Graphs Depth First Search

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1 Grp Dpt Frt Sr SFO 337 LAX DFW ORD - 1 -

2 Grp Sr Aort - 2 -

3 Outo Ø By unrtnn t tur, you ou to: q L rp orn to t orr n w vrt r ovr, xpor ro n n n pt-rt r. q Cy o t pt-rt r tr,, orwr n ro q Ipnt pt-rt r q Dontrt p ppton o pt-rt r - 3 -

4 Outn Ø Aort Ø Exp Ø Appton - 4 -

5 Outn Ø Aort Ø Exp Ø Appton - 5 -

6 Dpt Frt Sr () Ø I: q Contnu rn pr nto t rp, unt w t tu. q I t vn v v n xpor w tr to t vrtx ro w v w ovr. q Anoou to Eur tour or tr Ø U to p ov ny rp pro, nun q Intyn no tt r r ro p no v q Dttn y q Extrtn trony onnt oponnt q Topoo ort - 6 -

7 Ø T ort r to trty or xporn z q W r ntrton, ornr n n (vrtx) vt q W r orror ( ) trvr q W p tr o t pt to t ntrn (trt vrtx) y n o rop (ruron t) Dpt-Frt Sr - 7 -

8 Dpt-Frt Sr Inp u t: Grp G = ( V, E) (rt or unrt ) Ø Expor vry, trtn ro rnt vrt nry. Ø A oon vrtx ovr, xpor ro t. Ø Kp tr o pror y oourn vrt: q B: unovr vrt q R: ovr, ut not n (t xporn ro t) q Gry: n (Dovr vrytn r ro t)

9 Exp on Unrt Grp A A unxpor n xpor A A n unxpor B D E ovry C A A B D E B D E C C - 9 -

10 Exp (ont.) A A B D E B D E C C A A B D E B D E C C

11 Aort Pttrn (G) Pronton: G rp Potonton: vrt n G v n vt or vrtx u V[G] oor[u] = BLACK ntz vrtx or vrtx u V[G] oor[u] = BLACK yt unxpor -Vt(u)

12 Aort Pttrn -Vt (u) Pronton: vrtx u unovr Potonton: vrt r ro u v n pro oour[u] RED or v A[u] xpor (u,v) oor[v] = BLACK -Vt(v) oour[u] GRAY

13 Proprt o Proprty 1 -Vt(u) vt t vrt n n t onnt oponnt o u Proprty 2 A T ovry y -Vt(u) or pnnn tr o t onnt oponnt o u B C D E

14 Aort Pttrn (G) Pronton: G rp Potonton: vrt n G v n vt or vrtx u V[G] oor[u] = BLACK ntz vrtx or vrtx u V[G] oor[u] = BLACK yt unxpor -Vt(u) tot wor = θ(v )

15 Aort Pttrn -Vt (u) Pronton: vrtx u unovr Potonton: vrt r ro u v n pro oour[u] RED or v A[u] xpor (u,v) oor[v] = BLACK -Vt(v) oour[u] GRAY tot wor = A[v] = θ(e) v V Tu runnn t = θ(v + E) (un ny t trutur)

16 Vrnt o Dpt-Frt Sr Ø In ton to, or nt o n vrt wt oour, ty n wt ovry n nn t. Ø T n ntr tt nrnt wnvr vrtx n tt q ro unxpor to ovr q ro ovr to n Ø T ovry n nn t n tn u to ov otr rp pro (.., oputn trony-onnt oponnt) Inp u t: Grp G = ( V, E) (rt or unrt ) Output: 2 ttp on vrtx: v [ ] = ovry t. v [ ] = nn t. 1 v [ ] < v [ ] 2 V

17 Aort wt Dovry n Fn T (G) Pronton: G rp Potonton: vrt n G v n vt or vrtx u V[G] oor[u] = BLACK ntz vrtx t 0 or vrtx u V[G] oor[u] = BLACK yt unxpor -Vt(u)

18 Aort wt Dovry n Fn T -Vt (u) Pronton: vrtx u unovr Potonton: vrt r ro u v n pro oour[u] RED t t + 1 [u] t or v A[u] xpor (u,v) oor[v] = BLACK -Vt(v) oour[u] GRAY t t + 1 [u] t

19 Otr Vrnt o Dpt-Frt Sr Ø T Pttrn n o u to q Coput ort o pnnn tr (on or to vt) no n pror t π[u] q L n t rp orn to tr ro n t r ²Dovry tr, trvr to n unovr vrtx ²Forwr, trvr to nnt vrtx on t urrnt pnnn tr ²B, trvr to n ntor vrtx on t urrnt pnnn tr ²Cro, trvr to vrtx tt ry n ovr, ut not n ntor or nnt

20 En o Ltur Mr 27,

21 Outn Ø Aort Ø Exp Ø Appton

22 Not: St Lt-In Frt-Out (LIFO) Dovr Not Fn St <no,# >

23 1 Dovr Not Fn St <no,# > ,0

24 2 1 Dovr Not Fn St <no,# > ,0,1

25 2 1 Dovr Not Fn St <no,# > ,0,1,1

26 2 1 Dovr Not Fn St <no,# > ,0,1,1,1

27 2 1 Dovr Not Fn St <no,# > ,0,1,1,1,1

28 2 1 Dovr Not Fn St <no,# > 3 Pt on St 4 Dovry tr E ,1,1,1,1

29 2 1 Dovr Not Fn St <no,# > ,1,1,1

30 2 1 Dovr Not Fn St <no,# > ,0,2,1,1

31 Cro E to Fn no: []<[] Dovr Not Fn St <no,# >,1,2,1,1

32 2 1 Dovr Not Fn St <no,# > ,2,2,1,1

33 2 1 Dovr Not Fn St <no,# > ,0,3,2,1,1

34 2 1 Dovr Not Fn St <no,# > ,1,3,2,1,1

35 2 1 Dovr Not Fn St <no,# > ,3,2,1,1

36 2 1 Dovr Not Fn St <no,# > ,0,4,2,1,1

37 2 1 Dovr Not Fn St <no,# > ,0,1,4,2,1,1

38 B E to no on St: 2 1 Dovr Not Fn St <no,# > ,1,1,4,2,1,1

39 2 1 Dovr Not Fn St <no,# > ,0,2,1,4,2,1,1

40 2 1 Dovr Not Fn St <no,# > ,1,2,1,4,2,1,1

41 2 1 Dovr Not Fn St <no,# > ,2,1,4,2,1,1

42 2 1 Dovr Not Fn St <no,# > ,1,4,2,1,1

43 2 1 Dovr Not Fn St <no,# > ,4,2,1,1

44 2 1 Dovr Not Fn St <no,# > ,0,5,2,1,1

45 2 1 Dovr Not Fn St <no,# > ,1,5,2,1,1

46 2 1 Dovr Not Fn St <no,# > ,5,2,1,1

47 2 1 Dovr Not Fn St <no,# > ,2,1,1

48 Forwr E 2 1 Dovr Not Fn St <no,# > ,3,1,1

49 2 1 Dovr Not Fn St <no,# > ,1,1

50 2 1 Dovr Not Fn St <no,# > ,2,1

51 220 1 Dovr Not Fn St <no,# > ,1

52 220 1 Dovr Not Fn St <no,# > ,0,2

53 220 1 Dovr Not Fn St <no,# > ,1,2

54 220 1 Dovr Not Fn St <no,# > ,2,2

55 220 1 Dovr Not Fn St <no,# > ,0,3,2

56 220 1 Dovr Not Fn St <no,# > ,1,3,2

57 220 1 Dovr Not Fn St <no,# > ,3,2

58 220 1 Dovr Not Fn St <no,# > ,2

59 220 1 Dovr Not Fn St <no,# > ,3

60 Dovr Not Fn St <no,# > ,0,4

61 Dovr Not Fn St <no,# > ,1,4

62 Dovr Not Fn St <no,# > ,2,4

63 Dovr Not Fn St <no,# > ,3,4

64 Dovr Not Fn St <no,# > ,4

65 Tr E B E Forwr E Cro E Fn! Dovr Not Fn St <no,# >

66 Cton o E n 1. Tr r n t pt-rt ort G π. E (u, v) tr v w rt ovr y xporn (u, v). 2. B r to (u, v) onntn vrtx u to n ntor v n pt-rt tr. 3. Forwr r non-tr (u, v) onntn vrtx u to nnt v n pt-rt tr. 4. Cro r otr. Ty n o twn vrt n t pt-rt tr, on on vrtx not n ntor o t otr

67 Cton o E n 1. Tr : E (u, v) tr v w wn (u, v) trvr. Not tt [v] > [u]. 2. B : (u, v) v w r wn (u, v) trvr. Not tt [v] < [u]. 3. Forwr : (u, v) orwr v w ry wn (u, v) trvr n [v] > [u]. 4. Cro (u,v) ro v w ry wn (u, v) trvr n [v] < [u]. 127 Cyn n p to nty proprt o t rp,.., rp y y no

68 on Unrt Grp Ø In pt-rt r o n unrt rp, vry tr tr or. Ø Wy?

69 on Unrt Grp Ø Suppo tt (u,v) orwr or ro n o n unrt rp. Ø (u,v) orwr or ro wn v ry Fn (ry) wn ro u. Ø T n tt vrt r ro v v n xpor. Ø Sn w r urrnty nn u, u ut r. Ø Cry v r ro u. Ø Sn t rp unrt, u ut o r ro v. Ø Tu u ut ry v n Fn: u ut ry. Ø Contrton! u v

70 Outn Ø Aort Ø Exp Ø Appton

71 Appton 1: Pt Fnn Ø T pttrn n u to n pt twn two vn vrt u n z, on xt Ø W u t to p tr o t urrnt pt Ø I t tnton vrtx z nountr, w rturn t pt t ontnt o t t -Pt (u,z,t) Pronton: u n z r vrt n rp, t ontn urrnt pt Potonton: rturn tru pt ro u to z xt, t ontn pt oour[u] RED pu u onto t u = z rturn TRUE or v A[u] xpor (u,v) oor[v] = BLACK oour[u] GRAY pop u ro t rturn FALSE -Pt(v,z,t) rturn TRUE

72 Appton 2: Cy Fnn Ø T pttrn n u to trn wtr rp y. Ø I nountr, w rturn tru. -Cy (u) Pronton: u vrtx n rp G Potonton: rturn tru tr y r ro u. oour[u] RED or v A[u] xpor (u,v) oor[v] = RED rturn tru oor[v] = BLACK -Cy(v) rturn tru oour[u] GRAY rturn

73 Wy ut on rp wt y nrt? Ø Suppo tt vrtx n onnt oponnt S tt ontn y C. Ø Sn vrt n S r r ro, ty w vt y ro. Ø Lt v t rt vrtx n C r y ro. Ø Tr r two vrt u n w nt to v on t y C. Ø wo, uppo u xpor rt. Ø Sn w r ro u, w w vntuy ovr. u v w Ø Wn xporn w ny t, t - (w, v) w ovr

74 A4Q2: Cour Prrqut Ø In ot pot-onry pror, our v prrqut. Ø For xp, you nnot t EECS 3101 unt you v p. Ø How n w rprnt u yt o pnn? Ø A ntur o rt rp. q E vrtx rprnt our q E rt rprnt prrqut ²A rt ro Cour U to Cour V n tt Cour U ut tn or Cour V

75 A4Q2: Cour Prrqut Ø W o wnt to to n t norton or prtur our quy. Ø T our nur prov onvnnt y tt n u to ornz our ror n ort p, pnt nry r tr (. A3Q1). Ø Tu t n to rprnt our un ot ort p (or nt ) n rt rp (to rprnt pnn). Ø By torn rrn to t rt rp vrtx or our n t ort p, w n nty our pnn

76 A4Q2: Cour Prrqut Ky: 2011 Vu: Nur: 2011 N: Dt Strutur Vrtx: (K 1,V 1 ) (K 2,V 2 ) Sort Mp (K 3,V 3 ) Drt Grp

77 A4Q2: Cour Prrqut Ø It portnt tt t our prrqut rp rt y rp (DAG). Wy?

78 A4Q2: Cour Prrqut Ø In t quton, you r prov wt pntton o yt to rprnt our n pnn. Ø Mto or n our n ttn prrqut r prov. Ø You n ony wrt t to or n prrqut. Ø T to w u pt-rt-r ort (o prov) tt n u to prvnt t ton o prrqut tt ntrou y

79 A4Q2: Ipntton un nt.ttrutur Ø W u t TrMp to rprnt t ort p (. A3Q1). Ky: 2011 Vu: Nur: 2011 N: Dt Strutur Vrtx: Mp AtrtMp SortMp AtrtSortMp (K 1,V 1 ) Sort Mp (K 3,V 3 ) TrMp Entry MpEntry (K 2,V 2 )

80 A4Q2: Ipntton un nt.ttrutur Ø W u t AnyMpGrp to rprnt t rt rp. Ø T pntton u ProHMp, nr pro t, to rprnt t non n outon or vrtx. Drt Grp Mp Grp AtrtMp AnyMpGrp AtrtHMp ProHMp

81 Outn Ø Aort Ø Exp Ø Appton

82 Outo Ø By unrtnn t tur, you ou to: q L rp orn to t orr n w vrt r ovr, xpor ro n n n pt-rt r. q Cy o t pt-rt r tr,, orwr n ro q Ipnt pt-rt r q Dontrt p ppton o pt-rt r

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