Reachability. Directed DFS. Strong Connectivity Algorithm. Strong Connectivity. DFS tree rooted at v: vertices reachable from v via directed paths

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1 irt Grphs OR SFO FW LX JFK MI OS irph is rph whos s r ll irt Short or irt rph pplitions on-wy strts lihts tsk shulin irphs ( 12.) irt Grphs 1 irt Grphs 2 irph Proprtis rph G=(V,) suh tht h os in on irtion: (,) os rom to, ut not to. I G is simpl, m < n*(n-1). I w kp in-s n out-s in sprt jny lists, w n prorm listin o in-s n out-s in tim proportionl to thir siz. irph pplition Shulin: (,) mns tsk must omplt or n strt is21 is1 is11 is22 is is11 is2 is2 is121 is11 is11 is11 Th oo li irt Grphs irt Grphs

2 irt FS Rhility W n spiliz th trvrsl lorithms (FS n FS) to irphs y trvrsin s only lon thir irtion In th irt FS lorithm, w hv our typs o s isovry s k s orwr s ross s irt FS strtin vrtx s trmins th vrtis rhl rom s FS tr root t v: vrtis rhl rom v vi irt pths F F irt Grphs irt Grphs Stron onntivity h vrtx n rh ll othr vrtis Stron onntivity lorithm Pik vrtx v in G. Prorm FS rom v in G. I thr s w not visit, print no. Lt G G with s rvrs. Prorm FS rom v in G. I thr s w not visit, print no. ls, print ys. Runnin tim: O(n+m). G: G : irt Grphs irt Grphs

3 Mximl surphs suh tht h vrtx n rh ll othr vrtis in th surph n lso on in O(n+m) tim usin FS, ut is mor omplit (similr to ionntivity). Stronly onnt omponnts {,, } {,,, } irt Grphs Trnsitiv losur Givn irph G, th trnsitiv losur o G is th irph G* suh tht G* hs th sm vrtis s G i G hs irt pth rom u to v (u v), G* hs irt rom u to v Th trnsitiv losur provis rhility inormtion out irph irt Grphs 10 G G* omputin th Trnsitiv losur W n prorm FS strtin t h vrtx O(n(n+m)) I thr's wy to t rom to n rom to, thn thr's wy to t rom to. ltrntivly... Us ynmi prormmin: Th Floy-Wrshll lorithm irt Grphs 11 Floy-Wrshll Trnsitiv losur I #1: Numr th vrtis 1, 2,, n. I #2: onsir pths tht us only vrtis numr 1, 2,, k, s intrmit vrtis: Uss only vrtis numr 1,,k-1 i Uss only vrtis numr 1,,k ( this i it s not lry in) k irt Grphs 12 j Uss only vrtis numr 1,,k-1

4 Floy-Wrshll s lorithm Floy-Wrshll s lorithm numrs th vrtis o G s,, v n n omputs sris o irphs G 0,, G n G 0 =G G k hs irt (v i ) i G hs irt pth rom v i to v j with intrmit vrtis in th st {,, v k } W hv tht G n = G* In phs k, irph G k is omput rom G k 1 Runnin tim: O(n ), ssumin rjnt is O(1) (.., jny mtrix) lorithm FloyWrshll(G) Input irph G Output trnsitiv losur G* o G i 1 or ll v G.vrtis() not v s v i i i + 1 G 0 G or k 1 to n o G k G k 1 or i 1 to n (i k) o or j 1 to n (j i, k) o i G k 1.rjnt(v i, v k ) G k 1.rjnt(v k ) i G k.rjnt(v i ) G k.insrtirt(v i, k) xmpl v v G = G 0 = G 1 = G 2 v v v v v G v G = G = G* rturn G n irt Grphs 1 v irt Grphs 1 Gs n Topoloil Orrin irt yli rph (G) is irph tht hs no irt yls topoloil orrin o irph is numrin,, v n o th vrtis suh tht or vry (v i ), w hv i < j xmpl: in tsk shulin irph, topoloil orrin tsk squn tht stisis th prn onstrints Thorm irph mits topoloil orrin i n only i it is G irt Grphs 1 G G v v v Topoloil orrin o G Topoloil Sortin Numr vrtis, so tht (u,v) in implis u < v 1 wk up typil stunt y 2 t stuy omputr si. ply mk ookis or prossors np mor.s. writ.s. prorm 10 slp work out 11 rm out rphs irt Grphs 1

5 lorithm or Topoloil Sortin Not: This lorithm is irnt thn th on in Goorih-Tmssi Runnin tim: O(n + m). How? Mtho TopoloilSort(G) H G // Tmporry opy o G n G.numVrtis() whil H is not mpty o Lt v vrtx with no outoin s Ll v n n n - 1 Rmov v rom H irt Grphs 1 Topoloil Sortin lorithm usin FS Simult th lorithm y usin pth-irst srh lorithm topoloilfs(g) Input G Output topoloil orrin o G n G.numVrtis() or ll u G.vrtis() stll(u, UNXPLOR) or ll G.s() stll(, UNXPLOR) or ll v G.vrtis() i tll(v) = UNXPLOR topoloilfs(g, v) O(n+m) tim. lorithm topoloilfs(g, v) Input rph G n strt vrtx v o G Output llin o th vrtis o G in th onnt omponnt o v stll(v, VISIT) or ll G.inints(v) i tll() = UNXPLOR w opposit(v,) i tll(w) = UNXPLOR stll(, ISOVRY) topoloilfs(g, w) ls { is orwr or ross } Ll v with topoloil numr n n n - 1 irt Grphs 1 Topoloil Sortin xmpl Topoloil Sortin xmpl irt Grphs 1 irt Grphs 20

6 Topoloil Sortin xmpl Topoloil Sortin xmpl irt Grphs 21 irt Grphs 22 Topoloil Sortin xmpl Topoloil Sortin xmpl irt Grphs 2 irt Grphs 2

7 Topoloil Sortin xmpl Topoloil Sortin xmpl irt Grphs 2 irt Grphs 2 Topoloil Sortin xmpl Topoloil Sortin xmpl irt Grphs 2 irt Grphs 2