| 1 | package Graph::Base; |
| 2 | |
| 3 | use strict; |
| 4 | local $^W = 1; |
| 5 | |
| 6 | use vars qw(@ISA); |
| 7 | |
| 8 | =head1 NAME |
| 9 | |
| 10 | Graph::Base - graph base class |
| 11 | |
| 12 | =head1 SYNOPSIS |
| 13 | |
| 14 | use Graph::Directed; |
| 15 | use Graph::Undirected; |
| 16 | |
| 17 | $d1 = new Graph; |
| 18 | $d2 = new Graph::Directed; |
| 19 | $u = new Graph::Undirected; |
| 20 | |
| 21 | =head1 DESCRIPTION |
| 22 | |
| 23 | You create new graphs by calling the C<new> constructors of classes |
| 24 | C<Graph>, C<Graph::Directed>, and C<Graph::Undirected>. The classes |
| 25 | C<Graph> and C<Graph::Directed> are identical. After creating the |
| 26 | graph you can modify and explore the graph with following methods. |
| 27 | |
| 28 | =over 4 |
| 29 | |
| 30 | =cut |
| 31 | |
| 32 | require Exporter; |
| 33 | @ISA = qw(Exporter); |
| 34 | |
| 35 | =pod |
| 36 | |
| 37 | =item new |
| 38 | |
| 39 | $G = Graph->new(@V) |
| 40 | |
| 41 | Returns a new graph $G with the optional vertices @V. |
| 42 | |
| 43 | =cut |
| 44 | |
| 45 | sub new { |
| 46 | my $class = shift; |
| 47 | |
| 48 | my $G = { }; |
| 49 | |
| 50 | bless $G, $class; |
| 51 | |
| 52 | $G->add_vertices(@_) if @_; |
| 53 | |
| 54 | return $G; |
| 55 | } |
| 56 | |
| 57 | =pod |
| 58 | |
| 59 | =item add_vertices |
| 60 | |
| 61 | $G = $G->add_vertices(@v) |
| 62 | |
| 63 | Adds the vertices to the graph $G, returns the graph. |
| 64 | |
| 65 | =cut |
| 66 | |
| 67 | sub add_vertices { |
| 68 | my ($G, @v) = @_; |
| 69 | |
| 70 | @{ $G->{ V } }{ @v } = @v; |
| 71 | |
| 72 | return $G; |
| 73 | } |
| 74 | |
| 75 | =pod |
| 76 | |
| 77 | =item add_vertex |
| 78 | |
| 79 | $G = $G->add_vertex($v) |
| 80 | |
| 81 | Adds the vertex $v to the graph $G, returns the graph. |
| 82 | |
| 83 | =cut |
| 84 | |
| 85 | sub add_vertex { |
| 86 | my ($G, $v) = @_; |
| 87 | |
| 88 | return $G->add_vertices($v); |
| 89 | } |
| 90 | |
| 91 | =pod |
| 92 | |
| 93 | =item vertices |
| 94 | |
| 95 | @V = $G->vertices |
| 96 | |
| 97 | In list context returns the vertices @V of the graph $G. |
| 98 | In scalar context returns the number of the vertices. |
| 99 | |
| 100 | =cut |
| 101 | |
| 102 | sub vertices { |
| 103 | my $G = shift; |
| 104 | my @V = exists $G->{ V } ? sort values %{ $G->{ V } } : (); |
| 105 | |
| 106 | return @V; |
| 107 | } |
| 108 | |
| 109 | =pod |
| 110 | |
| 111 | =item has_vertices |
| 112 | |
| 113 | $G->has_vertices(@v) |
| 114 | |
| 115 | In list context returns a list which contains the vertex |
| 116 | of the vertices @v if the vertex exists in the graph $G |
| 117 | and undef if it doesn't. In scalar context returns the |
| 118 | number of the existing vertices. |
| 119 | |
| 120 | =cut |
| 121 | |
| 122 | sub has_vertices { |
| 123 | my $G = shift; |
| 124 | |
| 125 | return wantarray ? |
| 126 | map { exists $G->{ V }->{ $_ } ? $_ : undef } @_ : |
| 127 | grep { exists $G->{ V }->{ $_ } } @_ ; |
| 128 | } |
| 129 | |
| 130 | =pod |
| 131 | |
| 132 | =item has_vertex |
| 133 | |
| 134 | $b = $G->has_vertex($v) |
| 135 | |
| 136 | Returns true if the vertex $v exists in |
| 137 | the graph $G and false if it doesn't. |
| 138 | |
| 139 | =cut |
| 140 | |
| 141 | sub has_vertex { |
| 142 | my ($G, $v) = @_; |
| 143 | |
| 144 | return defined $v && exists $G->{ V } && exists $G->{ V }->{ $v }; |
| 145 | } |
| 146 | |
| 147 | =pod |
| 148 | |
| 149 | =item vertex |
| 150 | |
| 151 | $v = $G->has_vertex($v) |
| 152 | |
| 153 | Returns the vertex $v if the vertex exists in the graph $G |
| 154 | or undef if it doesn't. |
| 155 | |
| 156 | =cut |
| 157 | |
| 158 | sub vertex { |
| 159 | my ($G, $v) = @_; |
| 160 | |
| 161 | return defined $v && $G->{ V }->{ $v }; |
| 162 | } |
| 163 | |
| 164 | =pod |
| 165 | |
| 166 | =item directed |
| 167 | |
| 168 | $b = $G->directed($d) |
| 169 | |
| 170 | Set the directedness of the graph $G to $d or return the |
| 171 | current directedness. Directedness defaults to true. |
| 172 | |
| 173 | =cut |
| 174 | |
| 175 | sub directed { |
| 176 | my ($G, $d) = @_; |
| 177 | |
| 178 | if (defined $d) { |
| 179 | if ($d) { |
| 180 | my $o = $G->{ D }; # Old directedness. |
| 181 | |
| 182 | $G->{ D } = $d; |
| 183 | if (not $o) { |
| 184 | my @E = $G->edges; |
| 185 | |
| 186 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 187 | $G->add_edge($v, $u); |
| 188 | } |
| 189 | } |
| 190 | |
| 191 | return bless $G, 'Graph::Directed'; # Re-bless. |
| 192 | } else { |
| 193 | return $G->undirected(not $d); |
| 194 | } |
| 195 | } |
| 196 | |
| 197 | return $G->{ D }; |
| 198 | } |
| 199 | |
| 200 | =pod |
| 201 | |
| 202 | =item undirected |
| 203 | |
| 204 | $b = $G->undirected($d) |
| 205 | |
| 206 | Set the undirectedness of the graph $G to $u or return the |
| 207 | current undirectedness. Undirectedness defaults to false. |
| 208 | |
| 209 | =cut |
| 210 | |
| 211 | sub undirected { |
| 212 | my ($G, $u) = @_; |
| 213 | |
| 214 | $G->{ D } = 1 unless defined $G->{ D }; |
| 215 | |
| 216 | if (defined $u) { |
| 217 | if ($u) { |
| 218 | my $o = $G->{ D }; # Old directedness. |
| 219 | |
| 220 | $G->{ D } = not $u; |
| 221 | if ($o) { |
| 222 | my @E = $G->edges; |
| 223 | my %E; |
| 224 | |
| 225 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 226 | # Throw away duplicate edges. |
| 227 | $G->delete_edge($u, $v) if exists $E{$v}->{$u}; |
| 228 | $E{$u}->{$v}++; |
| 229 | } |
| 230 | } |
| 231 | |
| 232 | return bless $G, 'Graph::Undirected'; # Re-bless. |
| 233 | } else { |
| 234 | return $G->directed(not $u); |
| 235 | } |
| 236 | } |
| 237 | |
| 238 | return not $G->{ D }; |
| 239 | } |
| 240 | |
| 241 | =pod |
| 242 | |
| 243 | =item has_edge |
| 244 | |
| 245 | $b = $G->has_edge($u, $v) |
| 246 | |
| 247 | Return true if the graph $G has the edge between |
| 248 | the vertices $u, $v. |
| 249 | |
| 250 | =cut |
| 251 | |
| 252 | sub has_edge { |
| 253 | my ($G, $u, $v) = @_; |
| 254 | |
| 255 | return exists $G->{ Succ }->{ $u }->{ $v } || |
| 256 | ($G->undirected && exists $G->{ Succ }->{ $v }->{ $u }); |
| 257 | } |
| 258 | |
| 259 | =pod |
| 260 | |
| 261 | =item has_edges |
| 262 | |
| 263 | $G->has_edges($u1, $v1, $u2, $v2, ...) |
| 264 | |
| 265 | In list context returns a list which contains true for each |
| 266 | edge in the graph $G defined by the vertices $u1, $v1, ..., |
| 267 | and false for each non-existing edge. In scalar context |
| 268 | returns the number of the existing edges. |
| 269 | |
| 270 | =cut |
| 271 | |
| 272 | sub has_edges { |
| 273 | my $G = shift; |
| 274 | my @e; |
| 275 | |
| 276 | while (my ($u, $v) = splice(@_, 0, 2)) { |
| 277 | push @e, $G->has_edge($u, $v); |
| 278 | } |
| 279 | |
| 280 | return wantarray ? @e : grep { $_ } @e; |
| 281 | } |
| 282 | |
| 283 | =pod |
| 284 | |
| 285 | =item has_path |
| 286 | |
| 287 | $G->has_path($u, $v, ...) |
| 288 | |
| 289 | Return true if the graph $G has the cycle defined by |
| 290 | the vertices $u, $v, ..., false otherwise. |
| 291 | |
| 292 | =cut |
| 293 | |
| 294 | sub has_path { |
| 295 | my $G = shift; |
| 296 | my $u = shift; |
| 297 | |
| 298 | while (my $v = shift) { |
| 299 | return 0 unless $G->has_edge($u, $v); |
| 300 | $u = $v; |
| 301 | } |
| 302 | |
| 303 | return 1; |
| 304 | } |
| 305 | |
| 306 | =pod |
| 307 | |
| 308 | =item has_cycle |
| 309 | |
| 310 | $G->has_cycle($u, $v, ...) |
| 311 | |
| 312 | Return true if the graph $G has the cycle defined by |
| 313 | the vertices $u, $v, ...,false otherwise. |
| 314 | |
| 315 | =cut |
| 316 | |
| 317 | sub has_cycle { |
| 318 | my $G = shift; |
| 319 | |
| 320 | return $G->has_path(@_, $_[0]); # Just wrap around. |
| 321 | } |
| 322 | |
| 323 | # _union_vertex_set |
| 324 | # |
| 325 | # $G->_union_vertex_set($u, $v) |
| 326 | # |
| 327 | # (INTERNAL USE ONLY) |
| 328 | # Adds the vertices $u and $v in the graph $G to the same vertex set. |
| 329 | # |
| 330 | sub _union_vertex_set { |
| 331 | my ($G, $u, $v) = @_; |
| 332 | |
| 333 | my $su = $G->vertex_set( $u ); |
| 334 | my $sv = $G->vertex_set( $v ); |
| 335 | |
| 336 | return if $su eq $sv; |
| 337 | |
| 338 | my $ru = $G->{ VertexSetRank }->{ $su }; |
| 339 | my $rv = $G->{ VertexSetRank }->{ $sv }; |
| 340 | |
| 341 | if ( $ru < $rv ) { # Union by rank (weight balancing). |
| 342 | $G->{ VertexSetParent }->{ $su } = $sv; |
| 343 | } else { |
| 344 | $G->{ VertexSetParent }->{ $sv } = $su; |
| 345 | $G->{ VertexSetRank }->{ $sv }++ if $ru == $rv; |
| 346 | } |
| 347 | } |
| 348 | |
| 349 | =pod |
| 350 | |
| 351 | =item vertex_set |
| 352 | |
| 353 | $s = $G->vertex_set($v) |
| 354 | |
| 355 | Returns the vertex set of the vertex $v in the graph $G. |
| 356 | A "vertex set" is represented by its parent vertex. |
| 357 | |
| 358 | =cut |
| 359 | |
| 360 | sub vertex_set { |
| 361 | my ($G, $v) = @_; |
| 362 | |
| 363 | if ( exists $G->{ VertexSetParent }->{ $v } ) { |
| 364 | # Path compression. |
| 365 | $G->{ VertexSetParent }->{ $v } = |
| 366 | $G->vertex_set( $G->{ VertexSetParent }->{ $v } ) |
| 367 | if $v ne $G->{ VertexSetParent }->{ $v }; |
| 368 | } else { |
| 369 | $G->{ VertexSetParent }->{ $v } = $v; |
| 370 | $G->{ VertexSetRank }->{ $v } = 0; |
| 371 | } |
| 372 | |
| 373 | return $G->{ VertexSetParent }->{ $v }; |
| 374 | } |
| 375 | |
| 376 | =pod |
| 377 | |
| 378 | =item add_edge |
| 379 | |
| 380 | $G = $G->add_edge($u, $v) |
| 381 | |
| 382 | Adds the edge defined by the vertices $u, $v, to the graph $G. |
| 383 | Also implicitly adds the vertices. Returns the graph. |
| 384 | |
| 385 | =cut |
| 386 | |
| 387 | sub add_edge { |
| 388 | my ($G, $u, $v) = @_; |
| 389 | |
| 390 | $G->add_vertex($u); |
| 391 | $G->add_vertex($v); |
| 392 | $G->_union_vertex_set( $u, $v ); |
| 393 | push @{ $G->{ Succ }->{ $u }->{ $v } }, $v; |
| 394 | push @{ $G->{ Pred }->{ $v }->{ $u } }, $u; |
| 395 | |
| 396 | return $G; |
| 397 | } |
| 398 | |
| 399 | =pod |
| 400 | |
| 401 | =item add_edges |
| 402 | |
| 403 | $G = $G->add_edges($u1, $v1, $u2, $v2, ...) |
| 404 | |
| 405 | Adds the edge defined by the vertices $u1, $v1, ..., |
| 406 | to the graph $G. Also implicitly adds the vertices. |
| 407 | Returns the graph. |
| 408 | |
| 409 | =cut |
| 410 | |
| 411 | sub add_edges { |
| 412 | my $G = shift; |
| 413 | |
| 414 | while (my ($u, $v) = splice(@_, 0, 2)) { |
| 415 | $G->add_edge($u, $v); |
| 416 | } |
| 417 | |
| 418 | return $G; |
| 419 | } |
| 420 | |
| 421 | =pod |
| 422 | |
| 423 | =item add_path |
| 424 | |
| 425 | $G->add_path($u, $v, ...) |
| 426 | |
| 427 | Adds the path defined by the vertices $u, $v, ..., |
| 428 | to the graph $G. Also implicitly adds the vertices. |
| 429 | Returns the graph. |
| 430 | |
| 431 | =cut |
| 432 | |
| 433 | sub add_path { |
| 434 | my $G = shift; |
| 435 | my $u = shift; |
| 436 | |
| 437 | while (my $v = shift) { |
| 438 | $G->add_edge($u, $v); |
| 439 | $u = $v; |
| 440 | } |
| 441 | |
| 442 | return $G; |
| 443 | } |
| 444 | |
| 445 | =pod |
| 446 | |
| 447 | =item add_cycle |
| 448 | |
| 449 | $G = $G->add_cycle($u, $v, ...) |
| 450 | |
| 451 | Adds the cycle defined by the vertices $u, $v, ..., |
| 452 | to the graph $G. Also implicitly adds the vertices. |
| 453 | Returns the graph. |
| 454 | |
| 455 | =cut |
| 456 | |
| 457 | sub add_cycle { |
| 458 | my $G = shift; |
| 459 | |
| 460 | $G->add_path(@_, $_[0]); # Just wrap around. |
| 461 | } |
| 462 | |
| 463 | # _successors |
| 464 | # |
| 465 | # @s = $G->_successors($v) |
| 466 | # |
| 467 | # (INTERNAL USE ONLY, use only on directed graphs) |
| 468 | # Returns the successor vertices @s of the vertex |
| 469 | # in the graph $G. |
| 470 | # |
| 471 | sub _successors { |
| 472 | my ($G, $v) = @_; |
| 473 | |
| 474 | my @s = |
| 475 | defined $G->{ Succ }->{ $v } ? |
| 476 | map { @{ $G->{ Succ }->{ $v }->{ $_ } } } |
| 477 | sort keys %{ $G->{ Succ }->{ $v } } : |
| 478 | ( ); |
| 479 | |
| 480 | return @s; |
| 481 | } |
| 482 | |
| 483 | # _predecessors |
| 484 | # |
| 485 | # @p = $G->_predecessors($v) |
| 486 | # |
| 487 | # (INTERNAL USE ONLY, use only on directed graphs) |
| 488 | # Returns the predecessor vertices @p of the vertex $v |
| 489 | # in the graph $G. |
| 490 | # |
| 491 | sub _predecessors { |
| 492 | my ($G, $v) = @_; |
| 493 | |
| 494 | my @p = |
| 495 | defined $G->{ Pred }->{ $v } ? |
| 496 | map { @{ $G->{ Pred }->{ $v }->{ $_ } } } |
| 497 | sort keys %{ $G->{ Pred }->{ $v } } : |
| 498 | ( ); |
| 499 | |
| 500 | return @p; |
| 501 | } |
| 502 | |
| 503 | =pod |
| 504 | |
| 505 | =item neighbors |
| 506 | |
| 507 | @n = $G->neighbors($v) |
| 508 | |
| 509 | Returns the neighbor vertices of the vertex in the graph. |
| 510 | (Also 'neighbours' works.) |
| 511 | |
| 512 | =cut |
| 513 | |
| 514 | sub neighbors { |
| 515 | my ($G, $v) = @_; |
| 516 | |
| 517 | my @n = ($G->_successors($v), $G->_predecessors($v)); |
| 518 | |
| 519 | return @n; |
| 520 | } |
| 521 | |
| 522 | use vars '*neighbours'; |
| 523 | *neighbours = \&neighbors; # Keep both sides of the Atlantic happy. |
| 524 | |
| 525 | =pod |
| 526 | |
| 527 | =item successors |
| 528 | |
| 529 | @s = $G->successors($v) |
| 530 | |
| 531 | Returns the successor vertices of the vertex in the graph. |
| 532 | |
| 533 | =cut |
| 534 | |
| 535 | sub successors { |
| 536 | my ($G, $v) = @_; |
| 537 | |
| 538 | return $G->directed ? $G->_successors($v) : $G->neighbors($v); |
| 539 | } |
| 540 | |
| 541 | =pod |
| 542 | |
| 543 | =item predecessors |
| 544 | |
| 545 | @p = $G->predecessors($v) |
| 546 | |
| 547 | Returns the predecessor vertices of the vertex in the graph. |
| 548 | |
| 549 | =cut |
| 550 | |
| 551 | sub predecessors { |
| 552 | my ($G, $v) = @_; |
| 553 | |
| 554 | return $G->directed ? $G->_predecessors($v) : $G->neighbors($v); |
| 555 | } |
| 556 | |
| 557 | =pod |
| 558 | |
| 559 | =item out_edges |
| 560 | |
| 561 | @e = $G->out_edges($v) |
| 562 | |
| 563 | Returns the edges leading out of the vertex $v in the graph $G. |
| 564 | In list context returns the edges as ($start_vertex, $end_vertex) |
| 565 | pairs. In scalar context returns the number of the edges. |
| 566 | |
| 567 | =cut |
| 568 | |
| 569 | sub out_edges { |
| 570 | my ($G, $v) = @_; |
| 571 | |
| 572 | return () unless $G->has_vertex($v); |
| 573 | |
| 574 | my @e = $G->_edges($v, undef); |
| 575 | |
| 576 | return wantarray ? @e : @e / 2; |
| 577 | } |
| 578 | |
| 579 | =pod |
| 580 | |
| 581 | =item in_edges |
| 582 | |
| 583 | @e = $G->in_edges($v) |
| 584 | |
| 585 | Returns the edges leading into the vertex $v in the graph $G. |
| 586 | In list context returns the edges as ($start_vertex, $end_vertex) |
| 587 | pairs; in scalar context returns the number of the edges. |
| 588 | |
| 589 | =cut |
| 590 | |
| 591 | sub in_edges { |
| 592 | my ($G, $v) = @_; |
| 593 | |
| 594 | return () unless $G->has_vertex($v); |
| 595 | |
| 596 | my @e = $G->_edges(undef, $v); |
| 597 | |
| 598 | return wantarray ? @e : @e / 2; |
| 599 | } |
| 600 | |
| 601 | =pod |
| 602 | |
| 603 | =item edges |
| 604 | |
| 605 | @e = $G->edges($u, $v) |
| 606 | |
| 607 | Returns the edges between the vertices $u and $v, or if $v |
| 608 | is undefined, the edges leading into or out of the vertex $u, |
| 609 | or if $u is undefined, returns all the edges, of the graph $G. |
| 610 | In list context returns the edges as a list of |
| 611 | $start_vertex, $end_vertex pairs; in scalar context |
| 612 | returns the number of the edges. |
| 613 | |
| 614 | =cut |
| 615 | |
| 616 | sub edges { |
| 617 | my ($G, $u, $v) = @_; |
| 618 | |
| 619 | return () if defined $v and not $G->has_vertex($v); |
| 620 | |
| 621 | my @e = |
| 622 | defined $u ? |
| 623 | ( defined $v ? |
| 624 | $G->_edges($u, $v) : |
| 625 | ($G->in_edges($u), $G->out_edges($u)) ) : |
| 626 | $G->_edges; |
| 627 | |
| 628 | return wantarray ? @e : @e / 2; |
| 629 | } |
| 630 | |
| 631 | =pod |
| 632 | |
| 633 | =item delete_edge |
| 634 | |
| 635 | $G = $G->delete_edge($u, $v) |
| 636 | |
| 637 | Deletes an edge defined by the vertices $u, $v from the graph $G. |
| 638 | Note that the edge need not actually exist. |
| 639 | Returns the graph. |
| 640 | |
| 641 | =cut |
| 642 | |
| 643 | sub delete_edge { |
| 644 | my ($G, $u, $v) = @_; |
| 645 | |
| 646 | pop @{ $G->{ Succ }->{ $u }->{ $v } }; |
| 647 | pop @{ $G->{ Pred }->{ $v }->{ $u } }; |
| 648 | |
| 649 | delete $G->{ Succ }->{ $u }->{ $v } |
| 650 | unless @{ $G->{ Succ }->{ $u }->{ $v } }; |
| 651 | delete $G->{ Pred }->{ $v }->{ $u } |
| 652 | unless @{ $G->{ Pred }->{ $v }->{ $u } }; |
| 653 | |
| 654 | delete $G->{ Succ }->{ $u } |
| 655 | unless keys %{ $G->{ Succ }->{ $u } }; |
| 656 | delete $G->{ Pred }->{ $v } |
| 657 | unless keys %{ $G->{ Pred }->{ $v } }; |
| 658 | |
| 659 | return $G; |
| 660 | } |
| 661 | |
| 662 | =pod |
| 663 | |
| 664 | =item delete_edges |
| 665 | |
| 666 | $G = $G->delete_edges($u1, $v1, $u2, $v2, ..) |
| 667 | |
| 668 | Deletes edges defined by the vertices $u1, $v1, ..., |
| 669 | from the graph $G. |
| 670 | Note that the edges need not actually exist. |
| 671 | Returns the graph. |
| 672 | |
| 673 | =cut |
| 674 | |
| 675 | sub delete_edges { |
| 676 | my $G = shift; |
| 677 | |
| 678 | while (my ($u, $v) = splice(@_, 0, 2)) { |
| 679 | if (defined $v) { |
| 680 | $G->delete_edge($u, $v); |
| 681 | } else { |
| 682 | my @e = $G->edges($u); |
| 683 | |
| 684 | while (($u, $v) = splice(@e, 0, 2)) { |
| 685 | $G->delete_edge($u, $v); |
| 686 | } |
| 687 | } |
| 688 | } |
| 689 | |
| 690 | return $G; |
| 691 | } |
| 692 | |
| 693 | =pod |
| 694 | |
| 695 | =item delete_path |
| 696 | |
| 697 | $G = $G->delete_path($u, $v, ...) |
| 698 | |
| 699 | Deletes a path defined by the vertices $u, $v, ..., from the graph $G. |
| 700 | Note that the path need not actually exist. Returns the graph. |
| 701 | |
| 702 | =cut |
| 703 | |
| 704 | sub delete_path { |
| 705 | my $G = shift; |
| 706 | my $u = shift; |
| 707 | |
| 708 | while (my $v = shift) { |
| 709 | $G->delete_edge($u, $v); |
| 710 | $u = $v; |
| 711 | } |
| 712 | |
| 713 | return $G; |
| 714 | } |
| 715 | |
| 716 | =pod |
| 717 | |
| 718 | =item delete_cycle |
| 719 | |
| 720 | $G = $G->delete_cycle($u, $v, ...) |
| 721 | |
| 722 | Deletes a cycle defined by the vertices $u, $v, ..., from the graph $G. |
| 723 | Note that the cycle need not actually exist. Returns the graph. |
| 724 | |
| 725 | =cut |
| 726 | |
| 727 | sub delete_cycle { |
| 728 | my $G = shift; |
| 729 | |
| 730 | $G->delete_path(@_, $_[0]); # Just wrap around. |
| 731 | } |
| 732 | |
| 733 | =pod |
| 734 | |
| 735 | =item delete_vertex |
| 736 | |
| 737 | $G = $G->delete_vertex($v) |
| 738 | |
| 739 | Deletes the vertex $v and all its edges from the graph $G. |
| 740 | Note that the vertex need not actually exist. |
| 741 | Returns the graph. |
| 742 | |
| 743 | =cut |
| 744 | |
| 745 | sub delete_vertex { |
| 746 | my ($G, $v) = @_; |
| 747 | |
| 748 | $G->delete_edges($v); |
| 749 | |
| 750 | delete $G->{ V }->{ $v }; |
| 751 | |
| 752 | return $G; |
| 753 | } |
| 754 | |
| 755 | =pod |
| 756 | |
| 757 | =item delete_vertices |
| 758 | |
| 759 | $G = $G->delete_vertices(@v) |
| 760 | |
| 761 | Deletes the vertices @v and all their edges from the graph $G. |
| 762 | Note that the vertices need not actually exist. |
| 763 | Returns the graph. |
| 764 | |
| 765 | =cut |
| 766 | |
| 767 | sub delete_vertices { |
| 768 | my $G = shift; |
| 769 | |
| 770 | foreach my $v (@_) { |
| 771 | $G->delete_vertex($v); |
| 772 | } |
| 773 | |
| 774 | return $G; |
| 775 | } |
| 776 | |
| 777 | =pod |
| 778 | |
| 779 | =item in_degree |
| 780 | |
| 781 | $d = $G->in_degree($v) |
| 782 | |
| 783 | Returns the in-degree of the vertex $v in the graph $G, |
| 784 | or, if $v is undefined, the total in-degree of all the |
| 785 | vertices of the graph, or undef if the vertex doesn't |
| 786 | exist in the graph. |
| 787 | |
| 788 | =cut |
| 789 | |
| 790 | sub in_degree { |
| 791 | my ($G, $v) = @_; |
| 792 | |
| 793 | return undef unless $G->has_vertex($v); |
| 794 | |
| 795 | if ($G->directed) { |
| 796 | if (defined $v) { |
| 797 | return scalar $G->in_edges($v); |
| 798 | } else { |
| 799 | my $in = 0; |
| 800 | |
| 801 | foreach my $v ($G->vertices) { |
| 802 | $in += $G->in_degree($v); |
| 803 | } |
| 804 | |
| 805 | return $in; |
| 806 | } |
| 807 | } else { |
| 808 | return scalar $G->edges($v); |
| 809 | } |
| 810 | } |
| 811 | |
| 812 | =pod |
| 813 | |
| 814 | =item out_degree |
| 815 | |
| 816 | $d = $G->out_degree($v) |
| 817 | |
| 818 | Returns the out-degree of the vertex $v in the graph $G, |
| 819 | or, if $v is undefined, the total out-degree of all the |
| 820 | vertices of the graph, of undef if the vertex doesn't |
| 821 | exist in the graph. |
| 822 | |
| 823 | =cut |
| 824 | |
| 825 | sub out_degree { |
| 826 | my ($G, $v) = @_; |
| 827 | |
| 828 | return undef unless $G->has_vertex($v); |
| 829 | |
| 830 | if ($G->directed) { |
| 831 | if (defined $v) { |
| 832 | return scalar $G->out_edges($v); |
| 833 | } else { |
| 834 | my $out = 0; |
| 835 | |
| 836 | foreach my $v ($G->vertices) { |
| 837 | $out += $G->out_degree($v); |
| 838 | } |
| 839 | |
| 840 | return $out; |
| 841 | } |
| 842 | } else { |
| 843 | return scalar $G->edges($v); |
| 844 | } |
| 845 | } |
| 846 | |
| 847 | =pod |
| 848 | |
| 849 | =item degree |
| 850 | |
| 851 | $d = $G->degree($v) |
| 852 | |
| 853 | Returns the degree of the vertex $v in the graph $G |
| 854 | or, if $v is undefined, the total degree of all the |
| 855 | vertices of the graph, or undef if the vertex $v |
| 856 | doesn't exist in the graph. |
| 857 | |
| 858 | =cut |
| 859 | |
| 860 | sub degree { |
| 861 | my ($G, $v) = @_; |
| 862 | |
| 863 | if (defined $v) { |
| 864 | return undef unless $G->has_vertex($v); |
| 865 | |
| 866 | if ($G->directed) { |
| 867 | return $G->in_degree($v) - $G->out_degree($v); |
| 868 | } else { |
| 869 | return $G->edges($v); |
| 870 | } |
| 871 | } else { |
| 872 | if ($G->directed) { |
| 873 | return 0; |
| 874 | } else { |
| 875 | my $deg = 0; |
| 876 | |
| 877 | foreach my $v ($G->vertices) { |
| 878 | $deg += $G->degree($v); |
| 879 | } |
| 880 | |
| 881 | return $deg; |
| 882 | } |
| 883 | } |
| 884 | } |
| 885 | |
| 886 | =pod |
| 887 | |
| 888 | =item average_degree |
| 889 | |
| 890 | $d = $G->average_degree |
| 891 | |
| 892 | Returns the average degree of the vertices of the graph $G. |
| 893 | |
| 894 | =cut |
| 895 | |
| 896 | sub average_degree { |
| 897 | my $G = shift; |
| 898 | my $V = $G->vertices; |
| 899 | |
| 900 | return $V ? $G->degree / $V : 0; |
| 901 | } |
| 902 | |
| 903 | =pod |
| 904 | |
| 905 | =item is_source_vertex |
| 906 | |
| 907 | $b = $G->is_source_vertex($v) |
| 908 | |
| 909 | Returns true if the vertex $v is a source vertex of the graph $G. |
| 910 | |
| 911 | =cut |
| 912 | |
| 913 | sub is_source_vertex { |
| 914 | my ($G, $v) = @_; |
| 915 | |
| 916 | $G->in_degree($v) == 0 && $G->out_degree($v) > 0; |
| 917 | } |
| 918 | |
| 919 | =pod |
| 920 | |
| 921 | =item is_sink_vertex |
| 922 | |
| 923 | $b = $G->is_sink_vertex($v) |
| 924 | |
| 925 | Returns true if the vertex $v is a sink vertex of the graph $G. |
| 926 | |
| 927 | =cut |
| 928 | |
| 929 | sub is_sink_vertex { |
| 930 | my ($G, $v) = @_; |
| 931 | |
| 932 | $G->in_degree($v) > 0 && $G->out_degree($v) == 0; |
| 933 | } |
| 934 | |
| 935 | =pod |
| 936 | |
| 937 | =item is_isolated_vertex |
| 938 | |
| 939 | $b = $G->is_isolated_vertex($v) |
| 940 | |
| 941 | Returns true if the vertex $v is a isolated vertex of the graph $G. |
| 942 | |
| 943 | =cut |
| 944 | |
| 945 | sub is_isolated_vertex { |
| 946 | my ($G, $v) = @_; |
| 947 | |
| 948 | $G->in_degree($v) == 0 && $G->out_degree($v) == 0; |
| 949 | } |
| 950 | |
| 951 | =pod |
| 952 | |
| 953 | =item is_exterior_vertex |
| 954 | |
| 955 | $b = $G->is_exterior_vertex($v) |
| 956 | |
| 957 | Returns true if the vertex $v is a exterior vertex of the graph $G. |
| 958 | |
| 959 | =cut |
| 960 | |
| 961 | sub is_exterior_vertex { |
| 962 | my ($G, $v) = @_; |
| 963 | |
| 964 | $G->in_degree($v) == 0 xor $G->out_degree($v) == 0; |
| 965 | } |
| 966 | |
| 967 | =pod |
| 968 | |
| 969 | =item is_interior_vertex |
| 970 | |
| 971 | $b = $G->is_interior_vertex($v) |
| 972 | |
| 973 | Returns true if the vertex $v is a interior vertex of the graph $G. |
| 974 | |
| 975 | =cut |
| 976 | |
| 977 | sub is_interior_vertex { |
| 978 | my ($G, $v) = @_; |
| 979 | |
| 980 | $G->in_degree($v) && $G->out_degree($v); |
| 981 | } |
| 982 | |
| 983 | =pod |
| 984 | |
| 985 | =item is_self_loop_vertex |
| 986 | |
| 987 | $b = $G->is_self_loop_vertex($v) |
| 988 | |
| 989 | Returns true if the vertex $v is a self-loop vertex of the graph $G. |
| 990 | |
| 991 | =cut |
| 992 | |
| 993 | sub is_self_loop_vertex { |
| 994 | my ($G, $v) = @_; |
| 995 | |
| 996 | exists $G->{ Succ }->{ $v }->{ $v }; |
| 997 | } |
| 998 | |
| 999 | =pod |
| 1000 | |
| 1001 | =item source_vertices |
| 1002 | |
| 1003 | @s = $G->source_vertices |
| 1004 | |
| 1005 | Returns the source vertices @s of the graph $G. |
| 1006 | |
| 1007 | =cut |
| 1008 | |
| 1009 | sub source_vertices { |
| 1010 | my $G = shift; |
| 1011 | |
| 1012 | return grep { $G->is_source_vertex($_) } $G->vertices; |
| 1013 | } |
| 1014 | |
| 1015 | =pod |
| 1016 | |
| 1017 | =item sink_vertices |
| 1018 | |
| 1019 | @s = $G->sink_vertices |
| 1020 | |
| 1021 | Returns the sink vertices @s of the graph $G. |
| 1022 | |
| 1023 | =cut |
| 1024 | |
| 1025 | sub sink_vertices { |
| 1026 | my $G = shift; |
| 1027 | |
| 1028 | return grep { $G->is_sink_vertex($_) } $G->vertices; |
| 1029 | } |
| 1030 | |
| 1031 | =pod |
| 1032 | |
| 1033 | =item isolated_vertices |
| 1034 | |
| 1035 | @i = $G->isolated_vertices |
| 1036 | |
| 1037 | Returns the isolated vertices @i of the graph $G. |
| 1038 | |
| 1039 | =cut |
| 1040 | |
| 1041 | sub isolated_vertices { |
| 1042 | my $G = shift; |
| 1043 | |
| 1044 | return grep { $G->is_isolated_vertex($_) } $G->vertices; |
| 1045 | } |
| 1046 | |
| 1047 | =pod |
| 1048 | |
| 1049 | =item exterior_vertices |
| 1050 | |
| 1051 | @e = $G->exterior_vertices |
| 1052 | |
| 1053 | Returns the exterior vertices @e of the graph $G. |
| 1054 | |
| 1055 | =cut |
| 1056 | |
| 1057 | sub exterior_vertices { |
| 1058 | my $G = shift; |
| 1059 | |
| 1060 | return grep { $G->is_exterior_vertex($_) } $G->vertices; |
| 1061 | } |
| 1062 | |
| 1063 | =pod |
| 1064 | |
| 1065 | =item interior_vertices |
| 1066 | |
| 1067 | @i = $G->interior_vertices |
| 1068 | |
| 1069 | Returns the interior vertices @i of the graph $G. |
| 1070 | |
| 1071 | =cut |
| 1072 | |
| 1073 | sub interior_vertices { |
| 1074 | my $G = shift; |
| 1075 | |
| 1076 | return grep { $G->is_interior_vertex($_) } $G->vertices; |
| 1077 | } |
| 1078 | |
| 1079 | =pod |
| 1080 | |
| 1081 | =item self_loop_vertices |
| 1082 | |
| 1083 | @s = $G->self_loop_vertices |
| 1084 | |
| 1085 | Returns the self-loop vertices @s of the graph $G. |
| 1086 | |
| 1087 | =cut |
| 1088 | |
| 1089 | sub self_loop_vertices { |
| 1090 | my $G = shift; |
| 1091 | |
| 1092 | return grep { $G->is_self_loop_vertex($_) } $G->vertices; |
| 1093 | } |
| 1094 | |
| 1095 | =pod |
| 1096 | |
| 1097 | =item density_limits |
| 1098 | |
| 1099 | ($sparse, $dense, $complete) = $G->density_limits |
| 1100 | |
| 1101 | Returns the density limits for the number of edges |
| 1102 | in the graph $G. Note that reaching $complete edges |
| 1103 | does not really guarantee completeness because we |
| 1104 | can have multigraphs. The limit of sparse is less |
| 1105 | than 1/4 of the edges of the complete graph, the |
| 1106 | limit of dense is more than 3/4 of the edges of the |
| 1107 | complete graph. |
| 1108 | |
| 1109 | =cut |
| 1110 | |
| 1111 | sub density_limits { |
| 1112 | my $G = shift; |
| 1113 | |
| 1114 | my $V = $G->vertices; |
| 1115 | my $M = $V * ($V - 1); |
| 1116 | |
| 1117 | $M = $M / 2 if $G->undirected; |
| 1118 | |
| 1119 | return ($M/4, 3*$M/4, $M); |
| 1120 | } |
| 1121 | |
| 1122 | =pod |
| 1123 | |
| 1124 | =item density |
| 1125 | |
| 1126 | $d = $G->density |
| 1127 | |
| 1128 | Returns the density $d of the graph $G. |
| 1129 | |
| 1130 | =cut |
| 1131 | |
| 1132 | sub density { |
| 1133 | my $G = shift; |
| 1134 | my ($sparse, $dense, $complete) = $G->density_limits; |
| 1135 | |
| 1136 | return $complete ? $G->edges / $complete : 0; |
| 1137 | } |
| 1138 | |
| 1139 | =pod |
| 1140 | |
| 1141 | =item is_sparse |
| 1142 | |
| 1143 | $d = $G->is_sparse |
| 1144 | |
| 1145 | Returns true if the graph $G is sparse. |
| 1146 | |
| 1147 | =cut |
| 1148 | |
| 1149 | sub is_sparse { |
| 1150 | my $G = shift; |
| 1151 | my ($sparse, $dense, $complete) = $G->density_limits; |
| 1152 | |
| 1153 | return $complete ? $G->edges / $complete <= $dense : 1; |
| 1154 | } |
| 1155 | |
| 1156 | =pod |
| 1157 | |
| 1158 | =item is_dense |
| 1159 | |
| 1160 | $d = $G->is_dense |
| 1161 | |
| 1162 | Returns true if the graph $G is dense. |
| 1163 | |
| 1164 | =cut |
| 1165 | |
| 1166 | sub is_dense { |
| 1167 | my $G = shift; |
| 1168 | my ($sparse, $dense, $complete) = $G->density_limits; |
| 1169 | |
| 1170 | return $complete ? $G->edges / $complete >= $dense : 0; |
| 1171 | } |
| 1172 | |
| 1173 | =pod |
| 1174 | |
| 1175 | =item complete |
| 1176 | |
| 1177 | $C = $G->complete; |
| 1178 | |
| 1179 | Returns a new complete graph $C corresponding to the graph $G. |
| 1180 | |
| 1181 | =cut |
| 1182 | |
| 1183 | sub complete { |
| 1184 | my $G = shift; |
| 1185 | my $C = (ref $G)->new; |
| 1186 | my @V = $G->vertices; |
| 1187 | |
| 1188 | if ($G->directed) { |
| 1189 | foreach my $u (@V) { |
| 1190 | foreach my $v (@V) { |
| 1191 | $C->add_edge($u, $v) unless $u eq $v; |
| 1192 | } |
| 1193 | } |
| 1194 | } else { |
| 1195 | my %E; |
| 1196 | |
| 1197 | foreach my $u (@V) { |
| 1198 | foreach my $v (@V) { |
| 1199 | next if $u eq $v or $E{$u}->{$v} || $E{$v}->{$u}; |
| 1200 | $C->add_edge($u, $v); |
| 1201 | $E{$u}->{$v}++; |
| 1202 | $E{$v}->{$u}++; |
| 1203 | } |
| 1204 | } |
| 1205 | } |
| 1206 | |
| 1207 | $C->directed($G->directed); |
| 1208 | |
| 1209 | return $C; |
| 1210 | } |
| 1211 | |
| 1212 | =pod |
| 1213 | |
| 1214 | =item complement |
| 1215 | |
| 1216 | $C = $G->complement; |
| 1217 | |
| 1218 | Returns a new complement graph $C corresponding to the graph $G. |
| 1219 | |
| 1220 | =cut |
| 1221 | |
| 1222 | sub complement { |
| 1223 | my $G = shift; |
| 1224 | my $C = $G->complete; |
| 1225 | |
| 1226 | if (my @E = $G->edges) { |
| 1227 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 1228 | $C->delete_edge($u, $v); |
| 1229 | } |
| 1230 | } |
| 1231 | |
| 1232 | return $C; |
| 1233 | } |
| 1234 | |
| 1235 | =pod |
| 1236 | |
| 1237 | =item copy |
| 1238 | |
| 1239 | $C = $G->copy; |
| 1240 | |
| 1241 | Returns a new graph $C corresponding to the graph $G. |
| 1242 | |
| 1243 | =cut |
| 1244 | |
| 1245 | sub copy { |
| 1246 | my $G = shift; |
| 1247 | my $C = (ref $G)->new($G->vertices); |
| 1248 | |
| 1249 | if (my @E = $G->edges) { |
| 1250 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 1251 | $C->add_edge($u, $v); |
| 1252 | } |
| 1253 | } |
| 1254 | |
| 1255 | $C->directed($G->directed); |
| 1256 | |
| 1257 | return $C; |
| 1258 | } |
| 1259 | |
| 1260 | =pod |
| 1261 | |
| 1262 | =item transpose |
| 1263 | |
| 1264 | $T = $G->transpose; |
| 1265 | |
| 1266 | Returns a new transpose graph $T corresponding to the graph $G. |
| 1267 | |
| 1268 | =cut |
| 1269 | |
| 1270 | sub transpose { |
| 1271 | my $G = shift; |
| 1272 | |
| 1273 | return $G->copy if $G->undirected; |
| 1274 | |
| 1275 | my $T = (ref $G)->new($G->vertices); |
| 1276 | |
| 1277 | if (my @E = $G->edges) { |
| 1278 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 1279 | $T->add_edge($v, $u); |
| 1280 | } |
| 1281 | } |
| 1282 | |
| 1283 | return $T; |
| 1284 | } |
| 1285 | |
| 1286 | # _stringify |
| 1287 | # |
| 1288 | # $s = $G->_stringify($connector, $separator) |
| 1289 | # |
| 1290 | # (INTERNAL USE ONLY) |
| 1291 | # Returns a string representation of the graph $G. |
| 1292 | # The edges are represented by $connector and edges/isolated |
| 1293 | # vertices are represented by $separator. |
| 1294 | # |
| 1295 | sub _stringify { |
| 1296 | my ($G, $connector, $separator) = @_; |
| 1297 | my @E = $G->edges; |
| 1298 | my @e = map { [ $_ ] } $G->isolated_vertices; |
| 1299 | |
| 1300 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 1301 | push @e, [$u, $v]; |
| 1302 | } |
| 1303 | |
| 1304 | return join($separator, |
| 1305 | map { @$_ == 2 ? |
| 1306 | join($connector, $_->[0], $_->[1]) : |
| 1307 | $_->[0] } |
| 1308 | sort { $a->[0] cmp $b->[0] || @$a <=> @$b } @e); |
| 1309 | } |
| 1310 | |
| 1311 | =pod |
| 1312 | |
| 1313 | =item set_attribute |
| 1314 | |
| 1315 | $G->set_attribute($attribute, $value) |
| 1316 | $G->set_attribute($attribute, $v, $value) |
| 1317 | $G->set_attribute($attribute, $u, $v, $value) |
| 1318 | |
| 1319 | Sets the $attribute of graph/vertex/edge to $value |
| 1320 | but only if the vertex/edge already exists. Returns |
| 1321 | true if the attribute is set successfully, false if not. |
| 1322 | |
| 1323 | =cut |
| 1324 | |
| 1325 | sub set_attribute { |
| 1326 | my $G = shift; |
| 1327 | my $attribute = shift; |
| 1328 | my $value = pop; |
| 1329 | my ($u, $v) = @_; |
| 1330 | |
| 1331 | if (defined $u) { |
| 1332 | return 0 unless $G->has_vertex($u); |
| 1333 | if (defined $v) { |
| 1334 | return 0 unless $G->has_edge($u, $v); |
| 1335 | $G->{ Attr }->{ E }->{ $u }->{ $v }->{ $attribute } = $value; |
| 1336 | $G->{ Attr }->{ E }->{ $v }->{ $u }->{ $attribute } = $value |
| 1337 | if $G->undirected; |
| 1338 | } else { |
| 1339 | $G->{ Attr }->{ V }->{ $u }->{ $attribute } = $value; |
| 1340 | } |
| 1341 | } else { |
| 1342 | $G->{ Attr }->{ G }->{ $attribute } = $value; |
| 1343 | } |
| 1344 | |
| 1345 | return 1; |
| 1346 | } |
| 1347 | |
| 1348 | =pod |
| 1349 | |
| 1350 | =item get_attribute |
| 1351 | |
| 1352 | $value = $G->get_attribute($attribute) |
| 1353 | $value = $G->get_attribute($attribute, $v) |
| 1354 | $value = $G->get_attribute($attribute, $u, $v) |
| 1355 | |
| 1356 | Returns the $value of $attribute of graph/vertex/edge. |
| 1357 | |
| 1358 | =cut |
| 1359 | |
| 1360 | sub get_attribute { |
| 1361 | my $G = shift; |
| 1362 | my $attribute = shift; |
| 1363 | my ($u, $v) = @_; |
| 1364 | |
| 1365 | if (defined $u) { |
| 1366 | if (defined $v) { |
| 1367 | return undef |
| 1368 | unless exists $G->{ Attr }->{ E }; |
| 1369 | |
| 1370 | my $E = $G->{ Attr }->{ E }; |
| 1371 | |
| 1372 | if ( $G->directed ) { |
| 1373 | return $E->{ $u }->{ $v }->{ $attribute }; |
| 1374 | } else { |
| 1375 | return undef |
| 1376 | unless exists $G->{ Attr }->{ E }; |
| 1377 | |
| 1378 | return $E->{ $u }->{ $v }->{ $attribute } |
| 1379 | if exists $E->{ $u }->{ $v }->{ $attribute }; |
| 1380 | |
| 1381 | return $E->{ $v }->{ $u }->{ $attribute }; |
| 1382 | } |
| 1383 | } else { |
| 1384 | return $G->{ Attr }->{ V }->{ $u }->{ $attribute }; |
| 1385 | } |
| 1386 | } else { |
| 1387 | return $G->{ Attr }->{ G }->{ $attribute }; |
| 1388 | } |
| 1389 | } |
| 1390 | |
| 1391 | =pod |
| 1392 | |
| 1393 | =item has_attribute |
| 1394 | |
| 1395 | $value = $G->has_attribute($attribute) |
| 1396 | $value = $G->has_attribute($attribute, $v) |
| 1397 | $value = $G->has_attribute($attribute, $u, $v) |
| 1398 | |
| 1399 | Returns the $value of $attribute of graph/vertex/edge. |
| 1400 | |
| 1401 | =cut |
| 1402 | |
| 1403 | sub has_attribute { |
| 1404 | my $G = shift; |
| 1405 | my $attribute = shift; |
| 1406 | my ($u, $v) = @_; |
| 1407 | |
| 1408 | if (defined $u) { |
| 1409 | if (defined $v) { |
| 1410 | return undef |
| 1411 | unless exists $G->{ Attr }->{ E }; |
| 1412 | |
| 1413 | my $E = $G->{ Attr }->{ E }; |
| 1414 | |
| 1415 | if ( $G->directed ) { |
| 1416 | return exists $E->{ $u }->{ $v }->{ $attribute }; |
| 1417 | } else { |
| 1418 | return exists $E->{ $u }->{ $v }->{ $attribute } or |
| 1419 | exists $E->{ $v }->{ $u }->{ $attribute }; |
| 1420 | } |
| 1421 | } else { |
| 1422 | exists $G->{ Attr }->{ V }->{ $u }->{ $attribute }; |
| 1423 | } |
| 1424 | } else { |
| 1425 | exists $G->{ Attr } && |
| 1426 | exists $G->{ Attr }->{ G }->{ $attribute }; |
| 1427 | } |
| 1428 | } |
| 1429 | |
| 1430 | =pod |
| 1431 | |
| 1432 | =item get_attributes |
| 1433 | |
| 1434 | %attributes = $G->get_attributes() |
| 1435 | %attributes = $G->get_attributes($v) |
| 1436 | %attributes = $G->get_attributes($u, $v) |
| 1437 | |
| 1438 | Returns as a hash all the attribute names and values |
| 1439 | of graph/vertex/edge. |
| 1440 | |
| 1441 | =cut |
| 1442 | |
| 1443 | sub get_attributes { |
| 1444 | my $G = shift; |
| 1445 | my ($u, $v) = @_; |
| 1446 | |
| 1447 | return ( ) unless exists $G->{ Attr }; |
| 1448 | if (defined $u) { |
| 1449 | if (defined $v) { |
| 1450 | return exists $G->{ Attr }->{ E } && |
| 1451 | exists $G->{ Attr }->{ E }->{ $u } && |
| 1452 | exists $G->{ Attr }->{ E }->{ $u }->{ $v } ? |
| 1453 | %{ $G->{ Attr }->{ E }->{ $u }->{ $v } } : |
| 1454 | ( ); |
| 1455 | } else { |
| 1456 | return exists $G->{ Attr }->{ V } && |
| 1457 | exists $G->{ Attr }->{ V }->{ $u } ? |
| 1458 | %{ $G->{ Attr }->{ V }->{ $u } } : ( ); |
| 1459 | } |
| 1460 | } else { |
| 1461 | return exists $G->{ Attr }->{ G } ? |
| 1462 | %{ $G->{ Attr }->{ G } } : ( ); |
| 1463 | } |
| 1464 | } |
| 1465 | |
| 1466 | =pod |
| 1467 | |
| 1468 | =item delete_attribute |
| 1469 | |
| 1470 | $G->delete_attribute($attribute) |
| 1471 | $G->delete_attribute($attribute, $v) |
| 1472 | $G->delete_attribute($attribute, $u, $v) |
| 1473 | |
| 1474 | Deletes the $attribute of graph/vertex/edge. |
| 1475 | |
| 1476 | =cut |
| 1477 | |
| 1478 | sub delete_attribute { |
| 1479 | my $G = shift; |
| 1480 | my $attribute = shift; |
| 1481 | my ($u, $v) = @_; |
| 1482 | |
| 1483 | if (defined $u) { |
| 1484 | if (defined $v) { |
| 1485 | return undef |
| 1486 | unless exists $G->{ Attr }->{ E }; |
| 1487 | |
| 1488 | my $E = $G->{ Attr }->{ E }; |
| 1489 | |
| 1490 | if ( $G->directed ) { |
| 1491 | delete $E->{ $u }->{ $v }->{ $attribute }; |
| 1492 | } else { |
| 1493 | delete $E->{ $v }->{ $u }->{ $attribute }; |
| 1494 | delete $E->{ $v }->{ $u }->{ $attribute }; |
| 1495 | } |
| 1496 | } else { |
| 1497 | delete $G->{ Attr }->{ V }->{ $u }->{ $attribute }; |
| 1498 | } |
| 1499 | } else { |
| 1500 | delete $G->{ Attr }->{ G }->{ $attribute }; |
| 1501 | } |
| 1502 | } |
| 1503 | |
| 1504 | =pod |
| 1505 | |
| 1506 | =item delete_attributes |
| 1507 | |
| 1508 | $G->delete_attributes() |
| 1509 | $G->delete_attributes($v) |
| 1510 | $G->delete_attributes($u, $v) |
| 1511 | |
| 1512 | Deletes all the attributes of graph/vertex/edge. |
| 1513 | |
| 1514 | =cut |
| 1515 | |
| 1516 | sub delete_attributes { |
| 1517 | my $G = shift; |
| 1518 | my ($u, $v) = @_; |
| 1519 | |
| 1520 | if (defined $u) { |
| 1521 | if (defined $v) { |
| 1522 | delete $G->{ Attr }->{ E }->{ $u }->{ $v }; |
| 1523 | } else { |
| 1524 | delete $G->{ Attr }->{ V }->{ $u }; |
| 1525 | } |
| 1526 | } else { |
| 1527 | delete $G->{ Attr }->{ G }; |
| 1528 | } |
| 1529 | } |
| 1530 | |
| 1531 | =pod |
| 1532 | |
| 1533 | =item add_weighted_edge |
| 1534 | |
| 1535 | $G->add_weighted_edge($u, $w, $v, $a) |
| 1536 | |
| 1537 | Adds in the graph $G an edge from vertex $u to vertex $v |
| 1538 | and the edge attribute 'weight' set to $w. |
| 1539 | |
| 1540 | =cut |
| 1541 | |
| 1542 | sub add_weighted_edge { |
| 1543 | my ($G, $u, $w, $v, $a) = @_; |
| 1544 | |
| 1545 | $G->add_edge($u, $v); |
| 1546 | $G->set_attribute('weight', $u, $v, $w); |
| 1547 | } |
| 1548 | |
| 1549 | =pod |
| 1550 | |
| 1551 | =item add_weighted_edges |
| 1552 | |
| 1553 | $G->add_weighted_edges($u1, $w1, $v1, $u2, $w2, $v2, ...) |
| 1554 | |
| 1555 | Adds in the graph $G the weighted edges. |
| 1556 | |
| 1557 | =cut |
| 1558 | |
| 1559 | sub add_weighted_edges { |
| 1560 | my $G = shift; |
| 1561 | |
| 1562 | while (my ($u, $w, $v) = splice(@_, 0, 3)) { |
| 1563 | $G->add_weighted_edge($u, $w, $v); |
| 1564 | } |
| 1565 | } |
| 1566 | |
| 1567 | =pod |
| 1568 | |
| 1569 | =item add_weighted_path |
| 1570 | |
| 1571 | $G->add_weighted_path($v1, $w1, $v2, $w2, ..., $wnm1, $vn) |
| 1572 | |
| 1573 | Adds in the graph $G the n edges defined by the path $v1 ... $vn |
| 1574 | with the n-1 'weight' attributes $w1 ... $wnm1 |
| 1575 | |
| 1576 | =cut |
| 1577 | |
| 1578 | sub add_weighted_path { |
| 1579 | my $G = shift; |
| 1580 | my $u = shift; |
| 1581 | |
| 1582 | while (my ($w, $v) = splice(@_, 0, 2)) { |
| 1583 | $G->add_weighted_edge($u, $w, $v); |
| 1584 | $u = $v; |
| 1585 | } |
| 1586 | } |
| 1587 | |
| 1588 | =pod |
| 1589 | |
| 1590 | =item MST_Kruskal |
| 1591 | |
| 1592 | $MST = $G->MST_Kruskal; |
| 1593 | |
| 1594 | Returns Kruskal's Minimum Spanning Tree (as a graph) of |
| 1595 | the graph $G based on the 'weight' attributes of the edges. |
| 1596 | (Needs the ->vertex_set() method.) |
| 1597 | |
| 1598 | =cut |
| 1599 | |
| 1600 | sub MST_Kruskal { |
| 1601 | my $G = shift; |
| 1602 | my $MST = (ref $G)->new; |
| 1603 | my @E = $G->edges; |
| 1604 | my (@W, $u, $v, $w); |
| 1605 | |
| 1606 | while (($u, $v) = splice(@E, 0, 2)) { |
| 1607 | $w = $G->get_attribute('weight', $u, $v); |
| 1608 | next unless defined $w; # undef weight == infinitely heavy |
| 1609 | push @W, [ $u, $v, $w ]; |
| 1610 | } |
| 1611 | |
| 1612 | $MST->directed( $G->directed ); |
| 1613 | |
| 1614 | # Sort by weights. |
| 1615 | foreach my $e ( sort { $a->[ 2 ] <=> $b->[ 2 ] } @W ) { |
| 1616 | ($u, $v, $w) = @$e; |
| 1617 | $MST->add_weighted_edge( $u, $w, $v ) |
| 1618 | unless $MST->vertex_set( $u ) eq $MST->vertex_set( $v ); |
| 1619 | } |
| 1620 | |
| 1621 | return $MST; |
| 1622 | } |
| 1623 | |
| 1624 | =pod |
| 1625 | |
| 1626 | =item edge_classify |
| 1627 | |
| 1628 | @C = $G->edge_classify(%param) |
| 1629 | |
| 1630 | Returns the edge classification as a list where each element |
| 1631 | is a triplet [$u, $v, $class] the $u, $v being the vertices |
| 1632 | of an edge and $class being the class. The %param can be |
| 1633 | used to control the search. |
| 1634 | |
| 1635 | =cut |
| 1636 | |
| 1637 | sub edge_classify { |
| 1638 | my $G = shift; |
| 1639 | |
| 1640 | my $unseen_successor = |
| 1641 | sub { |
| 1642 | my ($u, $v, $T) = @_; |
| 1643 | |
| 1644 | # Freshly seen successors make for tree edges. |
| 1645 | push @{ $T->{ edge_class_list } }, |
| 1646 | [ $u, $v, 'tree' ]; |
| 1647 | }; |
| 1648 | my $seen_successor = |
| 1649 | sub { |
| 1650 | my ($u, $v, $T) = @_; |
| 1651 | |
| 1652 | my $class; |
| 1653 | |
| 1654 | if ( $T->{ G }->directed ) { |
| 1655 | $class = 'cross'; # Default for directed non-tree edges. |
| 1656 | |
| 1657 | unless ( exists $T->{ vertex_finished }->{ $v } ) { |
| 1658 | $class = 'back'; |
| 1659 | } elsif ( $T->{ vertex_found }->{ $u } < |
| 1660 | $T->{ vertex_found }->{ $v }) { |
| 1661 | $class = 'forward'; |
| 1662 | } |
| 1663 | } else { |
| 1664 | # No cross nor forward edges in |
| 1665 | # an undirected graph, by definition. |
| 1666 | $class = 'back'; |
| 1667 | } |
| 1668 | |
| 1669 | push @{ $T->{ edge_class_list } }, [ $u, $v, $class ]; |
| 1670 | }; |
| 1671 | use Graph::DFS; |
| 1672 | my $d = |
| 1673 | Graph::DFS-> |
| 1674 | new( $G, |
| 1675 | unseen_successor => $unseen_successor, |
| 1676 | seen_successor => $seen_successor, |
| 1677 | @_); |
| 1678 | |
| 1679 | $d->preorder; |
| 1680 | |
| 1681 | return @{ $d->{ edge_class_list } }; |
| 1682 | } |
| 1683 | |
| 1684 | =pod |
| 1685 | |
| 1686 | =item toposort |
| 1687 | |
| 1688 | @toposort = $G->toposort |
| 1689 | |
| 1690 | Returns the vertices of the graph $G sorted topologically. |
| 1691 | |
| 1692 | =cut |
| 1693 | |
| 1694 | sub toposort { |
| 1695 | my $G = shift; |
| 1696 | my $d = Graph::DFS->new($G); |
| 1697 | |
| 1698 | reverse $d->postorder; # That's it. |
| 1699 | } |
| 1700 | |
| 1701 | # _strongly_connected |
| 1702 | # |
| 1703 | # $s = $G->_strongly_connected |
| 1704 | # |
| 1705 | # (INTERNAL USE ONLY) |
| 1706 | # Returns a graph traversal object that can be used for |
| 1707 | # strong connection computations. |
| 1708 | # |
| 1709 | sub _strongly_connected { |
| 1710 | my $G = shift; |
| 1711 | my $T = $G->transpose; |
| 1712 | |
| 1713 | Graph::DFS-> |
| 1714 | new($T, |
| 1715 | # Pick the potential roots in their DFS postorder. |
| 1716 | strong_root_order => [ reverse Graph::DFS->new($G)->postorder ], |
| 1717 | get_next_root => |
| 1718 | sub { |
| 1719 | my ($T, %param) = @_; |
| 1720 | |
| 1721 | while (my $root = |
| 1722 | shift @{ $param{ strong_root_order } }) { |
| 1723 | return $root if exists $T->{ pool }->{ $root }; |
| 1724 | } |
| 1725 | } |
| 1726 | ); |
| 1727 | } |
| 1728 | |
| 1729 | =pod |
| 1730 | |
| 1731 | =item strongly_connected_components |
| 1732 | |
| 1733 | @S = $G->strongly_connected_components |
| 1734 | |
| 1735 | Returns the strongly connected components @S of the graph $G |
| 1736 | as a list of anonymous lists of vertices, each anonymous list |
| 1737 | containing the vertices belonging to one strongly connected |
| 1738 | component. |
| 1739 | |
| 1740 | =cut |
| 1741 | |
| 1742 | sub strongly_connected_components { |
| 1743 | my $G = shift; |
| 1744 | my $T = $G->_strongly_connected; |
| 1745 | my %R = $T->_vertex_roots; |
| 1746 | my @C; |
| 1747 | |
| 1748 | # Clump together vertices having identical root vertices. |
| 1749 | while (my ($v, $r) = each %R) { push @{ $C[ $r ] }, $v } |
| 1750 | |
| 1751 | return @C; |
| 1752 | } |
| 1753 | |
| 1754 | =pod |
| 1755 | |
| 1756 | =item strongly_connected_graph |
| 1757 | |
| 1758 | $T = $G->strongly_connected_graph |
| 1759 | |
| 1760 | Returns the strongly connected graph $T of the graph $G. |
| 1761 | The names of the strongly connected components are |
| 1762 | formed from their constituent vertices by concatenating |
| 1763 | their names by '+'-characters: "a" and "b" --> "a+b". |
| 1764 | |
| 1765 | =cut |
| 1766 | |
| 1767 | sub strongly_connected_graph { |
| 1768 | my $G = shift; |
| 1769 | my $C = (ref $G)->new; |
| 1770 | my $T = $G->_strongly_connected; |
| 1771 | my %R = $T->_vertex_roots; |
| 1772 | my @C; # We're not calling the strongly_connected_components() |
| 1773 | # method because we will need also the %R. |
| 1774 | |
| 1775 | # Create the strongly connected components. |
| 1776 | while (my ($v, $r) = each %R) { push @{ $C[$r] }, $v } |
| 1777 | foreach my $c (@C) { $c = join("+", @$c) } |
| 1778 | |
| 1779 | $C->directed( $G->directed ); |
| 1780 | |
| 1781 | my @E = $G->edges; |
| 1782 | |
| 1783 | # Copy the edges between strongly connected components. |
| 1784 | my $edge_cnt = 0; |
| 1785 | my %n; |
| 1786 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 1787 | if ($R{ $u } != $R{ $v }) { |
| 1788 | $C->add_edge( $C[ $R{ $u } ], $C[ $R{ $v } ] ); |
| 1789 | $edge_cnt++; |
| 1790 | } elsif ($edge_cnt == 0) { |
| 1791 | $n{ $u } = ''; |
| 1792 | } |
| 1793 | } |
| 1794 | if ($edge_cnt == 0) { |
| 1795 | $C->add_vertex(join("+", keys %n)); |
| 1796 | } |
| 1797 | |
| 1798 | return $C; |
| 1799 | } |
| 1800 | |
| 1801 | =pod |
| 1802 | |
| 1803 | =item APSP_Floyd_Warshall |
| 1804 | |
| 1805 | $APSP = $G->APSP_Floyd_Warshall |
| 1806 | |
| 1807 | Returns the All-pairs Shortest Paths graph of the graph $G |
| 1808 | computed using the Floyd-Warshall algorithm and the attribute |
| 1809 | 'weight' on the edges. |
| 1810 | The returned graph has an edge for each shortest path. |
| 1811 | An edge has attributes "weight" and "path"; for the length of |
| 1812 | the shortest path and for the path (an anonymous list) itself. |
| 1813 | |
| 1814 | =cut |
| 1815 | |
| 1816 | sub APSP_Floyd_Warshall { |
| 1817 | my $G = shift; |
| 1818 | |
| 1819 | my @V = $G->vertices; |
| 1820 | my @E = $G->edges; |
| 1821 | my (%V2I, @I2V); |
| 1822 | my (@P, @W); |
| 1823 | |
| 1824 | # Compute the vertex <-> index mappings. |
| 1825 | @V2I{ @V } = 0..$#V; |
| 1826 | @I2V[ 0..$#V ] = @V; |
| 1827 | |
| 1828 | # Initialize the predecessor matrix @P and the weight matrix @W. |
| 1829 | # (The graph is converted into adjacency-matrix representation.) |
| 1830 | # (The matrix is a list of lists.) |
| 1831 | foreach my $i ( 0..$#V ) { $W[ $i ][ $i ] = 0 } |
| 1832 | while ( my ($u, $v) = splice(@E, 0, 2) ) { |
| 1833 | my ( $ui, $vi ) = ( $V2I{ $u }, $V2I{ $v } ); |
| 1834 | $P[ $ui ][ $vi ] = $ui unless $ui == $vi; |
| 1835 | $W[ $ui ][ $vi ] = $G->get_attribute( 'weight', $u, $v ); |
| 1836 | } |
| 1837 | |
| 1838 | # Do the O(N**3) loop. |
| 1839 | for ( my $k = 0; $k < @V; $k++ ) { |
| 1840 | my (@nP, @nW); # new @P, new @W |
| 1841 | |
| 1842 | for ( my $i = 0; $i < @V; $i++ ) { |
| 1843 | for ( my $j = 0; $j < @V; $j++ ) { |
| 1844 | my $w_ij = $W[ $i ][ $j ]; |
| 1845 | my $w_ik_kj = $W[ $i ][ $k ] + $W[ $k ][ $j ] |
| 1846 | if defined $W[ $i ][ $k ] and |
| 1847 | defined $W[ $k ][ $j ]; |
| 1848 | |
| 1849 | # Choose the minimum of w_ij and w_ik_kj. |
| 1850 | if ( defined $w_ij ) { |
| 1851 | if ( defined $w_ik_kj ) { |
| 1852 | if ( $w_ij <= $w_ik_kj ) { |
| 1853 | $nP[ $i ][ $j ] = $P[ $i ][ $j ]; |
| 1854 | $nW[ $i ][ $j ] = $w_ij; |
| 1855 | } else { |
| 1856 | $nP[ $i ][ $j ] = $P[ $k ][ $j ]; |
| 1857 | $nW[ $i ][ $j ] = $w_ik_kj; |
| 1858 | } |
| 1859 | } else { |
| 1860 | $nP[ $i ][ $j ] = $P[ $i ][ $j ]; |
| 1861 | $nW[ $i ][ $j ] = $w_ij; |
| 1862 | } |
| 1863 | } elsif ( defined $w_ik_kj ) { |
| 1864 | $nP[ $i ][ $j ] = $P[ $k ][ $j ]; |
| 1865 | $nW[ $i ][ $j ] = $w_ik_kj; |
| 1866 | } |
| 1867 | } |
| 1868 | } |
| 1869 | |
| 1870 | @P = @nP; @W = @nW; # Update the predecessors and weights. |
| 1871 | } |
| 1872 | |
| 1873 | # Now construct the APSP graph. |
| 1874 | |
| 1875 | my $APSP = (ref $G)->new; |
| 1876 | |
| 1877 | $APSP->directed( $G->directed ); # Copy the directedness. |
| 1878 | |
| 1879 | # Convert the adjacency-matrix representation |
| 1880 | # into a Graph (adjacency-list representation). |
| 1881 | for ( my $i = 0; $i < @V; $i++ ) { |
| 1882 | my $iv = $I2V[ $i ]; |
| 1883 | |
| 1884 | for ( my $j = 0; $j < @V; $j++ ) { |
| 1885 | if ( $i == $j ) { |
| 1886 | $APSP->add_weighted_edge( $iv, 0, $iv ); |
| 1887 | $APSP->set_attribute("path", $iv, $iv, [ $iv ]); |
| 1888 | next; |
| 1889 | } |
| 1890 | next unless defined $W[ $i ][ $j ]; |
| 1891 | |
| 1892 | my $jv = $I2V[ $j ]; |
| 1893 | |
| 1894 | $APSP->add_weighted_edge( $iv, $W[ $i ][ $j ], $jv ); |
| 1895 | |
| 1896 | my @path = ( $jv ); |
| 1897 | if ( $P[ $i ][ $j ] != $i ) { |
| 1898 | my $k = $P[ $i ][ $j ]; # Walk back the path. |
| 1899 | |
| 1900 | while ( $k != $i ) { |
| 1901 | push @path, $I2V[ $k ]; |
| 1902 | $k = $P[ $i ][ $k ]; # Keep walking. |
| 1903 | } |
| 1904 | } |
| 1905 | $APSP->set_attribute( "path", $iv, $jv, [ $iv, reverse @path ] ); |
| 1906 | } |
| 1907 | } |
| 1908 | |
| 1909 | return $APSP; |
| 1910 | } |
| 1911 | |
| 1912 | =pod |
| 1913 | |
| 1914 | =item TransitiveClosure_Floyd_Warshall |
| 1915 | |
| 1916 | $TransitiveClosure = $G->TransitiveClosure_Floyd_Warshall |
| 1917 | |
| 1918 | Returns the Transitive Closure graph of the graph $G computed |
| 1919 | using the Floyd-Warshall algorithm. |
| 1920 | The resulting graph has an edge between each *ordered* pair of |
| 1921 | vertices in which the second vertex is reachable from the first. |
| 1922 | |
| 1923 | =cut |
| 1924 | |
| 1925 | sub TransitiveClosure_Floyd_Warshall { |
| 1926 | my $G = shift; |
| 1927 | |
| 1928 | my @V = $G->vertices; |
| 1929 | my @E = $G->edges; |
| 1930 | my (%V2I, @I2V); |
| 1931 | my @C = ( '' ) x @V; |
| 1932 | |
| 1933 | # Compute the vertex <-> index mappings. |
| 1934 | @V2I{ @V } = 0..$#V; |
| 1935 | @I2V[ 0..$#V ] = @V; |
| 1936 | |
| 1937 | # Initialize the closure matrix @C. |
| 1938 | # (The graph is converted into adjacency-matrix representation.) |
| 1939 | # (The matrix is a bit matrix. Well, a list of bit vectors.) |
| 1940 | foreach my $i ( 0..$#V ) { vec( $C[ $i ], $i, 1 ) = 1 } |
| 1941 | while ( my ($u, $v) = splice(@E, 0, 2) ) { |
| 1942 | vec( $C[ $V2I{ $u } ], $V2I{ $v }, 1 ) = 1 |
| 1943 | } |
| 1944 | |
| 1945 | # Do the O(N**3) loop. |
| 1946 | for ( my $k = 0; $k < @V; $k++ ) { |
| 1947 | my @nC = ( '' ) x @V; # new @C |
| 1948 | |
| 1949 | for ( my $i = 0; $i < @V; $i++ ) { |
| 1950 | for ( my $j = 0; $j < @V; $j++ ) { |
| 1951 | vec( $nC[ $i ], $j, 1 ) = |
| 1952 | vec( $C[ $i ], $j, 1 ) | |
| 1953 | vec( $C[ $i ], $k, 1 ) & vec( $C[ $k ], $j, 1 ); |
| 1954 | } |
| 1955 | } |
| 1956 | |
| 1957 | @C = @nC; # Update the closure. |
| 1958 | } |
| 1959 | |
| 1960 | # Now construct the TransitiveClosure graph. |
| 1961 | |
| 1962 | my $TransitiveClosure = (ref $G)->new; |
| 1963 | |
| 1964 | $TransitiveClosure->directed( $G->directed ); |
| 1965 | |
| 1966 | # Convert the (closure-)adjacency-matrix representation |
| 1967 | # into a Graph (adjacency-list representation). |
| 1968 | for ( my $i = 0; $i < @V; $i++ ) { |
| 1969 | for ( my $j = 0; $j < @V; $j++ ) { |
| 1970 | $TransitiveClosure->add_edge( $I2V[ $i ], $I2V[ $j ] ) |
| 1971 | if vec( $C[ $i ], $j, 1 ); |
| 1972 | } |
| 1973 | } |
| 1974 | |
| 1975 | return $TransitiveClosure; |
| 1976 | } |
| 1977 | |
| 1978 | =pod |
| 1979 | |
| 1980 | =item articulation points |
| 1981 | |
| 1982 | @A = $G->articulation_points(%param) |
| 1983 | |
| 1984 | Returns the articulation points (vertices) @A of the graph $G. |
| 1985 | The %param can be used to control the search. |
| 1986 | |
| 1987 | =cut |
| 1988 | |
| 1989 | sub articulation_points { |
| 1990 | my $G = shift; |
| 1991 | my $articulate = |
| 1992 | sub { |
| 1993 | my ( $u, $T ) = @_; |
| 1994 | |
| 1995 | my $ap = $T->{ vertex_found }->{ $u }; |
| 1996 | |
| 1997 | my @S = @{ $T->{ active_list } }; # Current stack. |
| 1998 | |
| 1999 | $T->{ articulation_point }->{ $u } = $ap |
| 2000 | unless exists $T->{ articulation_point }->{ $u }; |
| 2001 | |
| 2002 | # Walk back the stack marking the active DFS branch |
| 2003 | # (below $u) as belonging to the articulation point $ap. |
| 2004 | for ( my $i = 1; $i < @S; $i++ ) { |
| 2005 | my $v = $S[ -$i ]; |
| 2006 | |
| 2007 | last if $v eq $u; |
| 2008 | |
| 2009 | $T->{ articulation_point }->{ $v } = $ap |
| 2010 | if not exists $T->{ articulation_point }->{ $v } or |
| 2011 | $ap < $T->{ articulation_point }->{ $v }; |
| 2012 | } |
| 2013 | }; |
| 2014 | my $unseen_successor = |
| 2015 | sub { |
| 2016 | my ($u, $v, $T) = @_; |
| 2017 | |
| 2018 | # We need to know the number of children for root vertices. |
| 2019 | $T->{ articulation_children }->{ $u }++; |
| 2020 | }; |
| 2021 | my $seen_successor = |
| 2022 | sub { |
| 2023 | my ($u, $v, $T) = @_; |
| 2024 | |
| 2025 | # If the $v is still active, articulate it. |
| 2026 | $articulate->( $v, $T ) if exists $T->{ active_pool }->{ $v }; |
| 2027 | }; |
| 2028 | my $d = |
| 2029 | Graph::DFS->new($G, |
| 2030 | articulate => $articulate, |
| 2031 | unseen_successor => $unseen_successor, |
| 2032 | seen_successor => $seen_successor, |
| 2033 | ); |
| 2034 | |
| 2035 | $d->preorder; |
| 2036 | |
| 2037 | # Now we need to find (the indices of) unique articulation points |
| 2038 | # and map them back to vertices. |
| 2039 | |
| 2040 | my (%ap, @vf); |
| 2041 | |
| 2042 | foreach my $v ( $G->vertices ) { |
| 2043 | $ap{ $d->{ articulation_point }->{ $v } } = $v; |
| 2044 | $vf[ $d->{ vertex_found }->{ $v } ] = $v; |
| 2045 | } |
| 2046 | |
| 2047 | %ap = map { ( $vf[ $_ ], $_ ) } keys %ap; |
| 2048 | |
| 2049 | # DFS tree roots are articulation points only |
| 2050 | # iff they have more than one children. |
| 2051 | foreach my $r ( $d->roots ) { |
| 2052 | delete $ap{ $r } if $d->{ articulation_children }->{ $r } < 2; |
| 2053 | } |
| 2054 | |
| 2055 | keys %ap; |
| 2056 | } |
| 2057 | |
| 2058 | =pod |
| 2059 | |
| 2060 | =item is_biconnected |
| 2061 | |
| 2062 | $b = $G->is_biconnected |
| 2063 | |
| 2064 | Returns true is the graph $G is biconnected |
| 2065 | (has no articulation points), false otherwise. |
| 2066 | |
| 2067 | =cut |
| 2068 | |
| 2069 | sub is_biconnected { |
| 2070 | my $G = shift; |
| 2071 | |
| 2072 | return $G->articulation_points == 0; |
| 2073 | } |
| 2074 | |
| 2075 | =pod |
| 2076 | |
| 2077 | =item largest_out_degree |
| 2078 | |
| 2079 | $v = $G->largest_out_degree( @V ) |
| 2080 | |
| 2081 | Selects the vertex $v from the vertices @V having |
| 2082 | the largest out degree in the graph $G. |
| 2083 | |
| 2084 | =cut |
| 2085 | |
| 2086 | sub largest_out_degree { |
| 2087 | my $G = shift; |
| 2088 | my $L = shift; |
| 2089 | my $O = $G->out_degree($L); |
| 2090 | |
| 2091 | for my $e (@_) { |
| 2092 | my $o = $G->out_degree($e); |
| 2093 | if ($o > $O) { |
| 2094 | $L = $e; |
| 2095 | $O = $o; |
| 2096 | } |
| 2097 | } |
| 2098 | |
| 2099 | return $L; |
| 2100 | } |
| 2101 | |
| 2102 | # _heap_init |
| 2103 | # |
| 2104 | # $G->_heap_init($heap, $u, \%in_heap, \%weight, \%parent) |
| 2105 | # |
| 2106 | # (INTERNAL USE ONLY) |
| 2107 | # Initializes the $heap with the vertex $u as the initial |
| 2108 | # vertex, its weight being zero, and marking all vertices |
| 2109 | # of the graph $G to be $in_heap, |
| 2110 | # |
| 2111 | sub _heap_init { |
| 2112 | my ($G, $heap, $u, $in_heap, $W, $P) = @_; |
| 2113 | |
| 2114 | use Graph::HeapElem; |
| 2115 | |
| 2116 | foreach my $v ( $G->vertices ) { |
| 2117 | my $e = Graph::HeapElem->new( $v, $W, $P ); |
| 2118 | $heap->add( $e ); |
| 2119 | $in_heap->{ $v } = $e; |
| 2120 | } |
| 2121 | |
| 2122 | $W->{ $u } = 0; |
| 2123 | } |
| 2124 | |
| 2125 | =pod |
| 2126 | |
| 2127 | =item MST_Prim |
| 2128 | |
| 2129 | $MST = $G->MST_Prim($u) |
| 2130 | |
| 2131 | Returns Prim's Minimum Spanning Tree (as a graph) of |
| 2132 | the graph $G based on the 'weight' attributes of the edges. |
| 2133 | The optional start vertex is $u, if none is given, a hopefully |
| 2134 | good one (a vertex with a large out degree) is chosen. |
| 2135 | |
| 2136 | =cut |
| 2137 | |
| 2138 | sub MST_Prim { |
| 2139 | my ( $G, $u ) = @_; |
| 2140 | my $MST = (ref $G)->new; |
| 2141 | |
| 2142 | $u = $G->largest_out_degree( $G->vertices ) unless defined $u; |
| 2143 | |
| 2144 | use Heap::Fibonacci; |
| 2145 | my $heap = Heap::Fibonacci->new; |
| 2146 | my ( %in_heap, %weight, %parent ); |
| 2147 | |
| 2148 | $G->_heap_init( $heap, $u, \%in_heap, \%weight, \%parent ); |
| 2149 | |
| 2150 | # Walk the edges at the current BFS front |
| 2151 | # in the order of their increasing weight. |
| 2152 | while ( defined $heap->minimum ) { |
| 2153 | $u = $heap->extract_minimum; |
| 2154 | delete $in_heap{ $u->vertex }; |
| 2155 | |
| 2156 | # Now extend the BFS front. |
| 2157 | |
| 2158 | foreach my $v ( $G->successors( $u->vertex ) ) { |
| 2159 | if ( defined( $v = $in_heap{ $v } ) ) { |
| 2160 | my $nw = $G->get_attribute( 'weight', |
| 2161 | $u->vertex, $v->vertex ); |
| 2162 | my $ow = $v->weight; |
| 2163 | |
| 2164 | if ( not defined $ow or $nw < $ow ) { |
| 2165 | $v->weight( $nw ); |
| 2166 | $v->parent( $u->vertex ); |
| 2167 | $heap->decrease_key( $v ); |
| 2168 | } |
| 2169 | } |
| 2170 | } |
| 2171 | } |
| 2172 | |
| 2173 | foreach my $v ( $G->vertices ) { |
| 2174 | $MST->add_weighted_edge( $v, $weight{ $v }, $parent{ $v } ) |
| 2175 | if defined $parent{ $v }; |
| 2176 | } |
| 2177 | |
| 2178 | return $MST; |
| 2179 | } |
| 2180 | |
| 2181 | # _SSSP_construct |
| 2182 | # |
| 2183 | # $SSSP = $G->_SSSP_construct( $s, $W, $P ); |
| 2184 | # |
| 2185 | # (INTERNAL USE ONLY) |
| 2186 | # Return the SSSP($s) graph of graph $G based on the computed |
| 2187 | # anonymous hashes for weights and parents: $W and $P. |
| 2188 | # The vertices of the graph will have two attributes: "weight", |
| 2189 | # which tells the length of the shortest single-source path, |
| 2190 | # and "path", which is an anymous list containing the path. |
| 2191 | # |
| 2192 | sub _SSSP_construct { |
| 2193 | my ($G, $s, $W, $P ) = @_; |
| 2194 | my $SSSP = (ref $G)->new; |
| 2195 | |
| 2196 | foreach my $u ( $G->vertices ) { |
| 2197 | $SSSP->add_vertex( $u ); |
| 2198 | |
| 2199 | $SSSP->set_attribute( "weight", $u, $W->{ $u } || 0 ); |
| 2200 | |
| 2201 | my @path = ( $u ); |
| 2202 | if ( defined $P->{ $u } ) { |
| 2203 | $SSSP->add_edge($P->{ $u }, $u ); |
| 2204 | $SSSP->set_attribute( "weight", $P->{ $u }, $u, $G->get_attribute("weight",$P->{ $u }, $u) || 0 ); |
| 2205 | push @path, $P->{ $u }; |
| 2206 | if ( $P->{ $u } ne $s ) { |
| 2207 | my $v = $P->{ $u }; |
| 2208 | |
| 2209 | while ( defined $v && exists $P->{ $v } && $v ne $s ) { |
| 2210 | push @path, $P->{ $v }; |
| 2211 | $v = $P->{ $v }; |
| 2212 | } |
| 2213 | } |
| 2214 | } |
| 2215 | $SSSP->set_attribute( "path", $u, [ reverse @path ] ); |
| 2216 | } |
| 2217 | |
| 2218 | return $SSSP; |
| 2219 | } |
| 2220 | |
| 2221 | =pod |
| 2222 | |
| 2223 | =item SSSP_Dijkstra |
| 2224 | |
| 2225 | $SSSP = $G->SSSP_Dijkstra($s) |
| 2226 | |
| 2227 | Returns the Single-source Shortest Paths (as a graph) |
| 2228 | of the graph $G starting from the vertex $s using Dijktra's |
| 2229 | SSSP algorithm. |
| 2230 | |
| 2231 | =cut |
| 2232 | |
| 2233 | sub SSSP_Dijkstra { |
| 2234 | my ( $G, $s ) = @_; |
| 2235 | |
| 2236 | use Heap::Fibonacci; |
| 2237 | my $heap = Heap::Fibonacci->new; |
| 2238 | my ( %in_heap, %weight, %parent ); |
| 2239 | |
| 2240 | # The other weights are by default undef (infinite). |
| 2241 | $weight{ $s } = 0; |
| 2242 | |
| 2243 | $G->_heap_init($heap, $s, \%in_heap, \%weight, \%parent ); |
| 2244 | |
| 2245 | # Walk the edges at the current BFS front |
| 2246 | # in the order of their increasing weight. |
| 2247 | while ( defined $heap->minimum ) { |
| 2248 | my $u = $heap->extract_minimum; |
| 2249 | delete $in_heap{ $u->vertex }; |
| 2250 | |
| 2251 | # Now extend the BFS front. |
| 2252 | my $uw = $u->weight; |
| 2253 | |
| 2254 | foreach my $v ( $G->successors( $u->vertex ) ) { |
| 2255 | if ( defined( $v = $in_heap{ $v } ) ) { |
| 2256 | my $ow = $v->weight; |
| 2257 | my $nw = |
| 2258 | $G->get_attribute( 'weight', $u->vertex, $v->vertex ) + |
| 2259 | ($uw || 0); # The || 0 helps for undefined $uw. |
| 2260 | |
| 2261 | # Relax the edge $u - $v. |
| 2262 | if ( not defined $ow or $ow > $nw ) { |
| 2263 | $v->weight( $nw ); |
| 2264 | $v->parent( $u->vertex ); |
| 2265 | $heap->decrease_key( $v ); |
| 2266 | } |
| 2267 | } |
| 2268 | } |
| 2269 | } |
| 2270 | |
| 2271 | return $G->_SSSP_construct( $s, \%weight, \%parent ); |
| 2272 | } |
| 2273 | |
| 2274 | =pod |
| 2275 | |
| 2276 | =item SSSP_Bellman_Ford |
| 2277 | |
| 2278 | $SSSP = $G->SSSP_Bellman_Ford($s) |
| 2279 | |
| 2280 | Returns the Single-source Shortest Paths (as a graph) |
| 2281 | of the graph $G starting from the vertex $s using Bellman-Ford |
| 2282 | SSSP algorithm. If there are one or more negatively weighted |
| 2283 | cycles, returns undef. |
| 2284 | |
| 2285 | =cut |
| 2286 | |
| 2287 | sub SSSP_Bellman_Ford { |
| 2288 | my ( $G, $s ) = @_; |
| 2289 | my ( %weight, %parent ); |
| 2290 | |
| 2291 | $weight{ $s } = 0; |
| 2292 | |
| 2293 | my $V = $G->vertices; |
| 2294 | my @E = $G->edges; |
| 2295 | |
| 2296 | foreach ( 1..$V ) { # |V|-1 times (*not* |V| times) |
| 2297 | my @C = @E; # Copy. |
| 2298 | |
| 2299 | while (my ($u, $v) = splice(@C, 0, 2)) { |
| 2300 | my $ow = $weight{ $v }; |
| 2301 | my $nw = $G->get_attribute( 'weight', $u, $v ); |
| 2302 | |
| 2303 | $nw += $weight{ $u } if defined $weight{ $u }; |
| 2304 | # Relax the edge $u - $w. |
| 2305 | if ( not defined $ow or $ow > $nw ) { |
| 2306 | $weight{ $v } = $nw; |
| 2307 | $parent{ $v } = $u; |
| 2308 | } |
| 2309 | } |
| 2310 | } |
| 2311 | |
| 2312 | my $negative; |
| 2313 | |
| 2314 | # Warn about detected negative cycles. |
| 2315 | while (my ($u, $v) = splice(@E, 0, 2)) { |
| 2316 | if ( $weight{ $v } > |
| 2317 | $weight{ $u } + $G->get_attribute( 'weight', $u, $v ) ) { |
| 2318 | warn "SSSP_Bellman_Ford: negative cycle $u $v\n"; |
| 2319 | $negative++; |
| 2320 | } |
| 2321 | } |
| 2322 | |
| 2323 | # Bail out if found negative cycles. |
| 2324 | return undef if $negative; |
| 2325 | |
| 2326 | # Otherwise return the SSSP graph. |
| 2327 | return $G->_SSSP_construct( $s, \%weight, \%parent ); |
| 2328 | } |
| 2329 | |
| 2330 | =pod |
| 2331 | |
| 2332 | =item SSSP_DAG |
| 2333 | |
| 2334 | $SSSP = $G->SSSP_DAG($s) |
| 2335 | |
| 2336 | Returns the Single-source Shortest Paths (as a graph) |
| 2337 | of the DAG $G starting from vertex $s. |
| 2338 | |
| 2339 | =cut |
| 2340 | |
| 2341 | sub SSSP_DAG { |
| 2342 | my ( $G, $s ) = @_; |
| 2343 | my $SSSP = (ref $G)->new; |
| 2344 | |
| 2345 | my ( %weight, %parent ); |
| 2346 | |
| 2347 | $weight{ $s } = 0; |
| 2348 | |
| 2349 | # Because by definition there can be no cycles |
| 2350 | # we can freely explore each successor of each vertex. |
| 2351 | foreach my $u ( $G->toposort ) { |
| 2352 | foreach my $v ( $G->successors( $u ) ) { |
| 2353 | my $ow = $weight{ $v }; |
| 2354 | my $nw = $G->get_attribute( 'weight', $u, $v ); |
| 2355 | |
| 2356 | $nw += $weight{ $u } if defined $weight{ $u }; |
| 2357 | |
| 2358 | # Relax the edge $u - $v. |
| 2359 | if ( not defined $ow or $ow > $nw ) { |
| 2360 | $weight{ $v } = $nw; |
| 2361 | $parent{ $v } = $u; |
| 2362 | } |
| 2363 | } |
| 2364 | } |
| 2365 | |
| 2366 | return $G->_SSSP_construct( $s, \%weight, \%parent ); |
| 2367 | } |
| 2368 | |
| 2369 | =pod |
| 2370 | |
| 2371 | =item add_capacity_edge |
| 2372 | |
| 2373 | $G->add_capacity_edge($u, $w, $v, $a) |
| 2374 | |
| 2375 | Adds in the graph $G an edge from vertex $u to vertex $v |
| 2376 | and the edge attribute 'capacity' set to $w. |
| 2377 | |
| 2378 | =cut |
| 2379 | |
| 2380 | sub add_capacity_edge { |
| 2381 | my ($G, $u, $w, $v, $a) = @_; |
| 2382 | |
| 2383 | $G->add_edge($u, $v); |
| 2384 | $G->set_attribute('capacity', $u, $v, $w); |
| 2385 | } |
| 2386 | |
| 2387 | =pod |
| 2388 | |
| 2389 | =item add_capacity_edges |
| 2390 | |
| 2391 | $G->add_capacity_edges($u1, $w1, $v1, $u2, $w2, $v2, ...) |
| 2392 | |
| 2393 | Adds in the graph $G the capacity edges. |
| 2394 | |
| 2395 | =cut |
| 2396 | |
| 2397 | sub add_capacity_edges { |
| 2398 | my $G = shift; |
| 2399 | |
| 2400 | while (my ($u, $w, $v) = splice(@_, 0, 3)) { |
| 2401 | $G->add_capacity_edge($u, $w, $v); |
| 2402 | } |
| 2403 | } |
| 2404 | |
| 2405 | =pod |
| 2406 | |
| 2407 | =item add_capacity_path |
| 2408 | |
| 2409 | $G->add_capacity_path($v1, $w1, $v2, $w2, ..., $wnm1, $vn) |
| 2410 | |
| 2411 | Adds in the graph $G the n edges defined by the path $v1 ... $vn |
| 2412 | with the n-1 'capacity' attributes $w1 ... $wnm1 |
| 2413 | |
| 2414 | =cut |
| 2415 | |
| 2416 | sub add_capacity_path { |
| 2417 | my $G = shift; |
| 2418 | my $u = shift; |
| 2419 | |
| 2420 | while (my ($w, $v) = splice(@_, 0, 2)) { |
| 2421 | $G->add_capacity_edge($u, $w, $v); |
| 2422 | $u = $v; |
| 2423 | } |
| 2424 | } |
| 2425 | |
| 2426 | =pod |
| 2427 | |
| 2428 | =item Flow_Ford_Fulkerson |
| 2429 | |
| 2430 | $F = $G->Flow_Ford_Fulkerson($S) |
| 2431 | |
| 2432 | Returns the (maximal) flow network of the flow network $G, |
| 2433 | parametrized by the state $S. The $G must have 'capacity' |
| 2434 | attributes on its edges. $S->{ source } must contain the |
| 2435 | source vertex and $S->{ sink } the sink vertex, and |
| 2436 | most importantly $S->{ next_augmenting_path } must contain |
| 2437 | an anonymous subroutine which takes $F and $S as arguments |
| 2438 | and returns the next potential augmenting path. |
| 2439 | Flow_Ford_Fulkerson will do the augmenting. |
| 2440 | The result graph $F will have 'flow' and (residual) 'capacity' |
| 2441 | attributes on its edges. |
| 2442 | |
| 2443 | =cut |
| 2444 | |
| 2445 | sub Flow_Ford_Fulkerson { |
| 2446 | my ( $G, $S ) = @_; |
| 2447 | |
| 2448 | my $F = (ref $G)->new; # The flow network. |
| 2449 | my @E = $G->edges; |
| 2450 | my ( $u, $v ); |
| 2451 | |
| 2452 | # Copy the edges and the capacities, zero the flows. |
| 2453 | while (($u, $v) = splice(@E, 0, 2)) { |
| 2454 | $F->add_edge( $u, $v ); |
| 2455 | $F->set_attribute( 'capacity', $u, $v, |
| 2456 | $G->get_attribute( 'capacity', $u, $v ) || 0 ); |
| 2457 | $F->set_attribute( 'flow', $u, $v, 0 ); |
| 2458 | } |
| 2459 | |
| 2460 | # Walk the augmenting paths. |
| 2461 | while ( my $ap = $S->{ next_augmenting_path }->( $F, $S ) ) { |
| 2462 | my @aps = @$ap; # augmenting path segments |
| 2463 | my $apr; # augmenting path residual capacity |
| 2464 | my $psr; # path segment residual capacity |
| 2465 | |
| 2466 | # Find the minimum capacity of the path. |
| 2467 | for ( $u = shift @aps; @aps; $u = $v ) { |
| 2468 | $v = shift @aps; |
| 2469 | $psr = $F->get_attribute( 'capacity', $u, $v ) - |
| 2470 | $F->get_attribute( 'flow', $u, $v ); |
| 2471 | $apr = $psr |
| 2472 | if $psr >= 0 and ( not defined $apr or $psr < $apr ); |
| 2473 | } |
| 2474 | |
| 2475 | if ( $apr > 0 ) { # Augment the path. |
| 2476 | for ( @aps = @$ap, $u = shift @aps; @aps; $u = $v ) { |
| 2477 | $v = shift @aps; |
| 2478 | $F->set_attribute( 'flow', |
| 2479 | $u, $v, |
| 2480 | $F->get_attribute( 'flow', $u, $v ) + |
| 2481 | $apr ); |
| 2482 | } |
| 2483 | } |
| 2484 | } |
| 2485 | |
| 2486 | return $F; |
| 2487 | } |
| 2488 | |
| 2489 | =pod |
| 2490 | |
| 2491 | =item Flow_Edmonds_Karp |
| 2492 | |
| 2493 | $F = $G->Flow_Edmonds_Karp($source, $sink) |
| 2494 | |
| 2495 | Return the maximal flow network of the graph $G built |
| 2496 | using the Edmonds-Karp version of Ford-Fulkerson. |
| 2497 | The input graph $G must have 'capacity' attributes on |
| 2498 | its edges; resulting flow graph will have 'capacity' and 'flow' |
| 2499 | attributes on its edges. |
| 2500 | |
| 2501 | =cut |
| 2502 | |
| 2503 | sub Flow_Edmonds_Karp { |
| 2504 | my ( $G, $source, $sink ) = @_; |
| 2505 | |
| 2506 | my $S; |
| 2507 | |
| 2508 | $S->{ source } = $source; |
| 2509 | $S->{ sink } = $sink; |
| 2510 | $S->{ next_augmenting_path } = |
| 2511 | sub { |
| 2512 | my ( $F, $S ) = @_; |
| 2513 | |
| 2514 | my $source = $S->{ source }; |
| 2515 | my $sink = $S->{ sink }; |
| 2516 | |
| 2517 | # Initialize our "todo" heap. |
| 2518 | unless ( exists $S->{ todo } ) { |
| 2519 | # The first element is a hash recording the vertices |
| 2520 | # seen so far, the rest are the path from the source. |
| 2521 | push @{ $S->{ todo } }, |
| 2522 | [ { $source => 1 }, $source ]; |
| 2523 | } |
| 2524 | |
| 2525 | while ( @{ $S->{ todo } } ) { |
| 2526 | # $ap: The next augmenting path. |
| 2527 | my $ap = shift @{ $S->{ todo } }; |
| 2528 | my $sv = shift @$ap; # The seen vertices. |
| 2529 | my $v = $ap->[ -1 ]; # The last vertex of path. |
| 2530 | |
| 2531 | if ( $v eq $sink ) { |
| 2532 | return $ap; |
| 2533 | } else { |
| 2534 | foreach my $s ( $G->successors( $v ) ) { |
| 2535 | unless ( exists $sv->{ $s } ) { |
| 2536 | push @{ $S->{ todo } }, |
| 2537 | [ { %$sv, $s => 1 }, @$ap, $s ]; |
| 2538 | } |
| 2539 | } |
| 2540 | } |
| 2541 | } |
| 2542 | }; |
| 2543 | |
| 2544 | return $G->Flow_Ford_Fulkerson( $S ); |
| 2545 | } |
| 2546 | |
| 2547 | use overload 'eq' => \&eq; |
| 2548 | |
| 2549 | =pod |
| 2550 | |
| 2551 | =item eq |
| 2552 | |
| 2553 | $G->eq($H) |
| 2554 | |
| 2555 | Return true if the graphs (actually, their string representations) |
| 2556 | are identical. This means really identical: they must have identical |
| 2557 | vertex names and identical edges between the vertices, and they must |
| 2558 | be similarly directed. (Just isomorphism isn't enough.) |
| 2559 | |
| 2560 | =cut |
| 2561 | |
| 2562 | sub eq { |
| 2563 | my ($G, $H) = @_; |
| 2564 | |
| 2565 | return ref $H ? $G->stringify eq $H->stringify : $G->stringify eq $H; |
| 2566 | } |
| 2567 | |
| 2568 | =pod |
| 2569 | |
| 2570 | =back |
| 2571 | |
| 2572 | =head1 COPYRIGHT |
| 2573 | |
| 2574 | Copyright 1999, O'Reilly & Associates. |
| 2575 | |
| 2576 | This code is distributed under the same copyright terms as Perl itself. |
| 2577 | |
| 2578 | =cut |
| 2579 | |
| 2580 | 1; |