Biconnected Components

Transcription

1 Presetato for use wth the textbook, Algorthm Desg ad Applcatos, by M. T. Goodrch ad R. Tamassa, Wley, 2015 Bcoected Compoets SEA PVD ORD FCO SNA MIA 2015 Goodrch ad Tamassa Bcoectvty 1

2 Applcato: Networkg A computer etwork ca be modeled as a graph, where vertces are routers ad edges are etwork coectos betwee edges. A router ca be cosdered crtcal f t ca dscoect the etwork for that router to fal. It would be ce to detfy whch routers are crtcal. We ca do such a detfcato by solvg the bcoected compoets problem Goodrch ad Tamassa Bcoectvty 2

3 Separato Edges ad Vertces Deftos Let G be a coected graph A separato edge of G s a edge whose removal dscoects G A separato vertex of G s a vertex whose removal dscoects G Applcatos Separato edges ad vertces represet sgle pots of falure a etwork ad are crtcal to the operato of the etwork Example DFW, LGA ad LAX are separato vertces (DFW,LAX) s a separato edge SFO ORD PVD LGA HNL LAX DFW MIA 2015 Goodrch ad Tamassa Bcoectvty 3

4 Bcoected Graph Equvalet deftos of a bcoected graph G Graph G has o separato edges ad o separato vertces For ay two vertces u ad v of G, there are two dsjot smple paths betwee u ad v (.e., two smple paths betwee u ad v that share o other vertces or edges) For ay two vertces u ad v of G, there s a smple cycle cotag u ad v Example SFO ORD PVD HNL LAX DFW LGA MIA 2015 Goodrch ad Tamassa Bcoectvty 4

5 Bcoected Compoets Bcoected compoet of a graph G A maxmal bcoected subgraph of G, or A subgraph cosstg of a separato edge of G ad ts ed vertces Iteracto of bcoected compoets A edge belogs to exactly oe bcoected compoet A oseparato vertex belogs to exactly oe bcoected compoet A separato vertex belogs to two or more bcoected compoets Example of a graph wth four bcoected compoets HNL SFO LAX DFW ORD LGA MIA PVD RDU 2015 Goodrch ad Tamassa Bcoectvty 5

6 Equvalece Classes Gve a set S, a relato R o S s a set of ordered pars of elemets of S,.e., R s a subset of S S A equvalece relato R o S satsfes the followg propertes Reflexve: (x,x) R Symmetrc: (x,y) R (y,x) R Trastve: (x,y) R (y,z) R (x,z) R A equvalece relato R o S duces a partto of the elemets of S to equvalece classes Example (coectvty relato amog the vertces of a graph): Let V be the set of vertces of a graph G Defe the relato C = {(v,w) V V such that G has a path from v to w} Relato C s a equvalece relato The equvalece classes of relato C are the vertces each coected compoet of graph G 2015 Goodrch ad Tamassa Bcoectvty 6

7 Lk Relato Edges e ad f of coected graph G are lked f e = f, or G has a smple cycle cotag e ad f Theorem: The lk relato o the edges of a graph s a equvalece relato Proof Sketch: The reflexve ad symmetrc propertes follow from the defto For the trastve property, cosder two smple cycles sharg a edge 2015 Goodrch ad Tamassa Bcoectvty 7 a b c d Equvalece classes of lked edges: {a} {b, c, d, e, f} {g,, j} a b c d e f e f g g j j

8 Lk Compoets The lk compoets of a coected graph G are the equvalece classes of edges wth respect to the lk relato A bcoected compoet of G s the subgraph of G duced by a equvalece class of lked edges A separato edge s a sgle-elemet equvalece class of lked edges A separato vertex has cdet edges at least two dstct equvalece classes of lked edge HNL SFO LAX DFW ORD LGA MIA PVD RDU 2015 Goodrch ad Tamassa Bcoectvty 8

9 Auxlary Graph Auxlary graph B for a coected graph G Assocated wth a DFS traversal of G c b g d e f h j The vertces of B are the edges of G a For each back edge e of G, B has edges (e,f 1 ), (e,f 2 ),, (e,f k ), where f 1, f 2,, f k are the dscovery edges of G that form a smple cycle wth e Its coected compoets correspod to the the lk compoets of G a DFS o graph G g e h b f d c Auxlary graph B j 2015 Goodrch ad Tamassa Bcoectvty 9

10 Auxlary Graph (cot.) I the worst case, the umber of edges of the auxlary graph s proportoal to m DFS o graph G Auxlary graph B 2015 Goodrch ad Tamassa Bcoectvty 10

11 A O(m)-Tme Algorthm Lemma: The coected compoets of the auxlary graph B correspod to the lk compoets of the graph G that duced B. Ths lemma yelds the followg O(m)-tme algorthm for computg all the lk compoets of a graph G wth vertces ad m edges: 2015 Goodrch ad Tamassa Bcoectvty 11

12 A Lear-Tme Algorthm 2015 Goodrch ad Tamassa Bcoectvty

13 Aalyss wth the Proxy Graph, F Proxy graph F for a coected graph G Spag forest of the auxlary graph B Has m vertces ad O(m) edges Ca be costructed O( + m) tme Its coected compoets (trees) correspod to the the lk compoets of G Gve a graph G wth vertces ad m edges, we ca compute the followg O( + m) tme: The bcoected compoets of G The separato vertces of G The separato edges of G c a h g b e j d f a DFS o graph G g e h b f d c Proxy graph F j 2015 Goodrch ad Tamassa Bcoectvty 13