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1 Grps Dpt Frst Sr SFO ORD LAX DFW

2 Grp Sr Aortms

3 Outn DFS Aortm DFS Exmp DFS Apptons

4 Outn DFS Aortm DFS Exmp DFS Apptons

5 I: Dpt Frst Sr (DFS) Contnu srn pr nto t rp, unt w t stu. I t s vn v v bn xpor w btr to t vrtx rom w v ws sovr. Anoous to Eur tour or trs Us to p sov mny rp probms, nun Nos tt r rb rom sp no v Dtton o ys Extrton o strony onnt omponnts Topoo sorts

6 T DFS ortm s smr to ss strty or xporn mz W mr ntrston, ornr n n (vrtx) vst W mr orror ( ) trvrs W p tr o t pt b to t ntrn (strt vrtx) by mns o rop (rurson st) Dpt-Frst Sr

7 Dpt-Frst Sr Inp u t: Grp G ( V, E ) (rt or unrt) Expor vry, strtn rom rnt vrts nssry. As soon s vrtx sovr, xpor rom t. Kp tr o prorss by oourn vrts: B: unsovr vrts R: sovr, but not ns (st xporn rom t) Gry: ns (oun vrytn rb rom t).

8 DFS Exmp on Unrt Grp A A unxpor bn xpor A A ns unxpor B D E sovry b C A A B D E B D E C C

9 Exmp (ont.) A A B D E B D E C C A A B D E B D E C C

10 DFS Aortm Pttrn DFS(G) Pronton: G s rp Postonton: vrts n G v bn vst or vrtx u ÎV[G] oor[u] = BLACK ntz vrtx or vrtx u ÎV[G] oor[u] = BLACK s yt unxpor DFS-Vst(u)

11 DFS Aortm Pttrn DFS-Vst (u) Pronton: vrtx u s unsovr Postonton: vrts rb rom u v bn pross oour[u] RED or v ÎA[u] xpor (u,v) oor[v] = BLACK DFS-Vst(v) oour[u] GRAY

12 Proprts o DFS Proprty 1 DFS-Vst(u) vsts t vrts n s n t onnt omponnt o u Proprty 2 A T sovry s b by DFS-Vst(u) orm spnnn tr o t onnt omponnt o u B C D E

13 DFS Aortm Pttrn DFS(G) Pronton: G s rp Postonton: vrts n G v bn vst or vrtx u ÎV[G] oor[u] = BLACK ntz vrtx or vrtx u ÎV[G] oor[u] = BLACK s yt unxpor DFS-Vst(u) tot wor = q(v )

14 DFS Aortm Pttrn DFS-Vst (u) Pronton: vrtx u s unsovr Postonton: vrts rb rom u v bn pross oour[u] RED or v ÎA[u] xpor (u,v) oor[v] = BLACK DFS-Vst(v) oour[u] GRAY Tus runnn tm = q(v + E) (ssumn ny st strutur) tot wor å = A[v] = q(e) v ÎV

15 Vrnts o Dpt-Frst Sr In ton to, or nst o bn vrts wt oours, ty n b b wt sovry n nsn tms. Tm s n ntr tt s nrmnt wnvr vrtx ns stt rom unxpor to sovr rom sovr to ns Ts sovry n nsn tms n tn b us to sov otr rp probms (.., omputn strony-onnt omponnts) Inp u t: Grp G ( V, E ) (rt or unrt) Output: 2 tmstmps on vrtx: v [ ] sovry tm. v [ ] nsn tm. 1 [ v ] [ v ] 2 V

16 DFS Aortm wt Dsovry n Fns Tms DFS(G) Pronton: G s rp Postonton: vrts n G v bn vst or vrtx u ÎV[G] oor[u] = BLACK ntz vrtx tm 0 or vrtx u ÎV[G] oor[u] = BLACK s yt unxpor DFS-Vst(u)

17 DFS Aortm wt Dsovry n Fns Tms DFS-Vst (u) Pronton: vrtx u s unsovr Postonton: vrts rb rom u v bn pross oour[u] RED tm tm + 1 [u] tm or v ÎA[u] xpor (u,v) oor[v] = BLACK DFS-Vst(v) oour[u] GRAY tm tm + 1 [u] tm

18 Otr Vrnts o Dpt-Frst Sr T DFS Pttrn n so b us to Comput orst o spnnn trs (on or to DFSvst) no n prssor st π[u] Lb s n t rp orn to tr ro n t sr (s txtboo) Tr s, trvrs to n unsovr vrtx Forwr s, trvrs to snnt vrtx on t urrnt spnnn tr B s, trvrs to n nstor vrtx on t urrnt spnnn tr Cross s, trvrs to vrtx tt s ry bn sovr, but s not n nstor or snnt

19 Outn DFS Aortm DFS Exmp DFS Apptons

20 DFS s Not: St s Lst-In Frst-Out (LIFO) Foun Not Hn St <no,# s> b m

21 s 1 DFS b Foun Not Hn St <no,# s> m s,0

22 2 s 1 DFS b Foun Not Hn St <no,# s> m,0 s,1

23 2 s 1 DFS b Foun Not Hn St <no,# s> 3 m,0,1 s,1

24 2 s 1 DFS b Foun Not Hn St <no,# s> 3 4 m,0,1,1 s,1

25 2 s 1 DFS b Foun Not Hn St <no,# s> m,0,1,1,1 s,1

26 2 s 1 DFS b Foun Not Hn St <no,# s> 3 Pt on St 4 Tr E 56 m,1,1,1 s,1

27 2 s 1 DFS b Foun Not Hn St <no,# s> m,1,1 s,1

28 2 s 1 DFS b Foun Not Hn St <no,# s> m,0,2,1 s,1

29 Cross E to n no: []<[] s 1 DFS b m Foun Not Hn St <no,# s>,1,2,1 s,1

30 2 s 1 DFS b Foun Not Hn St <no,# s> m,2,2,1 s,1

31 2 s 1 DFS b Foun Not Hn St <no,# s> m,0,3,2,1 s,1

32 2 s 1 DFS b Foun Not Hn St <no,# s> m,1,3,2,1 s,1

33 2 s 1 DFS b Foun Not Hn St <no,# s> m,3,2,1 s,1

34 2 s 1 DFS b Foun Not Hn St <no,# s> m,0,4,2,1 s,1

35 2 s 1 DFS b Foun Not Hn St <no,# s> m 12,0,1,4,2,1 s,1

36 B E to no on St: 2 s 1 DFS b Foun Not Hn St <no,# s> m 12,1,1,4,2,1 s,1

37 2 s 1 DFS b Foun Not Hn St <no,# s> m m,0,2,1,4,2,1 s,1

38 2 s 1 DFS b Foun Not Hn St <no,# s> m m,1,2,1,4,2,1 s,1

39 2 s 1 DFS b Foun Not Hn St <no,# s> m ,2,1,4,2,1 s,1

40 2 s 1 DFS b Foun Not Hn St <no,# s> m ,1,4,2,1 s,1

41 2 s 1 DFS b Foun Not Hn St <no,# s> m ,4,2,1 s,1

42 2 s 1 DFS b Foun Not Hn St <no,# s> m ,0,5,2,1 s,1

43 2 s 1 DFS b Foun Not Hn St <no,# s> m ,1,5,2,1 s,1

44 2 s 1 DFS b Foun Not Hn St <no,# s> m ,5,2,1 s,1

45 2 s 1 DFS b Foun Not Hn St <no,# s> m ,2,1 s,1

46 Forwr E 2 s 1 DFS b Foun Not Hn St <no,# s> m ,3,1 s,1

47 2 s 1 DFS b Foun Not Hn St <no,# s> m ,1 s,1

48 2 s 1 DFS b Foun Not Hn St <no,# s> m ,2 s,1

49 220 s 1 DFS b Foun Not Hn St <no,# s> m s,1

50 220 s 1 DFS b Foun Not Hn St <no,# s> m ,0 s,2

51 220 s 1 DFS b Foun Not Hn St <no,# s> m ,1 s,2

52 220 s 1 DFS b Foun Not Hn St <no,# s> m ,2 s,2

53 220 s 1 DFS b Foun Not Hn St <no,# s> m ,0,3 s,2

54 220 s 1 DFS b Foun Not Hn St <no,# s> m ,1,3 s,2

55 220 s 1 DFS b Foun Not Hn St <no,# s> m ,3 s,2

56 220 s 1 DFS b Foun Not Hn St <no,# s> m s,2

57 220 s 1 DFS b Foun Not Hn St <no,# s> m s,3

58 220 s 1 DFS b 25 Foun Not Hn St <no,# s> m b,0 s,4

59 220 s 1 DFS b 25 Foun Not Hn St <no,# s> m b,1 s,4

60 220 s 1 DFS b 25 Foun Not Hn St <no,# s> m b,2 s,4

61 220 s 1 DFS b 25 Foun Not Hn St <no,# s> m b,3 s,4

62 220 s 1 DFS b 2526 Foun Not Hn St <no,# s> m s,4

63 Tr Es B Es Forwr Es Cross Es s DFS 1116 Fns! 910 b 2526 m Foun Not Hn St <no,# s>

64 Csston o Es n DFS 1. Tr s r s n t pt-rst orst G π. E (u, v) s tr v ws rst sovr by xporn (u, v). 2. B s r tos s (u, v) onntn vrtx u to n nstor v n pt-rst tr. 3. Forwr s r non-tr s (u, v) onntn vrtx u to snnt v n pt-rst tr. 4. Cross s r otr s. Ty n o btwn vrts n t sm pt-rst tr, s on s on vrtx s not n nstor o t otr. s b m

65 Csston o Es n DFS 1. Tr s: E (u, v) s tr v ws b wn (u, v) trvrs. 2. B s: (u, v) s b v ws r wn (u, v) trvrs. 3. Forwr s: (u, v) s orwr v ws ry wn (u, v) trvrs n [v] > [u]. 4. Cross s (u,v) s ross v ws ry wn (u, v) trvrs n [v] < [u]. Cssyn s n p to nty proprts o t rp,.., rp s y DFS ys no b s s b m

66 DFS on Unrt Grps In pt-rst sr o n unrt rp, vry s tr tr or b. Wy?

67 DFS on Unrt Grps Suppos tt (u,v) s orwr or ross n DFS o n unrt rp. (u,v) s orwr or ross wn v s ry n (ry) wn ss rom u. Ts mns tt vrts rb rom v v bn xpor. Sn w r urrnty nn u, u must b r. Cry v s rb rom u. Sn t rp s unrt, u must so b rb rom v. Tus u must ry v bn n: u must b ry. Contrton! u v

68 Outn DFS Aortm DFS Exmp DFS Apptons

69 DFS Appton 1: Pt Fnn T DFS pttrn n b us to n pt btwn two vn vrts u n z, on xsts W us st to p tr o t urrnt pt I t stnton vrtx z s nountr, w rturn t pt s t ontnts o t st DFS-Pt (u,z,st) Pronton: u n z r vrts n rp, st ontns urrnt pt Postonton: rturns tru pt rom u to z xsts, st ontns pt oour[u] RED pus u onto st u = z rturn TRUE or v ÎA[u] xpor (u,v) oor[v] = BLACK oour[u] GRAY pop u rom st rturn FALSE DFS-Pt(v,z,st) rturn TRUE

70 DFS Appton 2: Cy Fnn T DFS pttrn n b us to trmn wtr rp s y. I b s nountr, w rturn tru. DFS-Cy (u) Pronton: u s vrtx n rp G Postonton: rturns tru tr s y rb rom u. oour[u] RED or v ÎA[u] xpor (u,v) oor[v] = RED b rturn tru s oor[v] = BLACK DFS-Cy(v) rturn tru oour[u] GRAY rturn s

71 Wy must DFS on rp wt y nrt b? Suppos tt vrtx s s n onnt omponnt S tt ontns y C. Sn vrts n S r rb rom s, ty w b vst by DFS rom s. Lt v b t rst vrtx n C r by DFS rom s. Tr r two vrts u n w nt to v on t y C. wo, suppos u s xpor rst. Sn w s rb rom u, w w vntuy b sovr. Wn xporn w s ny st, t b- (w, v) w b sovr. u s v w

72 DFS Appton 3. Topoo Sortn (.., puttn tss n nr orr) Not: T txtboo so srbs brtrst TopooSort ortm (Ston )

73 DAGs n Topoo Orrn A rt y rp (DAG) s rp tt s no rt ys A topoo orrn o rp s numbrn D E v 1,, v n o t vrts su tt or vry (v, v ), w v Exmp: n ts sun rp, topoo orrn s ts squn tt stss t prn onstrnts Torm A rp mts topoo orrn n ony t s DAG v 2 v 1 B A B A C D C DAG G v 4 v 5 v 3 E Topoo orrn o G

74 Topoo (Lnr) Orr unrwr sos pnts sos unrwr pnts sos sos sos unrwr pnts sos

75 Topoo (Lnr) Orr unrwr sos pnts sos sos sos pnts unrwr Inv Orr

76 Aortm or Topoo Sortn Not: Ts ortm s rnt tn t on n Goor-Tmss Mto TopooSort(G) H G Tmporry opy o G n G.numVrts() w H s not mpty o Lt v b vrtx wt no outon s Lb v n n n - 1 Rmov v rom H s w s s nvovn v

77 Lnr Orr b Pr-Conton: A Drt Ay Grp (DAG) Post-Conton: Fn on v nr orr Aortm: Fn trmn no (sn). Put t st n squn. Dt rom rp & rpt Runnn tm: V å = O V 2 =1 ( ) O( V ).. Cn w o bttr?

78 b Lnr Orr A: DFS Foun Not Hn St..

79 b Lnr Orr A: DFS Foun Not Hn St Wn no s popp o st, nsrt t ront o nry-orr to o st. Lnr Orr:..

80 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,

81 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,

82 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,,

83 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,,,

84 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,,,

85 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,,,,

86 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,,,,,

87 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,,,,,,

88 b Lnr Orr A: DFS Foun Not Hn St b Lnr Orr:,,,,,,,

89 b Lnr Orr A: DFS Foun Not Hn St b Lnr Orr:,,,,,,,,

90 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr: b,,,,,,,,,

91 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr: b,,,,,,,,,

92 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,b,,,,,,,,,

93 b Lnr Orr A: DFS Foun Not Hn St Lnr Orr:,,b,,,,,,,,, Don!

94 DFS Aortm or Topoo Sort Ms sns. But ow o w prov tt t wors?

95 u v Lnr Orr Proo: Consr Cs 1: u os on st rst bor v. Bus o, v os on bor u oms o v oms o bor u oms o v os tr u n orr. Foun Not Hn St u v u v

96 u v Lnr Orr Proo: Consr Cs 1: u os on st rst bor v. Cs 2: v os on st rst bor u. v oms o bor u os on. v os tr u n orr. Foun Not Hn St u v u v

97 Lnr Orr Proo: Consr Cs 1: u os on st rst bor v. Cs 2: v os on st rst bor u. v oms o bor u os on. Cs 3: v os on st rst bor u. u os on bor v oms o. Pn: u os tr v n orr. Cy mns nr orr s mpossb Foun Not Hn St u v T nos n t st orm pt strtn t s. u v v u

98 b Lnr Orr A: DFS Anyss: (V+E) Foun Not Hn St Lnr Orr:,,b,,,,,,,,, Don!

99 DFS Appton 3. Topoo Sort Topoo-Sort(G) Pronton: G s rp Postonton: vrts n G v bn pus onto st n rvrs nr orr or vrtx u ÎV[G] oor[u] = BLACK ntz vrtx or vrtx u ÎV[G] oor[u] = BLACK s yt unxpor Topoo-Sort-Vst(u)

100 DFS Appton 3. Topoo Sort Topoo-Sort-Vst (u) Pronton: vrtx u s unsovr Postonton: u n vrts rb rom u v bn pus onto st n rvrs nr orr oour[u] RED or v ÎA[u] xpor (u,v) oor[v] = BLACK Topoo-Sort-Vst(v) pus u onto st oour[u] GRAY

101 Outn DFS Aortm DFS Exmp DFS Apptons