CMSC 451: Lecture 4 Bridges and 2-Edge Connectivity Thursday, Sep 7, 2017

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1 Rn: Not ovr n or rns. CMSC 451: Ltr 4 Brs n 2-E Conntvty Trsy, Sp 7, 2017 Hr-Orr Grp Conntvty: (T ollown mtrl ppls only to nrt rps!) Lt G = (V, E) n onnt nrt rp. W otn ssm tt or rps r onnt, t somtms t s srl to v r r o onntvty. For xmpl, rp n sonnt tro t rmovl o snl or vrtx, t onntvty s rtr rl. Hr r som ntons: Br: Any wos rmovl rslts n sonnt rp (s F. 1). 2-E Connt: A rp s 2- onnt t ontns no rs (s F. 1). In nrl rp s k- onnt t rmovl o ny k 1 s rslts n onnt rp. Hr r lso vrtx-s qvlnts: Ct Vrtx: Any vrtx wos rmovl (totr wt t rmovl o ny nnt s) rslts n sonnt rp (s F. 1). Bonnt: A rp s onnt t ontns no t vrts (s F. 1()). In nrl rp s k-onnt t rmovl o ny k 1 vrts rslts n onnt rp. Not onnt nor 2-E Connt t vrtx 2-E Connt Bonnt t vrtx r () F. 1: Hr-orr onntvty n rps W wll prsnt n O(n + m)-tm lortm or omptn ll t rs o n nrt rp. (T lortm n mo to ompt t t vrts s wll.) Alto w wll not prov t, sl t ot 2- onnt rps s tt twn vry pr o stnt vrts, tr r t lst two -sont pts twn tm. (Ts s onsqn o mor nrl rslt ll Mnr s Torm.) I rp s onnt, tn or vry pr o s tr s smpl yl (tt s yl tt os not rpt ny vrts) tt ontns ot s. Ltr 4 1 Fll 2017

2 Fnn Brs tro DFS: An ovos, t slow, wy or omptn rs wol to lt n tn pply DFS to trmn wtr t rsltn rp rmns onnt. Howvr, ts wol tk O(m(n + m)) tm. W wll s tt t s possl to nty ll t rs wt snl pplton o DFS n O(n + m) tm. W ssm tt G s onnt, w mpls tt tr s snl DFS tr. Rll tt t DFS tr onssts o two typs o s: tr s, w onnt prnt wt ts l n t DFS tr, n k s, w onnt (non-prnt) nstor wt (non-l) snnt. Sppos tt w r rrntly prossn vrtx n DFSvst, n w s n (, v) on to nor v o. I ts s k (tt s, v s n nstor o ) tn (, v) nnot r, s t tr s twn n v prov son wy to onnt ts vrts. Tror, w my lmt onsrton to wn (, v) s tr, tt s, v s not yt n sovr, n so w wll nvok DFSvst(v). Wl w r on ts, w wll kp trk o t k s n t str root t v. Osrv tt ll ts k s rmn ntrly wtn ts str (s F. 2) tn (, v) s r, sn ts rmovl ompltly sonnts ts str rom t rst o t tr. On t otr n, tr s vn snl k ln ot rom ts str, tn (, v) s not r. S k mst o rom wtn t str to propr nstor o v. By t Prntss Lmm, ts mns tt t ls to vrtx wos sovry tm s strtly smllr tn v s sovry tm. (Rll vrtx s nstors r sovr or t.) In smmry, w v stls t ollown lm. (, v) s r v (, v) s not r v F. 2: Contons or vrtx to t vrtx. Clm: An (, v) s r n only t s tr n (ssmn tt s t prnt o v) tr s no k wtn v s str tt ls to vrtx wos sovry tm s strtly smllr tn v s sovry tm. Trkn Bk Es: T ov lm provs s wt strtrl rtrzton o rs. How n w sn n lortm tt tsts ts onton? To o ts, w wll ntro n xlry qntty, w wll ompt s t DFS rns. W n Low[] = mn([], mn{[w] : k (v, w) wr v s snnt o. Ltr 4 2 Fll 2017

3 Not tt w s t trm snnt n t nonstrt sns, tt s, v my tsl. Inttvly, Low[] s t losst to t root tt yo n t n t tr y tkn ny on k rom tr or ny o ts snnts. (Bwr o ts notton: Low mns low sovry tm, not low n or rwn o t DFS tr. In t Low[] tns to n t tr, n t sns o n los to t root.) Also not tt yo my onsr ny snnt o, t yo my only ollow on k. To ompt Low[] w s t ollown smpl rls: Sppos tt w r prormn DFS on t vrtx. Intlzton: Low[] = []. Bk (, v): Low[] = mn(low[], [v]). Explnton: W v tt nw k omn ot o. I ts os to lowr -vl tn t prvos k tn mk ts t nw Low (s F. 3). Tr (, v): Low[] = mn(low[], Low[v]). Explnton: Sn v s n t str root t ny snl k lvn t tr root t v s snl k or t tr root t (s F. 3). Bk v [v] Low[] mn(low[], [v]) Tr Low[v] v Low[] mn(low[], Low[v]) F. 3: Ct Vrts n t nton o Low[]. T o lok low sows ow to ompt Low[] or ll vrts. W o not ot omptn ns tms, sn ty r not n or or prposs. Not tt tr s stlty n trmnn wtr n s k. Clrly, s n mst o to prvosly sovr vrtx (w s wy t s n t ls-ls), t w lso n to k tt ts vrtx s not s prnt. Rll tt vry o n nrt rp s rlt tw n t ny lst (wt v s nor o n s nor o v). W n to k tt w r not smply sn t tr n, t rom t l k to t prnt. To o ts, w k v s not s prnt, tt s v pr[]. Osrv tt on Low[] s ompt or ll vrts, w n tst wtr vn tr (pr[v], v) s r y tstn wtr tr s no k n v s str on to n rlr sovr vrtx, tt s, [v] = Low[v]. (Atlly, t tst s mor ntrlly stt s [v] Low[v], t y nton Low[v] [v], so tstn or qlty s qvlnt.) T nl o s sown n t o lok low. Ltr 4 3 Fll 2017

4 DFSvst() { mrk[] = sovr Low[] = [] = ++tm or (v n A()) { (mrk[v] == nsovr) { pr[v] = DFSvst(v) Low[] = mn(low[], Low[v]) ls (v!= pr[]) { Low[] = mn(low[], [v]) Moton o DFSvst or Low omptton // st sovry tm n nt Low // (,v) s tr // v s prnt s // pt Low[] // (,v) s k // pt Low[] nallbrs(g) { tm = 0 or ( n V) mrk[] = nsovr // ntlz Comptn Brs v DFS or ( n V) (mrk[] == nsovr) DFSVst() or (v n V) { = pr[v] (!= nll n [v] == Low[v]) otpt (, v) s n r // nsovr vrtx? //...strt nw sr r // k or t rs 2:2 1:1 8:8 [] : Low[] 4:4 5:4 3:2 9:8 7:2 10:8 Brs: (, ) (, ) (, ) 6:4 () F. 4: Comptn rs v DFS. Ltr 4 4 Fll 2017

5 Wrpp: W v sown ow to ompt rs n n nrt rp. Tr r nmr o ntrstn prolms tt w stll v not sss. Frst, w lm tt t s possl to pt ts lortm to ompt t vrts s wll. (T omptton o Low s t sm, t rnt onton s ppl to trmn w vrts r t vrts.) Son, rp ls to 2- onnt, t my srl to prtton t vrts o t rp nto 2- onnt omponnts. For xmpl, n F. 4() t omponnts onsst o {,,, {,,, {, n {,,. Ts n on y smpl xtnson s wll. (T vrts r stor on stk, n wnvr r s tt, w pop o n pproprt sst o t stk. W wll lv t tls s n xrs.) A smlr ppro n ppl to omptn t onnt omponnts o rp, w s prtton o t st o t rp. Ltr 4 5 Fll 2017

Depth First Search. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong

Depth First Search. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong Dprtmnt o Computr Sn n Ennrn Cns Unvrsty o Hon Kon W v lry lrn rt rst sr (BFS). Toy, w wll suss ts sstr vrson : t pt rst sr (DFS) lortm. Our susson wll on n ous on rt rps, us t xtnson to unrt rps s strtorwr.