Grp Sr Aortms 1
Grp 2 No ~ ty or omputr E ~ ro or t Unrt or Drt A surprsny r numr o omputton proms n xprss s rp proms.
3 Drt n Unrt Grps () A rt 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. () An unrt 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. () T surp o t rp n prt () nu y t vrtx st {1,2,3,6}.
4 Trs Tr Forst Grp wt Cy A tr s onnt, y, unrt rp. A orst s st o trs (not nssry onnt)
5 Runnn Tm o Grp Aortms Runnn tm otn unton o ot V n E. For onvnn, rop t. n symptot notton,.. O(V+E).
6 Rprsnttons: Unrt Grps Sp ompxty: Tm to n nours o vrtx u : Tm to trmn ( uv, ) E: Any Lst θ ( V + E) θ(r( u)) θ(r( u)) Any Mtrx 2 θ( V ) θ ( V ) θ (1)
7 Rprsnttons: Drt Grps Sp ompxty: Tm to n nours o vrtx u : Tm to trmn ( uv, ) E: Any Lst θ ( V + E) θ(r( u)) θ(r( u)) Any Mtrx 2 θ( V ) θ ( V ) θ (1)
8 Brt-Frst Sr Go: To rovr t sortst pts rom sour no s to otr r nos v n rp. T nt o pt n t pts tmsvs r rturn. Nots: Tr r n xponnt numr o poss pts Ts prom s rr or nr rps tn trs us o ys!? s
9 Brt-Frst Sr Input: Grp G = ( V, E) (rt or unrt) n sour vrtx s V. Output: v [ ] = sortst pt stn δ( sv, ) rom sto v, v V. π[ v] = u su tt ( u, v) s st on sortst pt rom s to v. I: sn out sr wv rom s. Kp tr o prorss y oourn vrts: Unsovr vrts r oour Just sovr vrts (on t wvront) r oour r. Prvousy sovr vrts (n wvront) r oour ry.
s BFS Frst-In Frst-Out (FIFO) quu stors ust sovr vrts Not Hn Quu m 10
s BFS =0 m Not Hn Quu s =0 11
s BFS =0 =1 m Not Hn Quu =0 =1 12
s BFS =0 =1 m Not Hn Quu =1 13
s BFS =0 =1 m Not Hn Quu =2 =1 =2 14
s BFS =0 =1 m Not Hn Quu =2 m =1 =2 15
s BFS =0 =1 m Not Hn Quu =2 m =1 =2 16
s BFS =0 =1 m Not Hn Quu =2 m =1 =2 17
s BFS =0 =1 m Not Hn Quu =2 m =2 18
s BFS =0 =1 19 m =3 Not Hn Quu =2 m =2 =3
s BFS =0 =1 20 m =3 Not Hn Quu =2 m =2 =3
s BFS =0 =1 21 m =3 Not Hn Quu =2 =2 =3
s BFS =0 =1 22 m =3 Not Hn Quu =2 =2 =3
s BFS =0 =1 23 m =3 Not Hn Quu =2 =2 =3
s BFS =0 =1 24 m =3 Not Hn Quu =2 =3
s =4 BFS =0 =1 25 m =3 Not Hn Quu =2 =3 =4
s =4 BFS =0 =1 26 m =3 Not Hn Quu =2 =3 =4
s =4 BFS =0 =1 27 m =3 Not Hn Quu =2 =3 =4
s =4 BFS =0 =1 28 m =3 Not Hn Quu =2 =4
s =4 BFS =0 =1 29 m =3 Not Hn Quu =2 =4 =5
30 Brt-Frst Sr Aortm Q s FIFO quu. BLACK E vrtx ssn nt vu t most on. RED Q ontns vrts wt vus {,,, +1,, +1} vus ssn r monotony nrsn ovr tm. BLACK RED GRAY
31 Brt-Frst-Sr s Gry Vrts r n: n orr o tr sovry (FIFO quu) Smst vus rst
Corrtnss 32 Bs 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 +1.
33 Corrtnss Vrts r sovr n orr o tr stn rom t sour vrtx s. Wn w sovr v, ow o w now tr s not sortr pt to v? Bus tr ws, w wou ry v sovr t! s u v
Corrtnss 34 Input: Grp G = ( V, E) (rt or unrt) n sour vrtx s V. Output: v [ ] = stn rom sto v, v V. π[ v] = u su 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 Cm 1. s nvr too sm: [ v] δ( s, v) v V Proo: Tr xsts pt rom s to v o nt [ v]. By Inuton: Suppos t s tru or vrts tus r sovr ( r n ry). v s sovr rom som nt vrtx u n n. v [ ] = u [ ] + 1 δ(, s u) + 1 δ(, sv) s u v sn vrtx v s ssn vu xty on, t oows tt o n xt, v [ ] δ ( s, v) v V.
36 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, ) 'sovr' ( r or r y) v V BLACK RED δ(, su) + 1 δ(, sv) GRAY
37 Cm 2. s nvr too : [ v] δ( s, v) v V Proo y ontrton: Suppos on or mor vrts rv vu rtr tn δ. Lt v t vrtx wt mnmum δ( s, v) tt rvs su vu. Suppos tt v s sovr n ssn ts vu wn vrtx x s quu. Lt u v's prssor 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 R : vrts r quu n nrsn orr o v u. u ws quu or x. [ v] = [ u] + 1 =δ( s, v) Contrton!
38 Corrtnss Cm 1. s nvr too sm: [ v] δ( s, v) v V Cm 2. s nvr too : [ v] δ( s, v) v V s ust rt: [ v] = δ( s, v) v V
Prorss? 39 On vry trton on vrtx s pross (turns ry). BLACK RED BLACK RED GRAY
40 Runnn Tm E vrtx s nquu t most on O( V) E ntry n t ny sts s snn t most on Tus run tm s OV ( + E). O(E) BLACK RED BLACK RED GRAY
41 Optm Sustrutur Proprty T sortst pt prom s t optm sustrutur proprty: Evry supt o sortst pt s sortst pt. sortst pt s u v sortst pt sortst pt T optm sustrutur proprty s mr o ot ry n ynm prormmn ortms. ows us to omput ot sortst pt stn n t sortst pts tmsvs y storn ony on vu n on prssor vu pr vrtx.
Rovrn t Sortst Pt 42 For no v, stor prssor o v n π(v). s u π(v) v Prssor o v s π(v) = u.
43 Rovrn t Sortst Pt PRINT-PATH( G, s, v ) Pronton: s n v r vrts o rp G Postonton: t vrts on t sortst pt rom s to v v n 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
Coours r tuy not rqur 44
45 Dpt Frst Sr (DFS) I: Contnu srn pr nto t rp, unt w t stu. I t s vn v v n xpor w tr to t vrtx rom w v ws sovr. Dos not rovr sortst pts, ut n usu or xtrtn otr proprts o rp,.., Topoo sorts Dtton o ys Extrton o strony onnt omponnts
46 Dpt-Frst Sr Inp u t: Grp G = ( V, E) (rt or unrt ) Output: 2 tmstmps on vrtx: v [ ] = sovry tm. v [ ] = nsn tm. Expor vry, strtn rom rnt vrts nssry. As soon s vrtx sovr, xpor rom t. Kp tr o prorss y oourn vrts: B: unsovr vrts 1 v [ ] < v [ ] 2 V R: sovr, ut not ns (st xporn rom t) Gry: ns (oun vrytn r rom t).
DFS s Not: St s Lst-In Frst-Out (LIFO) Not Hn St <no,# s> 47 m
s 1 DFS Not Hn St <no,# s> 48 m s,0
2 s 1 DFS Not Hn St <no,# s> 49 m,0 s,1
2 s 1 DFS Not Hn St <no,# s> 3 50 m,0,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 4 51 m,0,1,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 4 5 52 m,0,1,1,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 Pt on St 4 Tr E 56 53 m,1,1,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 54 m,1,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 55 m,0,2,1 s,1
Cross E to n no: []<[] 3 47 2 8 56 s DFS 1 56 m Not Hn St <no,# s>,1,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 57 m,2,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 58 9 m,0,3,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 59 9 m,1,3,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 60 910 m,3,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 11 61 910 m,0,4,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 11 62 910 m 12,0,1,4,2,1 s,1
B E to no on St: 2 s 1 DFS Not Hn St <no,# s> 3 47 8 56 11 63 910 m 12,1,1,4,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 11 64 910 m 13 12 m,0,2,1,4,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 11 65 910 m 13 12 m,1,2,1,4,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 11 66 910 m 12 1314,2,1,4,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 11 67 910 m 1215 1314,1,4,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 56 8 1116 68 910 m 1215 1314,4,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 17 56 8 1116 69 910 m 1215 1314,0,5,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 17 56 8 1116 70 910 m 1215 1314,1,5,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 1718 56 8 1116 71 910 m 1215 1314,5,2,1 s,1
2 s 1 DFS Not Hn St <no,# s> 3 47 1718 56 819 1116 72 910 m 1215 1314,2,1 s,1
Forwr E 2 s 1 DFS Not Hn St <no,# s> 3 47 1718 56 819 1116 73 910 m 1215 1314,3,1 s,1
2 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 1116 74 910 m 1215 1314,1 s,1
2 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 1116 75 910 m 1215 1314,2 s,1
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 1116 76 910 m 1215 1314 s,1
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 21 1116 77 910 m 1215 1314,0 s,2
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 21 1116 78 910 m 1215 1314,1 s,2
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 21 1116 79 910 m 1215 1314,2 s,2
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 21 1116 80 910 m 22 1215 1314,0,3 s,2
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 21 1116 81 910 m 22 1215 1314,1,3 s,2
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 21 1116 82 910 m 2223 1215 1314,3 s,2
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 2124 1116 83 910 m 2223 1215 1314 s,2
220 s 1 DFS Not Hn St <no,# s> 319 47 1718 56 819 2124 1116 84 910 m 2223 1215 1314 s,3
220 s 1 DFS 25 Not Hn St <no,# s> 319 47 1718 56 819 2124 1116 85 910 m 2223 1215 1314,0 s,4
220 s 1 DFS 25 Not Hn St <no,# s> 319 47 1718 56 819 2124 1116 86 910 m 2223 1215 1314,1 s,4
220 s 1 DFS 25 Not Hn St <no,# s> 319 47 1718 56 819 2124 1116 87 910 m 2223 1215 1314,2 s,4
220 s 1 DFS 25 Not Hn St <no,# s> 319 47 1718 56 819 2124 1116 88 910 m 2223 1215 1314,3 s,4
220 s 1 DFS 2526 Not Hn St <no,# s> 319 47 1718 56 819 2124 1116 89 910 m 2223 1215 1314 s,4
319 47 Tr Es B Es Forwr Es Cross Es 220 1718 56 819 s 127 2124 DFS 1116 90 Fns! 910 2526 m Not Hn St <no,# s> 2223 1215 1314
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 y 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 twn vrts n t sm pt-rst tr, s on s on vrtx s not n nstor o t otr. 319 47 220 1718 56 819 s 127 2124 1116 2526 m 2223 1215 1314 91 910
Csston o Es n DFS 1. Tr s: E (u, v) s tr v ws wn (u, v) trvrs. 2. B s: (u, v) s 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 s. 319 47 220 1718 56 819 s 127 2124 1116 2526 m 2223 1215 1314 92 910
93 Unrt Grps In pt-rst sr o n unrt rp, vry s tr tr or. Wy?
94 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 r rom v v n xpor. Sn w r urrnty nn u, u must r. Cry v s r rom u. Sn t rp s unrt, u must so r rom v. Tus u must ry v n n: u must ry. Contrton! u v
95 DFS(G) Dpt-Frst Sr Aortm BLACK BLACK DFS-Vst ( u ) Pronton: vrtx u s unsovr Postonton: vrts r rom u v n pross RED BLACK BLACK GRAY GRAY
96 DFS(G) Dpt-Frst Sr Aortm BLACK DFS-Vst ( u ) BLACK Pronton: vrtx u s unsovr tot wor = θ( V ) Tus runnn tm = θ ( V + E) Postonton: vrts r rom u v n pross RED BLACK BLACK tot wor = A [ v] = θ( E ) vv GRAY GRAY
Topoo Sortn (.., puttn tss n nr orr) An ppton o Dpt-Frst Sr
98 Lnr Orr unrwr sos pnts sos unrwr pnts sos sos sos unrwr pnts sos
99 Lnr Orr unrwr sos pnts sos Too mny vo ms?
Lnr Orr Pronton: 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 Θ(V) Θ(V 2 ) 100.. W n o ttr!
Lnr Orr A: DFS Not Hn St 101..
Lnr Orr A: DFS 102 Not Hn St Wn no s popp o st, nsrt t ront o nry-orr to o st. Lnr Orr:..
Lnr Orr A: DFS Not Hn St Lnr Orr: 103,
Lnr Orr A: DFS Not Hn St Lnr Orr: 104,,
Lnr Orr A: DFS Not Hn St Lnr Orr: 105,,,
Lnr Orr A: DFS Not Hn St Lnr Orr: 106,,,,
Lnr Orr A: DFS Not Hn St Lnr Orr: 107,,,,
Lnr Orr A: DFS Not Hn St Lnr Orr: 108,,,,,
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,, 109
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,, 110
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,, 111
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,, 112
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,, 113
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,, 114
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,,, 115
Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,,,, Don! 116
Lnr Orr Proo: Consr Cs 1: u os on st rst or v. Bus o, v os on or u oms o v oms o or u oms o v os tr u n orr. Not Hn St v u u v 117 u v
Lnr Orr Proo: Consr Cs 1: u os on st rst or v. Cs 2: v os on st rst or u. v oms o or u os on. v os tr u n orr. Not Hn St v u u v 118 u v
Lnr Orr Proo: Consr Cs 1: u os on st rst or v. Cs 2: v os on st rst or u. v oms o or u os on. Cs 3: v os on st rst or u. u os on or v oms o. Pn: u os tr v n orr. Cy mns nr orr s mposs 119 Not Hn St u v T nos n t st orm pt strtn t s. u v v u
Lnr Orr A: DFS Anyss: Θ(V+E) Not Hn St Lnr Orr:,,,,,,,,,,, Don! 120
Sortst Pts Rvst
122 B to Sortst Pt BFS ns t sortst pts rom sour no s to vry vrtx v n t rp. Hr, t nt o pt s smpy t numr o s on t pt. But wt s v rnt osts? s δ (, sv) = 3 δ (, sv) = 12 v s 1 2 3 7 2 5 v
Sn-Sour (Wt) Sortst Pts
124 T Prom Wt s t sortst rvn rout rom Toronto to Ottw? (.. MAPQust, Goo Mps) Input: Drt Grp G = ( V, E) E wts w: E Wt o pt p =< v, v,..., v > = w( v, v) 0 1 1 = 1 Sortst-pt wt rom u to v : p δ ( uv, ) = mn{ w( p) : u L v} pt u L v, otrws. Sortst pt rom u to v s ny pt p su tt w( p) = δ( u, v).
125 Exmp Sn-sour sortst pt sr nus sr tr root t s. Ts tr, n n t pts tmsvs, r not nssry unqu.
126 Sortst pt vrnts Sn-sour sortst-pts prom: t sortst pt rom s to vrtx v. (.. BFS) Sn-stnton sortst-pts prom: Fn sortst pt to vn stnton vrtx t rom vrtx v. Sn-pr sortst-pt prom: Fn sortst pt rom u to v or vn vrts u n v. A-prs sortst-pts prom: Fn sortst pt rom u to v or vry pr o vrts u n v.
127 Ntv-wt s OK, s on s no ntv-wt ys r r rom t sour. I w v ntv-wt y, w n ust p on roun t, n t w(s, v) = or v on t y. But OK t ntv-wt y s not r rom t sour. Som ortms wor ony tr r no ntv-wt s n t rp.
128 Optm sustrutur Lmm: Any supt o sortst pt s sortst pt Proo: Cut n pst. Suppos ts pt p s sortst pt rom u to v. Tn δ ( uv, ) = w( p) = w( p ) + w( p ) + w( p ). Now suppos tr xsts sortr pt x L y. Tn w( pʹ ) < w( p ). xy Construt p ʹ : xy ux xy yv Tn w( pʹ ) = w( p ) + w( pʹ ) + w( p ) < w( p ) + w( p ) + w( p ) = w( p). ux xy yv So p wsn't sortst pt tr! ʹ p xy ux xy yv
129 Cys Sortst pts n t ontn ys: Ary ru out ntv-wt ys. Postv-wt: w n t sortr pt y omttn t y. Zro-wt: no rson to us tm ssum tt our soutons won t us tm.
130 Output o sn-sour sortst-pt ortm For vrtx v n V: [v] = δ(s, v). Inty, [v]=. Ru s ortm prorsss. But wys mntn [v] δ(s, v). C [v] sortst-pt stmt. π[v] = prssor o v on sortst pt rom s. I no prssor, π[v] = NIL. π nus tr sortst-pt tr.
131 Intzton A sortst-pts ortms strt wt t sm ntzton: INIT-SINGLE-SOURCE(V, s) or v n V o [v] [s] 0 π[v] NIL
132 Rxn n Cn w mprov sortst-pt stmt or v y 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
133 Gnr sn-sour sortst-pt strty 1. Strt y n INIT-SINGLE-SOURCE 2. Rx Es Aortms r n t orr n w s r tn n ow mny tms s rx.
134 Exmp: Sn-sour sortst pts n rt y rp (DAG) Sn rp s DAG, w r urnt no ntv-wt ys.
135 Aortm Tm: Θ ( V + E)
Exmp 136
Exmp 137
Exmp 138
Exmp 139
Exmp 140
Exmp 141
142 Corrtnss: Pt rxton proprty (Lmm 24.15) Lt p =< v, v,..., v > sortst pt rom s = v to v. 0 1 0 I w rx, n orr, ( v, v ), ( v, v ),..., ( v, v ), 0 1 1 2-1 vn ntrmx wt otr rxtons, tn v [ ] = δ( s, v).
143 Corrtnss o DAG Sortst Pt Aortm Bus w pross vrts n topooy sort orr, s o ny pt r rx n orr o pprn n t pt. Es on ny sortst pt r rx n orr. By pt-rxton proprty, orrt.
144 Exmp: Dstr s ortm Apps to nr wt rt rp (my ontn ys). But wts must non-ntv. Essnty wt vrson o BFS. Inst o FIFO quu, uss prorty quu. Kys r sortst-pt wts ([v]). Mntn 2 sts o vrts: S = vrts wos n sortst-pt wts r trmn. Q = prorty quu = V-S.
145 Dstr s ortm Dstr s ortm n vw s ry, sn t wys ooss t tst vrtx n V S to to S.
146 Dstr s ortm: Anyss Anyss: Usn mnp, quu oprtons ts O(oV) tm OV ( ) O(o V ) OV ( ) trtons O(o V ) O( E) trtons Runnn Tm s OE ( o V)
147 Exmp Ky: Wt Gry Not Hnn B Hn
Exmp 148
Exmp 149
Exmp 150
Exmp 151
Exmp 152
153 Corrtnss 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. Mntnn: N to sow tt [u] = δ(s, u) wn u s to S n trton. [u] os not n on u s to S.
154 Corrtnss o Dstr s Aortm: Uppr Boun Proprty Uppr Boun Proprty: 1. v [ ] δ( sv, ) v V 2. On v [ ] = δ( sv, ), t osn't n Proo: By nuton. Bs Cs : v [ ] δ( sv, ) v V mmty tr ntzton, sn s [ ] = 0 = δ( ss, ) v [ ] = v s Inutv Stp: Suppos x [ ] δ( sx, ) x V Suppos w rx ( uv, ). I v [ ] ns, tn v [ ] = u [ ] + wuv (, ) δ(, su) + wuv (, ) δ(, sv)
155 Corrtnss o Dstr s Aortm C m : Wn u s to S, [ u] = δ( s, u) Proo y Contrton: Lt u t rst vrtx to S su tt u [ ] δ( su, ) wn us. Lt y rst vrtx n V S on sortst pt to u Lt x t prssor o y on t sortst pt to u C m: y [ ] = δ( sy, ) wn us to S. Proo: x [ ] = δ( sx, ), sn x S. ( xy, ) ws rx wn xws to S y [ ] = δ( sx, ) + wxy (, ) = δ( sy, ) Hn
156 Corrtnss o Dstr s Aortm Tus y [ ] = δ( sy, ) wn us to S. y [ ] = δ( sy, ) δ( su, ) u [ ] (uppr oun 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) Consquns: Tr s sortst pt to u su tt t prssor o u π[ u] S wn u s to S. T pt trou y n ony sortst pt w[ p ] = 0. 2 Hn π [ u]
157 Corrtnss o Dstr s ortm Loop nvrnt: [v] = δ(s, v) or v n S. Mntnn: N to sow tt [u] = δ(s, u) wn u s to S n trton. [u] os not n on u s to S. Rx(u,v,w) n ony rs v [ ]. By t uppr oun proprty, v [ ] δ( sv, ). Tus on v [ ] = δ( sv, ), t w not n.?