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1 Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous prolms I Cn ptur ntitis, rltionships twn thm, th gr of th rltionship, t. I This hptr ovrs sis in grph thory, inluing rprsnttion, n lgorithms for si grph-thorti prolms I W ll uil on ths ltr this smstr Typs of Grphs Typs of Grphs (2) I A (simpl, or unirt) grph G =(V, E) onsists of V, nonmpty st of vrtis n E st of unorr pirs of istint vrtis ll gs A D B E C V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)} I A irt grph (igrph) G =(V, E) onsists of V, nonmpty st of vrtis n E st of orr pirs of istint vrtis ll gs Typs of Grphs (3) Rprsnttions of Grphs I A wight grph is n unirt or irt grph with th itionl proprty tht h g hs ssoit with it rl numr w() ll its wight I Two ommon wys of rprsnting grph: Ajny list n jny mtrix I Lt G =(V, E) grph with n vrtis n m gs

2 Ajny List Ajny Mtrix I For h vrtx v 2 V, stor list of vrtis jnt to v I For wight grphs, informtion to h no I How muh is sp rquir for storg? I Us n n n mtrix M, whrm(i, j) =1if(i, j) is n g, 0 othrwis I If G wight, stor wights in th mtrix, using 1 for non-gs I How muh is sp rquir for storg? Brth-First Srh (BFS) BFS(G, s) I Givn grph G =(V, E) (irtorunirt)nsour no s 2 V, BFS systmtilly visits vry vrtx tht is rhl from s I Uss quu t strutur to srh in rth-first mnnr I Crts strutur ll BFS tr suh tht for h vrtx v 2 V, th istn (numr of gs) from s to v in tr is shortst pth in G I Initiliz h no s olor to whit I As no is visit, olor it to gry () in quu), thn lk () finish) 1 for h vrtx u 2 V \{s} o 2 olor[u] =whit 3 [u] =1 4 [u] =nil 5 n 6 olor[s] =gry 7 [s] =0 8 [s] =nil 9 Q = ; 10 Enquu(Q, s) 11 whil Q 6= ; o 12 u = Dquu(Q) 13 for h v 2 Aj[u] o 14 if olor[v] == whit thn 15 olor[v] =gry 16 [v] =[u] [v] =u 18 Enquu(Q, v) n 21 olor[u] =lk 22 n BFS Exmpl BFS Exmpl (2)

3 BFS Proprtis Dpth-First Srh (DFS) I Wht is th running tim? I Hint: How mny tims will no nquu? I Aftr th n of th lgorithm, [v] = shortst istn from s to v ) Solvs unwight shortst pths I Cn print th pth from s to v y rursivly following [v], [ [v]], t. I If [v] == 1, thnv not rhl from s ) Solvs rhility I Anothr grph trvrsl lgorithm I Unlik BFS, this on follows pth s p s possil for ktrking I Whr BFS is quu-lik, DFS is stk-lik I Trks oth isovry tim n finishing tim of h no, whih will om in hny ltr DFS(G) DFS-Visit(u) 1 for h vrtx u 2 V o 2 olor[u] =whit 3 [u] =nil 4 n 5 tim =0 6 for h vrtx u 2 V o 7 if olor[u] ==whit thn 8 DFS-Visit(u) 9 10 n 1 olor[u] =gry 2 tim = tim +1 3 [u] =tim 4 for h v 2 Aj[u] o 5 if olor[v] ==whit thn 6 [v] =u 7 DFS-Visit(v) 8 9 n 10 olor[u] =lk 11 f [u] =tim = tim +1 DFS Exmpl DFS Exmpl (2)

4 DFS Proprtis I Tim omplxity sm s BFS: ( V + E ) I Vrtx u is propr snnt of vrtx v in th DF tr i [v] < [u] < f [u] < f [v] ) Prnthsis strutur: If on prints (u whn isovring u n u) whn finishing u, thnprinttxtwillwll-formprnthsiz sntn DFS Proprtis (2) I Clssifition of gs into groups I A tr g is on in th pth-first forst I A k g (u, v) onnts vrtx u to its nstor v in th DF tr (inlus slf-loops) I A forwr g is nontr g onnting no to on of its DF tr snnts I A ross g gos twn non-nstrl gs within DF tr or twn DF trs I S lls in DFS xmpl I Exmpl us of this proprty: A grph hs yl i DFS isovrs k g (pplition: lok ttion) I Whn DFS first xplors n g (u, v), look t v s olor: I olor[v] == whit implis tr g I olor[v] ==gry implis k g I olor[v] ==lk implis forwr or ross g Applition: Topologil Sort Applition: Topologil Sort (2) A irt yli grph (g) n rprsnt prns: n g (x, y) implis tht vnt/tivity x must our for y A topologil sort of g G is n linr orring of its vrtis suh tht if G ontins n g (u, v), thn u pprs for v in th orring Topologil Sort Algorithm Applition: Strongly Connt Componnts 1. Cll DFS lgorithm on g G 2. As h vrtx is finish, insrt it to th front of link list 3. Rturn th link list of vrtis I Thus topologil sort is sning sort of vrtis s on DFS finishing tims I Why os it work? I Whn no is finish, it hs no unxplor outgoing gs; i.. ll its snnt nos r lry finish n insrt t ltr spot in finl sort Givn irt grph G =(V, E), strongly onnt omponnt (SCC) of G is mximl st of vrtis C V suh tht for vry pir of vrtis u, v 2 Cuis rhl from v n v is rhl from u Wht r th SCCs of th ov grph?

5 Trnspos Grph SCC Algorithm I Our lgorithm for fining SCCs of G pns on th trnspos of G, not G T I G T is simply G with gs rvrs I Ft: G T n G hv sm SCCs. Why? 1. Cll DFS lgorithm on G 2. Comput G T 3. Cll DFS lgorithm on G T, looping through vrtis in orr of rsing finishing tims from first DFS ll 4. Eh DFS tr in son DFS run is n SCC in G SCC Algorithm Exmpl SCC Algorithm Exmpl (2) Aftr first roun of DFS: Aftr son roun of DFS: Whih no is first on to visit in son DFS? SCC Algorithm Anlysis I Wht is its tim omplxity? I How os it work? 1. Lt x no with highst finishing tim in first DFS 2. In G T, x s omponnt C hs no gs to ny othr omponnt (Lmm 22.14), so th son DFS s tr gs fin xtly x s omponnt 3. Now lt x 0 th nxt no xplor in nw omponnt C 0 4. Th only gs from C 0 to nothr omponnt r to nos in C, soth DFS tr gs fin xtly th omponnt for x 0 5. An so on...

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)} s s of s Computr Sin & Enginring 423/823 Dsign n Anlysis of Ltur 03 (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) s of s s r strt t typs tht r pplil to numrous prolms Cn ptur ntitis, rltionships twn

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