Graph Search Algorithms CSE

Transcription

1 Grp Src Aortms 1

2 Grp b No ~ cty or computr E ~ ro or t cb c Unrct or Drct A surprsny r numbr o computton probms cn b xprss s rp probms. 2

3 Drct n Unrct Grps () A rct rp G = (V, E), wr V = {1,2,3,4,5,6} n E = {(1,2), (2,2), (2,4), (2,5), (4,1), (4,5), (5,4), (6,3)}. T (2,2) s s-oop. (b) An unrct rp G = (V,E), wr V = {1,2,3,4,5,6} n E = {(1,2), (1,5), (2,5), (3,6)}. T vrtx 4 s sot. (c) T subrp o t rp n prt () nuc by t vrtx st {1,2,3,6}. 3

4 Trs Tr Forst Grp wt Cyc A tr s connct, cycc, unrct rp. A orst s st o trs (not ncssry connct) 4

5 Runnn Tm o Grp Aortms Runnn tm otn uncton o bot V n E. For convnnc, rop t. n symptotc notton,.. O(V+E). 5

6 En o Lctur 10 Apr 6, 2009

7 Rprsnttons: Unrct Grps Spc compxty: Tm to n nbours o vrtx u : Acncy Lst θ ( V + E) θ (r( u)) Acncy Mtrx 2 θ ( V ) θ ( V ) Tm to trmn ( uv, ) E: θ (r( u)) 7 θ (1)

8 Rprsnttons: Drct Grps Spc compxty: Tm to n nbours o vrtx u : Acncy Lst θ ( V + E) θ (r( u)) Acncy Mtrx 2 θ ( V ) θ ( V ) Tm to trmn ( uv, ) E: θ (r( u)) 8 θ (1)

9 Brt-Frst Src Go: To rcovr t sortst pts rom sourc no s to otr rcb nos v n rp. T nt o c pt n t pts tmsvs r rturn. Nots: Tr r n xponnt numbr o possb pts Ts probm s rr or nr rps tn trs bcus o cycs!? 9 s

10 Brt-Frst Src Input: Grp G = ( V, E ) (rct or unrct) n sourc vrtx s V. Output: v [ ] = sortst pt stnc δ ( sv, ) rom sto v, v V. π[ v] = u suc tt ( u, v) s st on sortst pt rom s to v. I: sn out src wv rom s. Kp trc o prorss by coourn vrtcs: Unscovr vrtcs r coour bc Just scovr vrtcs (on t wvront) r coour r. Prvousy scovr vrtcs (bn wvront) r coour ry. 10

11 s BFS Frst-In Frst-Out (FIFO) quu stors ust scovr vrtcs b Foun Not Hn Quu c m 11

12 c s BFS =0 b m Foun Not Hn Quu s =0 12

13 c s BFS =0 =1 b m Foun Not Hn Quu =0 =1 b 13

14 c s BFS =0 =1 b m Foun Not Hn Quu b =1 14

15 c s BFS =0 =1 b m Foun Not Hn Quu =2 b c =1 =2 15

16 c s BFS =0 =1 b m Foun Not Hn Quu =2 b c m =1 =2 16

17 c s BFS =0 =1 b m Foun Not Hn Quu =2 b c m =1 =2 17

18 c s BFS =0 =1 b m Foun Not Hn Quu =2 c m =1 =2 18

19 c s BFS =0 =1 b m Foun Not Hn Quu =2 c m =2 19

20 c s BFS =0 =1 b 20 m =3 Foun Not Hn Quu =2 m =2 =3

21 c s BFS =0 =1 b 21 m =3 Foun Not Hn Quu =2 m =2 =3

22 c s BFS =0 =1 b 22 m =3 Foun Not Hn Quu =2 =2 =3

23 c s BFS =0 =1 b 23 m =3 Foun Not Hn Quu =2 =2 =3

24 c s BFS =0 =1 b 24 m =3 Foun Not Hn Quu =2 =2 =3

25 c s BFS =0 =1 b 25 m =3 Foun Not Hn Quu =2 =3

26 c s =4 BFS =0 =1 b 26 m =3 Foun Not Hn Quu =2 =3 =4

27 c s =4 BFS =0 =1 b 27 m =3 Foun Not Hn Quu =2 =3 =4

28 c s =4 BFS =0 =1 b 28 m =3 Foun Not Hn Quu =2 =3 =4

29 c s =4 BFS =0 =1 b 29 m =3 Foun Not Hn Quu =2 =4

30 c s =4 BFS =0 =1 b 30 m =3 Foun Not Hn Quu =2 =4 =5

31 Brt-Frst Src Aortm Q s FIFO quu. BLACK Ec vrtx ssn nt vu t most onc. RED Q contns vrtcs wt vus {,,, +1,, +1} vus ssn r monotoncy ncrsn ovr tm. BLACK RED GRAY 31

32 Brt-Frst-Src s Gry Vrtcs r n: n orr o tr scovry (FIFO quu) Smst vus rst 32

33 Corrctnss Bsc Stps: s u v T sortst pt to u s nt & tr s n rom u to v Tr s pt to v wt nt

34 Corrctnss Vrtcs r scovr n orr o tr stnc rom t sourc vrtx s. Wn w scovr v, ow o w now tr s not sortr pt to v? Bcus tr ws, w wou ry v scovr t! s u v 34

35 Corrctnss Input: Grp G = ( V, E ) (rct or unrct) n sourc vrtx s V. Output: v [] = stnc rom sto v, v V. π[ v] = u suc tt ( u, v) s st on sortst pt rom s to v. Two-stp proo: On xt: 1. v [ ] δ ( sv, ) v V 2. v [ ] > δ ( sv, ) v V 35

36 Cm 1. s nvr too sm: [ v] δ ( s, v) v V Proo: Tr xsts pt rom s to v o nt [ v]. By Inucton: Suppos t s tru or vrtcs tus r scovr ( r n ry). v s scovr rom som cnt vrtx u bn n. v [ ] = u [ ] + 1 δ (, s u) + 1 δ (, sv) s u v snc c vrtx v s ssn vu xcty onc, t oows tt o n xt, v [ ] δ ( s, v) v V. 36

37 Cm 1. s nvr too sm: [ v] δ ( s, v) v V Proo: Tr xsts pt rom s to v o nt [ v]. BLACK RED s u v <LI>: v [ ] δ ( sv, ) 'scovr' ( r or r y) v V BLACK RED δ ( su, ) + 1 δ ( sv, ) GRAY 37

38 Cm 2. s nvr too b: [ v] δ ( s, v) v V Proo by contrcton: Suppos on or mor vrtcs rcv vu rtr tn δ. Lt v b t vrtx wt mnmum δ ( s, v) tt rcvs suc vu. Suppos tt v s scovr n ssn ts vu wn vrtx x s quu. Lt u b v's prcssor on sortst pt rom s to v. Tn δ (, s v) < [] v δ (, s v) 1 < [] v 1 u [ ] < x [ ] s x [ ] = v [ ] 1 [ u] = δ ( s, v ) 1 x u v Rc: vrtcs r quu n ncrsn orr o v u. u ws quu bor x. [ v] = [ u] + 1 =δ ( s, v) Contrcton! 38

39 Corrctnss Cm 1. s nvr too sm: [ v] δ ( s, v) v V Cm 2. s nvr too b: [ v] δ ( s, v) v V s ust rt: [ v] = δ ( s, v) v V 39

40 Prorss? On vry trton on vrtx s procss (turns ry). BLACK RED BLACK RED GRAY 40

41 Runnn Tm Ec vrtx s nquu t most onc OV ( ) Ec ntry n t cncy sts s scnn t most onc O(E) Tus run tm s OV ( + E). BLACK RED BLACK RED GRAY 41

42 Optm Substructur Proprty T sortst pt probm s t optm substructur proprty: Evry subpt o sortst pt s sortst pt. sortst pt s u v sortst pt sortst pt T optm substructur proprty s mr o bot ry n ynmc prormmn ortms. ows us to comput bot sortst pt stnc n t sortst pts tmsvs by storn ony on vu n on prcssor vu pr vrtx. 42

43 Rcovrn t Sortst Pt For c no v, stor prcssor o v n π(v). s u π(v) v Prcssor o v s π(v) = u. 43

44 Rcovrn t Sortst Pt PRINT-PATH( G, s, v ) Prconton: s n v r vrtcs o rp G Postconton: t vrtcs on t sortst pt rom s to v v bn prnt n orr v = s tn prnt s s π[] v = NI L tn prnt " no pt rom" s "to" v "xsts" s PRINT-PATH( G, s, π[ v]) prnt v 44

45 Coours r ctuy not rqur 45

46 Dpt Frst Src (DFS) I: Contnu srcn pr nto t rp, unt w t stuc. I t s vn v v bn xpor w bctrc to t vrtx rom wc v ws scovr. Dos not rcovr sortst pts, but cn b usu or xtrctn otr proprts o rp,.., Topooc sorts Dtcton o cycs Extrcton o strony connct componnts 46

47 Dpt-Frst Src Inp u t: Grp G = ( V, E ) (rct or unrct) Output: 2 tmstmps on c vrtx: v [] = scovry tm. v [] = nsn tm. Expor vry, strtn rom rnt vrtcs ncssry. As soon s vrtx scovr, xpor rom t. Kp trc o prorss by coourn vrtcs: Bc: unscovr vrtcs 1 v [ ] < v [ ] 2 V R: scovr, but not ns (st xporn rom t) Gry: ns (oun vrytn rcb rom t). 47

48 DFS s Not: Stc s Lst-In Frst-Out (LIFO) b Foun Not Hn Stc <no,# s> c 48 m

49 s 1 DFS b Foun Not Hn Stc <no,# s> c 49 m s,0

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

51 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c 51 m c,0,1 s,1

52 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c 4 52 m,0 c,1,1 s,1

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

54 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c Pt on Stc 4 Tr E m,1 c,1,1 s,1

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

56 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m,0 c,2,1 s,1

57 Cross E to n no: []<[] 3 c s 1 DFS 57 b m Foun Not Hn Stc <no,# s>,1 c,2,1 s,1

58 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m,2 c,2,1 s,1

59 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m,0,3 c,2,1 s,1

60 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m,1,3 c,2,1 s,1

61 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m,3 c,2,1 s,1

62 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m,0,4 c,2,1 s,1

63 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m 12,0,1,4 c,2,1 s,1

64 Bc E to no on Stc: 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m 12,1,1,4 c,2,1 s,1

65 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m m,0,2,1,4 c,2,1 s,1

66 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m m,1,2,1,4 c,2,1 s,1

67 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m ,2,1,4 c,2,1 s,1

68 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m ,1,4 c,2,1 s,1

69 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m ,4 c,2,1 s,1

70 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m ,0,5 c,2,1 s,1

71 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m ,1,5 c,2,1 s,1

72 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m ,5 c,2,1 s,1

73 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m c,2,1 s,1

74 Forwr E 2 s 1 DFS b Foun Not Hn Stc <no,# s> 3 c m c,3,1 s,1

75 2 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,1 s,1

76 2 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,2 s,1

77 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m s,1

78 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,0 s,2

79 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,1 s,2

80 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,2 s,2

81 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,0,3 s,2

82 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,1,3 s,2

83 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m ,3 s,2

84 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m s,2

85 220 s 1 DFS b Foun Not Hn Stc <no,# s> 319 c m s,3

86 220 s 1 DFS b 25 Foun Not Hn Stc <no,# s> 319 c m b,0 s,4

87 220 s 1 DFS b 25 Foun Not Hn Stc <no,# s> 319 c m b,1 s,4

88 220 s 1 DFS b 25 Foun Not Hn Stc <no,# s> 319 c m b,2 s,4

89 220 s 1 DFS b 25 Foun Not Hn Stc <no,# s> 319 c m b,3 s,4

90 220 s 1 DFS b 2526 Foun Not Hn Stc <no,# s> 319 c m s,4

91 319 c 47 Tr Es Bc Es Forwr Es Cross Es s DFS Fns! 910 b 2526 m Foun Not Hn Stc <no,# s>

92 Csscton o Es n DFS 1. Tr s r s n t pt-rst orst G π. E (u, v) s tr v ws rst scovr by xporn (u, v). 2. Bc s r tos s (u, v) connctn vrtx u to n ncstor v n pt-rst tr. 3. Forwr s r non-tr s (u, v) connctn vrtx u to scnnt v n pt-rst tr. 4. Cross s r otr s. Ty cn o btwn vrtcs n t sm pt-rst tr, s on s on vrtx s not n ncstor o t otr. 319 c s b 2526 m

93 Csscton o Es n DFS 1. Tr s: E (u, v) s tr v ws bc wn (u, v) trvrs. 2. Bc s: (u, v) s bc 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 cross v ws ry wn (u, v) trvrs n [v] < [u]. Cssyn s cn p to nty proprts o t rp,.., rp s cycc DFS ys no bc s. 319 c s b 2526 m

94 Unrct Grps In pt-rst src o n unrct rp, vry s tr tr or bc. Wy? 94

95 Unrct Grps Suppos tt (u,v) s orwr or cross n DFS o n unrct rp. (u,v) s orwr or cross wn v s ry n (ry) wn ccss rom u. Ts mns tt vrtcs rcb rom v v bn xpor. Snc w r currnty nn u, u must b r. Cry v s rcb rom u. Snc t rp s unrct, u must so b rcb rom v. Tus u must ry v bn n: u must b ry. Contrcton! u v 95

96 En o Lctur 11 Apr 8, 2009

97 DFS(G) Dpt-Frst Src Aortm BLACK BLACK DFS-Vst ( u ) Prconton: vrtx u s unscovr Postconton: vrtcs rcb rom u v bn procss RED BLACK BLACK GRAY GRAY 97

98 DFS(G) Dpt-Frst Src Aortm BLACK tot wor = θ( V ) BLACK Tus runnn tm = θ ( V + E ) DFS-Vst ( u ) Prconton: vrtx u s unscovr Postconton: vrtcs rcb rom u v bn procss RED BLACK BLACK tot wor = A[ v ] = θ( E ) v V GRAY GRAY 98

99 Topooc Sortn (.., puttn tss n nr orr) An ppcton o Dpt-Frst Src

100 Lnr Orr unrwr socs pnts sos unrwr pnts socs sos 100 socs unrwr pnts sos

101 Lnr Orr unrwr socs pnts sos Too mny vo ms? 101

102 Lnr Orr b c Prconton: A Drct Acycc Grp (DAG) Post Conton: Fn on v nr orr Aortm: Fn trmn no (sn). Put t st n squnc. Dt rom rp & rpt Θ(V) Θ(V 2 ) W cn o bttr!

103 b c Lnr Orr A: DFS Foun Not Hn Stc 103..

104 b c Lnr Orr A: DFS 104 Foun Not Hn Stc Wn no s popp o stc, nsrt t ront o nry-orr to o st. Lnr Orr:..

105 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: 105,

106 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: 106,,

107 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: 107,,,

108 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: 108,,,,

109 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: 109,,,,

110 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: 110,,,,,

111 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr:,,,,,, 111

112 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr:,,,,,,, 112

113 b c Lnr Orr A: DFS Foun Not Hn Stc c b Lnr Orr:,,,,,,, 113

114 b c Lnr Orr A: DFS Foun Not Hn Stc b Lnr Orr: c,,,,,,,, 114

115 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: b,c,,,,,,,, 115

116 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr: b,c,,,,,,,, 116

117 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr:,b,c,,,,,,,, 117

118 b c Lnr Orr A: DFS Foun Not Hn Stc Lnr Orr:,,b,c,,,,,,,, Don! 118

119 u v Lnr Orr Proo: Consr c Cs 1: u os on stc rst bor v. Bcus o, v os on bor u coms o v coms o bor u coms o v os tr u n orr. Foun Not Hn Stc u v 119 u v

120 u v Lnr Orr Proo: Consr c Cs 1: u os on stc rst bor v. Cs 2: v os on stc rst bor u. v coms o bor u os on. v os tr u n orr. Foun Not Hn Stc u v 120 u v

121 Lnr Orr Proo: Consr c Cs 1: u os on stc rst bor v. Cs 2: v os on stc rst bor u. v coms o bor u os on. Cs 3: v os on stc rst bor u. u os on bor v coms o. Pnc: u os tr v n orr. Cyc mns nr orr s mpossb Foun Not Hn Stc u v T nos n t stc orm pt strtn t s. u v v u 121

122 b c Lnr Orr A: DFS Anyss: Θ(V+E) Foun Not Hn Stc Lnr Orr:,,b,c,,,,,,,, Don! 122

123 Sortst Pts Rvst

124 Bc to Sortst Pt BFS ns t sortst pts rom sourc no s to vry vrtx v n t rp. Hr, t nt o pt s smpy t numbr o s on t pt. But wt s v rnt costs? s δ (, sv) = 3 δ (, sv) = 12 v s v 124

125 Sn-Sourc (Wt) Sortst Pts

126 T Probm Wt s t sortst rvn rout rom Toronto to Ottw? (.. MAPQust, Goo Mps) Input: Drct Grp G = ( V, E) E wts w : E Wt o pt p =< v, v,..., v > = w( v, v) = 1 Sortst-pt wt rom u to v : p δ ( uv, ) = mn{ w( p) : u v} pt u v, otrws. Sortst pt rom u to v s ny pt p suc tt w( p) = δ ( u, v). 126

127 Exmp Sn-sourc sortst pt src nucs src tr root t s. Ts tr, n nc t pts tmsvs, r not ncssry unqu. 127

128 Sortst pt vrnts Sn-sourc sortst-pts probm: t sortst pt rom s to c vrtx v. (.. BFS) Sn-stnton sortst-pts probm: Fn sortst pt to vn stnton vrtx t rom c vrtx v. Sn-pr sortst-pt probm: Fn sortst pt rom u to v or vn vrtcs u n v. A-prs sortst-pts probm: Fn sortst pt rom u to v or vry pr o vrtcs u n v. 128

129 Ntv-wt s OK, s on s no ntv-wt cycs r rcb rom t sourc. I w v ntv-wt cyc, w cn ust p on roun t, n t w(s, v) = or v on t cyc. But OK t ntv-wt cyc s not rcb rom t sourc. Som ortms wor ony tr r no ntv-wt s n t rp. 129

130 Optm substructur Lmm: Any subpt o sortst pt s sortst pt Proo: Cut n pst. Suppos ts pt p s sortst pt rom u to v. Tn δ ( u, v) = w( p) = w( p ) + w( p ) + w( p ). Now suppos tr xsts sortr pt x y. Tn w( p ) < w( p ). xy Construct p : xy ux xy yv Tn w( p ) = w( p ) + w( p ) + w( p ) < w( p ) + w( p ) + w( p ) = w( p). ux xy yv So p wsn't sortst pt tr! p xy ux xy yv 130

131 Cycs Sortst pts cn t contn cycs: Ary ru out ntv-wt cycs. Postv-wt: w cn t sortr pt by omttn t cyc. Zro-wt: no rson to us tm ssum tt our soutons won t us tm. 131

132 Output o sn-sourc sortst-pt ortm For c vrtx v n V: [v] = δ(s, v). Inty, [v]=. Ruc s ortm prorsss. But wys mntn [v] δ(s, v). C [v] sortst-pt stmt. π[v] = prcssor o v on sortst pt rom s. I no prcssor, π[v] = NIL. π nucs tr sortst-pt tr. 132

133 Intzton A sortst-pts ortms strt wt t sm ntzton: INIT-SINGLE-SOURCE(V, s) or c v n V o [v] π[v] NIL [s] 0 133

134 Rxn n Cn w mprov sortst-pt stmt or v by on trou u n tn (u,v)? RELAX(u, v,w) [v] > [u] + w(u, v) tn [v] [u] + w(u, v) π[v] u 134

135 Gnr sn-sourc sortst-pt strty 1. Strt by cn INIT-SINGLE-SOURCE 2. Rx Es Aortms r n t orr n wc s r tn n ow mny tms c s rx. 135

136 Exmp: Sn-sourc sortst pts n rct cycc rp (DAG) Snc rp s DAG, w r urnt no ntv-wt cycs. 136

137 Aortm Tm: Θ ( V + E) 137

138 Exmp 138

139 Exmp 139

140 Exmp 140

141 Exmp 141

142 Exmp 142

143 Exmp 143

144 Corrctnss: Pt rxton proprty (Lmm 24.15) Lt p =< v, v,..., v > b sortst pt rom s = v to v I w rx, n orr, ( v, v ), ( v, v ),..., ( v, v ), vn ntrmx wt otr rxtons, tn v [ ] = δ ( s, v). 144

145 Corrctnss o DAG Sortst Pt Aortm Bcus w procss vrtcs n topoocy sort orr, s o ny pt r rx n orr o pprnc n t pt. Es on ny sortst pt r rx n orr. By pt-rxton proprty, corrct. 145

146 Exmp: Dstr s ortm Apps to nr wt rct rp (my contn cycs). But wts must b non-ntv. Essnty wt vrson o BFS. Inst o FIFO quu, uss prorty quu. Kys r sortst-pt wts ([v]). Mntn 2 sts o vrtcs: S = vrtcs wos n sortst-pt wts r trmn. Q = prorty quu = V-S. 146

147 Dstr s ortm Dstr s ortm cn b vw s ry, snc t wys cooss t tst vrtx n V S to to S. 147

148 Dstr s ortm: Anyss Anyss: Usn mnp, quu oprtons ts O(oV) tm OV ( ) O(o V ) O( V) trtons Runnn Tm s OE ( o V) O(o V ) O( E) trtons 148

149 Exmp Ky: Wt Not Foun Gry Hnn Bc Hn 149

150 Exmp 150

151 Exmp 151

152 Exmp 152

153 Exmp 153

154 Exmp 154

155 Corrctnss o Dstr s ortm Loop nvrnt: [v] = δ(s, v) or v n S. Intzton: Inty, S s mpty, so trvy tru. Trmnton: At n, Q s mpty S = V [v] = δ(s, v) or v n V. Mntnnc: N to sow tt [u] = δ(s, u) wn u s to S n c trton. [u] os not cn onc u s to S. 155

156 Corrctnss o Dstr s Aortm: Uppr Boun Proprty Uppr Boun Proprty: 1. v [ ] δ ( sv, ) v V 2. Onc v [ ] = δ ( sv, ), t osn't cn Proo: By nucton. Bs Cs : v [ ] δ ( sv, ) v V mmty tr ntzton, snc s [ ] = 0 = δ ( ss, ) v [ ] = v s Inuctv Stp: Suppos x [ ] δ ( sx, ) x V Suppos w rx ( uv, ). I v [ ] cns, tn v [ ] = u [ ] + wuv (, ) δ (, su) + wuv (, ) δ (, sv) 156

157 Corrctnss o Dstr s Aortm C m : Wn u s to S, [ u] = δ ( s, u) Proo by Contrcton: Lt u b t rst vrtx to S suc tt u [ ] δ ( su, ) wn us. Lt y b rst vrtx n V S on sortst pt to u Lt x b t prcssor o y on t sortst pt to u C m: y [ ] = δ ( sy, ) wn us to S. Proo: x [ ] = δ ( sx, ), snc x S. ( x, y) ws rx wn x ws to S y [ ] = δ (, sx) + wxy (, ) = δ (, sy) Hn 157

158 Corrctnss o Dstr s Aortm Tus y [ ] = δ ( sy, ) wn us to S. y [ ] = δ ( sy, ) δ ( su, ) u [ ] (uppr boun proprty) But u [ ] y [ ] wn u to S Tus y [ ] = δ ( sy, ) = δ ( su, ) = u [ ]! Tus wn u s to S, [ u] = δ ( s, u) Consquncs: Tr s sortst pt to u suc tt t prcssor o u π[ u] S wn u s to S. T pt trou y cn ony b sortst pt w[ p ] = 0. 2 Hn π[ u] 158

159 Corrctnss o Dstr s ortm Loop nvrnt: [v] = δ(s, v) or v n S. Mntnnc: N to sow tt [u] = δ(s, u) wn u s to S n c trton. [u] os not cn onc u s to S. Rx(u,v,w) cn ony crs v [ ]. By t uppr boun proprty, v [ ] δ ( sv, ). Tus onc v [ ] = δ ( sv, ), t w not b cn.? 159