertex separation of a cicuit:: |
| 625 | |
| 626 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 627 | sage: LO = LinearOrdering() |
| 628 | sage: G = digraphs.Circuit(6) |
| 629 | sage: L = [u for u in G.vertices()] |
| 630 | sage: LO.width_of_vertex_separation(G, L) |
| 631 | 1 |
| 632 | |
| 633 | |
| 634 | TEST: |
| 635 | |
| 636 | Giving anything else than a Graph or a DiGraph:: |
| 637 | |
| 638 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 639 | sage: LO = LinearOrdering() |
| 640 | sage: LO.width_of_vertex_separation(2, []) |
| 641 | Traceback (most recent call last): |
| 642 | ... |
| 643 | ValueError: The input parameter must be a Graph or a DiGraph. |
| 644 | |
| 645 | Giving a vertex ordering on a different set of vertices:: |
| 646 | |
| 647 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 648 | sage: LO = LinearOrdering() |
| 649 | sage: G = digraphs.Circuit(6) |
| 650 | sage: LO.width_of_vertex_separation(G, [1, 2, 3]) |
| 651 | Traceback (most recent call last): |
| 652 | ... |
| 653 | ValueError: The input parameter is not a valid vertex ordering. |
| 654 | """ |
| 655 | # If G is a graph, we instead evaluate the path decomposition |
| 656 | from sage.graphs.graph import Graph |
| 657 | if isinstance(G, Graph): |
| 658 | return self.width_of_path_decomposition(G, L) |
| 659 | |
| 660 | if not self.is_valid(G, L): |
| 661 | raise ValueError("The input parameter is not a valid vertex ordering.") |
| 662 | |
| 663 | vsL = 0 |
| 664 | S = [] |
| 665 | neighbors_of_S_in_V_minus_S = [] |
| 666 | |
| 667 | for u in L: |
| 668 | |
| 669 | # We remove u from the list of neighbors of S |
| 670 | if u in neighbors_of_S_in_V_minus_S: |
| 671 | neighbors_of_S_in_V_minus_S.remove(u) |
| 672 | |
| 673 | # We add vertex u to the set S |
| 674 | S += [u] |
| 675 | |
| 676 | # We add the out-neighbors of u to the list of neighbors of S |
| 677 | for v in G.neighbors_out(u): |
| 678 | if (not v in S) and (not v in neighbors_of_S_in_V_minus_S): |
| 679 | neighbors_of_S_in_V_minus_S += [v] |
| 680 | |
| 681 | # We update the cost of the vertex separation |
| 682 | vsL = max( vsL, len(neighbors_of_S_in_V_minus_S) ) |
| 683 | |
| 684 | return vsL |
| 685 | |
| 686 | |
| 687 | def __vertex_separation_MILP__(self, D, integrality = False, verbosity = 0): |
| 688 | r""" |
| 689 | Implements an MILP for the vertex separation |
| 690 | |
| 691 | INPUTS: |
| 692 | |
| 693 | - ``D`` -- a digraph |
| 694 | |
| 695 | OUTPUT: |
| 696 | |
| 697 | A pair ``(cost, ordering)`` representing the optimal ordering of the |
| 698 | vertices and its cost. |
| 699 | |
| 700 | |
| 701 | VARIABLES: |
| 702 | |
| 703 | x[v,t] = 1 if and only if the working lightpath of connection v is torn down at epoch t |
| 704 | So if an agent is put on vertex v at step t |
| 705 | |
| 706 | y[v,t] = 1 if and only if the new lightpath of connection v is set up at epoch t |
| 707 | So if v is processed at step t |
| 708 | |
| 709 | u[v,t] = 1 if and only if the connection corresponding to v is disrupted at epoch t |
| 710 | So if there is an agent on v at step t |
| 711 | |
| 712 | z Objective = max number of concurrently disrupted connections |
| 713 | |
| 714 | CONSTANT: |
| 715 | T maximum number of epoch/steps. T <= 2.N |
| 716 | |
| 717 | TERMINOLOGY: |
| 718 | An epoch is the period of time between the establishment of 2 new lightpaths |
| 719 | So between the processing of 2 nodes |
| 720 | => 1 epoch per connection => #epochs = T = N |
| 721 | |
| 722 | CONSTRAINTS: |
| 723 | (1) Minimize z |
| 724 | (2) x[v][t] <= x[v][t+1] for all v in V, and for t:=0..T-2 |
| 725 | (3) y[v][t] <= y[v][t+1] for all v in V, and for t:=0..T-2 |
| 726 | (miss) y[v][t] <= x[v][t] for all v in V, and for all t:=0..T-1 |
| 727 | to ensure that once a vertex is processed, it has already been covered.... |
| 728 | (4) y[v][t] <= x[w][t] for all v in V, for all w in N^+(v), and for all t:=0..T-1 |
| 729 | (5) sum_{v in V} y[v][0] <= 1 |
| 730 | (6) sum_{v in V} y[v][t+1] <= sum_{v in V} y[v][t] + 1 for t:=0..T-2 |
| 731 | (7) sum_{v in V} y[v][T-1] = |V| |
| 732 | (8) u[v][t] >= x[v][t]-y[v][t] for all v in V, and for all t:=0..T-1 |
| 733 | (9) z >= sum_{v in V} u[v][t] for all t:=0..T-1 |
| 734 | (10) 0 <= x[v][t] and u[v][t] <= 1 |
| 735 | Meaning both are between 0 and 1 |
| 736 | (11) y[v][t] in {0,1} |
| 737 | (12) 0 <= z <= |V| |
| 738 | """ |
| 739 | from sage.numerical.mip import MixedIntegerLinearProgram, Sum, MIPSolverException |
| 740 | p = MixedIntegerLinearProgram( maximization = False ) |
| 741 | |
| 742 | if integrality: |
| 743 | x = p.new_variable( integer = True, dim = 2 ) |
| 744 | u = p.new_variable( integer = True, dim = 2 ) |
| 745 | else: |
| 746 | x = p.new_variable( dim = 2 ) |
| 747 | u = p.new_variable( dim = 2 ) |
| 748 | y = p.new_variable( integer = True, dim = 2 ) |
| 749 | z = p.new_variable( integer = True, dim = 1 ) |
| 750 | |
| 751 | N = D.num_verts() |
| 752 | T=N |
| 753 | |
| 754 | # (2) x[v][t] <= x[v][t+1] for all v in V, and for t:=0..T-2 |
| 755 | # (3) y[v][t] <= y[v][t+1] for all v in V, and for t:=0..T-2 |
| 756 | for v in D.vertices(): |
| 757 | for t in xrange(T-1): |
| 758 | p.add_constraint( x[v][t] - x[v][t+1], max = 0 ) |
| 759 | p.add_constraint( y[v][t] - y[v][t+1], max = 0 ) |
| 760 | |
| 761 | # (miss) y[v][t] <= x[v][t] for all v in V, and for all t:=0..T-1 |
| 762 | # for v in D.vertices(): |
| 763 | # for t in range(T): |
| 764 | # p.add_constraint(y[v][t]-x[v][t],max=0) |
| 765 | # |
| 766 | # (4) y[v][t] <= x[w][t] for all v in V, for all w in N^+(v), and for all t:=0..T-1 |
| 767 | for v in D.vertices(): |
| 768 | for w in D.neighbors_out(v): |
| 769 | for t in xrange(T): |
| 770 | p.add_constraint( y[v][t] - x[w][t], max = 0 ) |
| 771 | |
| 772 | # (5) sum_{v in V} y[v][0] <= 1 |
| 773 | p.add_constraint( Sum( y[v][0] for v in D.vertices() ), max = 1 ) |
| 774 | |
| 775 | # (6) sum_{v in V} y[v][t+1] <= sum_{v in V} y[v][t] + 1 for t:=0..T-2 |
| 776 | for t in xrange(T-1): |
| 777 | p.add_constraint( Sum( y[v][t+1] - y[v][t] for v in D.vertices() ), max = 1 ) |
| 778 | |
| 779 | # (7) sum_{v in V} y[v][T-1] = |V| |
| 780 | p.add_constraint( Sum( y[v][T-1] for v in D.vertices() ), min = N ) |
| 781 | p.add_constraint( Sum( y[v][T-1] for v in D.vertices() ), max = N ) |
| 782 | |
| 783 | # (8) u[v][t] >= x[v][t]-y[v][t] for all v in V, and for all t:=0..T-1 |
| 784 | for v in D.vertices(): |
| 785 | for t in xrange(T): |
| 786 | p.add_constraint( x[v][t] - y[v][t] - u[v][t], max = 0 ) |
| 787 | |
| 788 | # (9) z >= sum_{v in V} u[v][t] for all t:=0..T-1 |
| 789 | for t in xrange(T): |
| 790 | p.add_constraint( Sum( u[v][t] for v in D.vertices() ) - z['z'], max = 0 ) |
| 791 | |
| 792 | # (10) 0 <= x[v][t] and u[v][t] <= 1 |
| 793 | # (11) y[v][t] in {0,1} |
| 794 | for v in D.vertices(): |
| 795 | for t in xrange(T): |
| 796 | p.add_constraint( x[v][t], min = 0 ) |
| 797 | p.add_constraint( x[v][t], max = 1 ) |
| 798 | p.add_constraint( u[v][t], min = 0 ) |
| 799 | p.add_constraint( u[v][t], max = 1 ) |
| 800 | p.set_binary( y[v][t] ) |
| 801 | # (12) 0 <= z <= |V| |
| 802 | p.add_constraint( z['z'], min = 0 ) |
| 803 | p.add_constraint( z['z'], max = N ) |
| 804 | |
| 805 | # (1) Minimize z |
| 806 | p.set_objective( z['z'] ) |
| 807 | |
| 808 | try: |
| 809 | obj = p.solve( log=verbosity ) |
| 810 | |
| 811 | taby = p.get_values( y ) |
| 812 | tabz = p.get_values( z ) |
| 813 | # since exactly one vertex is processed per epoch, we can reconstruct the sequence |
| 814 | seq = [] |
| 815 | for t in xrange(T): |
| 816 | for v in D.vertices(): |
| 817 | if (taby[v][t] > 0) and (not v in seq): |
| 818 | seq += [v] |
| 819 | vs = int(round( tabz['z'] )); |
| 820 | |
| 821 | except MIPSolverException: |
| 822 | print "VSC NOT working with fractional stuff\n",D.edges(labels=False) |
| 823 | print " => try with integral version" |
| 824 | (vs,seq)=self.__vertex_separation_MILP__(D, True, verbosity) |
| 825 | print vs,seq |
| 826 | raise ValueError("Unbounded or unexpected error") |
| 827 | |
| 828 | del p |
| 829 | return vs,seq; |
| 830 | |
| 831 | |
| 832 | |
| 833 | |
| 834 | |
| 835 | def vertex_separation(self, G, method = 'exp'): |
| 836 | r""" |
| 837 | Returns an optimal ordering of the vertices and its cost for |
| 838 | vertex-separation. |
| 839 | |
| 840 | INPUT: |
| 841 | |
| 842 | - ``G`` -- a digraph |
| 843 | |
| 844 | - ``method`` (string) -- the method to use |
| 845 | |
| 846 | - 'exp' (default) -- Uses an algorithm with time and space |
| 847 | complexity in ``2^n``. Because of its current implementation, this |
| 848 | algorithm only works on graphs on less than 32 vertices. This can |
| 849 | be changed to 54 if necessary, but 32 vertices already require 4GB |
| 850 | of memory. |
| 851 | |
| 852 | - 'MILP' -- Uses an MILP implementation. |
| 853 | |
| 854 | OUTPUT: |
| 855 | |
| 856 | A pair ``(cost, ordering)`` representing the optimal ordering of the |
| 857 | vertices and its cost. |
| 858 | |
| 859 | EXAMPLE: |
| 860 | |
| 861 | The vertex separation of a circuit is equal to 1:: |
| 862 | |
| 863 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 864 | sage: LO = LinearOrdering() |
| 865 | sage: g = digraphs.Circuit(6) |
| 866 | sage: LO.vertex_separation(g) |
| 867 | (1, [0, 1, 2, 3, 4, 5]) |
| 868 | |
| 869 | TEST: |
| 870 | |
| 871 | Given anything else than a DiGraph:: |
| 872 | |
| 873 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 874 | sage: LO = LinearOrdering() |
| 875 | sage: g = graphs.CycleGraph(6) |
| 876 | sage: LO.vertex_separation(g) |
| 877 | Traceback (most recent call last): |
| 878 | ... |
| 879 | ValueError: The input parameter must be a DiGraph. |
| 880 | """ |
| 881 | from sage.graphs.graph import DiGraph |
| 882 | if not isinstance(G, DiGraph): |
| 883 | raise ValueError("The input parameter must be a DiGraph.") |
| 884 | |
| 885 | if method == 'MILP': |
| 886 | return self.__vertex_separation_MILP__(G) |
| 887 | else: |
| 888 | # default method |
| 889 | from sage.graphs.graph_decompositions import vertex_separation |
| 890 | return vertex_separation.vertex_separation(G) |
| 891 | |
| 892 | def path_decomposition(self, G, method = 'exp'): |
| 893 | r""" |
| 894 | Returns the pathwidth of the given graph and the ordering of the |
| 895 | vertices resulting in a corresponding path decomposition. |
| 896 | |
| 897 | INPUT: |
| 898 | |
| 899 | - ``G`` -- a graph |
| 900 | |
| 901 | - ``method`` (string) -- the method to use |
| 902 | |
| 903 | - 'exp' (default) -- Uses an algorithm with time and space |
| 904 | complexity in ``2^n``. Because of its current implementation, this |
| 905 | algorithm only works on graphs on less than 32 vertices. This can |
| 906 | be changed to 54 if necessary, but 32 vertices already require 4GB |
| 907 | of memory. |
| 908 | |
| 909 | - 'MILP' -- Uses an MILP implementation. |
| 910 | |
| 911 | OUTPUT: |
| 912 | |
| 913 | A pair ``(cost, ordering)`` representing the optimal ordering of the |
| 914 | vertices and its cost. |
| 915 | |
| 916 | EXAMPLE: |
| 917 | |
| 918 | The pathwidth of a circuit is equal to 2:: |
| 919 | |
| 920 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 921 | sage: LO = LinearOrdering() |
| 922 | sage: g = graphs.CycleGraph(6) |
| 923 | sage: LO.path_decomposition(g) |
| 924 | (2, [0, 1, 2, 3, 4, 5]) |
| 925 | |
| 926 | TEST: |
| 927 | |
| 928 | Given anything else than a Graph:: |
| 929 | |
| 930 | sage: from sage.graphs.graph_decompositions.linear_ordering import LinearOrdering |
| 931 | sage: LO = LinearOrdering() |
| 932 | sage: g = digraphs.Circuit(6) |
| 933 | sage: LO.path_decomposition(g) |
| 934 | Traceback (most recent call last): |
| 935 | ... |
| 936 | ValueError: The input parameter must be a Graph. |
| 937 | """ |
| 938 | from sage.graphs.graph import Graph |
| 939 | if not isinstance(G, Graph): |
| 940 | raise ValueError("The input parameter must be a Graph.") |
| 941 | |
| 942 | if method == 'MILP': |
| 943 | from sage.graphs.graph import DiGraph |
| 944 | D = DiGraph( G.edges(labels = None) + [(v,u) for u,v in G.edges(labels = None)] ) |
| 945 | return self.__vertex_separation_MILP__(D) |
| 946 | else: |
| 947 | # default method |
| 948 | from sage.graphs.graph_decompositions import vertex_separation |
| 949 | return vertex_separation.path_decomposition(G) |
| 950 | |
| 951 | |