Graph Search Algorithms
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- Norma Taylor
- 5 years ago
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1 Grp Sr Aortms 1
2 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 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 4 Trs Tr Forst Grp wt Cy A tr s onnt, y, unrt rp. A orst s st o trs (not nssry onnt)
5 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 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 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 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 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.
10 s BFS Frst-In Frst-Out (FIFO) quu stors ust sovr vrts Not Hn Quu m 10
11 s BFS =0 m Not Hn Quu s =0 11
12 s BFS =0 =1 m Not Hn Quu =0 =1 12
13 s BFS =0 =1 m Not Hn Quu =1 13
14 s BFS =0 =1 m Not Hn Quu =2 =1 =2 14
15 s BFS =0 =1 m Not Hn Quu =2 m =1 =2 15
16 s BFS =0 =1 m Not Hn Quu =2 m =1 =2 16
17 s BFS =0 =1 m Not Hn Quu =2 m =1 =2 17
18 s BFS =0 =1 m Not Hn Quu =2 m =2 18
19 s BFS =0 =1 19 m =3 Not Hn Quu =2 m =2 =3
20 s BFS =0 =1 20 m =3 Not Hn Quu =2 m =2 =3
21 s BFS =0 =1 21 m =3 Not Hn Quu =2 =2 =3
22 s BFS =0 =1 22 m =3 Not Hn Quu =2 =2 =3
23 s BFS =0 =1 23 m =3 Not Hn Quu =2 =2 =3
24 s BFS =0 =1 24 m =3 Not Hn Quu =2 =3
25 s =4 BFS =0 =1 25 m =3 Not Hn Quu =2 =3 =4
26 s =4 BFS =0 =1 26 m =3 Not Hn Quu =2 =3 =4
27 s =4 BFS =0 =1 27 m =3 Not Hn Quu =2 =3 =4
28 s =4 BFS =0 =1 28 m =3 Not Hn Quu =2 =4
29 s =4 BFS =0 =1 29 m =3 Not Hn Quu =2 =4 =5
30 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 31 Brt-Frst-Sr s Gry Vrts r n: n orr o tr sovry (FIFO quu) Smst vus rst
32 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 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
34 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 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 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 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 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
39 Prorss? 39 On vry trton on vrtx s pross (turns ry). BLACK RED BLACK RED GRAY
40 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 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.
42 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 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
44 Coours r tuy not rqur 44
45 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 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).
47 DFS s Not: St s Lst-In Frst-Out (LIFO) Not Hn St <no,# s> 47 m
48 s 1 DFS Not Hn St <no,# s> 48 m s,0
49 2 s 1 DFS Not Hn St <no,# s> 49 m,0 s,1
50 2 s 1 DFS Not Hn St <no,# s> 3 50 m,0,1 s,1
51 2 s 1 DFS Not Hn St <no,# s> m,0,1,1 s,1
52 2 s 1 DFS Not Hn St <no,# s> m,0,1,1,1 s,1
53 2 s 1 DFS Not Hn St <no,# s> 3 Pt on St 4 Tr E m,1,1,1 s,1
54 2 s 1 DFS Not Hn St <no,# s> m,1,1 s,1
55 2 s 1 DFS Not Hn St <no,# s> m,0,2,1 s,1
56 Cross E to n no: []<[] s DFS 1 56 m Not Hn St <no,# s>,1,2,1 s,1
57 2 s 1 DFS Not Hn St <no,# s> m,2,2,1 s,1
58 2 s 1 DFS Not Hn St <no,# s> m,0,3,2,1 s,1
59 2 s 1 DFS Not Hn St <no,# s> m,1,3,2,1 s,1
60 2 s 1 DFS Not Hn St <no,# s> m,3,2,1 s,1
61 2 s 1 DFS Not Hn St <no,# s> m,0,4,2,1 s,1
62 2 s 1 DFS Not Hn St <no,# s> m 12,0,1,4,2,1 s,1
63 B E to no on St: 2 s 1 DFS Not Hn St <no,# s> m 12,1,1,4,2,1 s,1
64 2 s 1 DFS Not Hn St <no,# s> m m,0,2,1,4,2,1 s,1
65 2 s 1 DFS Not Hn St <no,# s> m m,1,2,1,4,2,1 s,1
66 2 s 1 DFS Not Hn St <no,# s> m ,2,1,4,2,1 s,1
67 2 s 1 DFS Not Hn St <no,# s> m ,1,4,2,1 s,1
68 2 s 1 DFS Not Hn St <no,# s> m ,4,2,1 s,1
69 2 s 1 DFS Not Hn St <no,# s> m ,0,5,2,1 s,1
70 2 s 1 DFS Not Hn St <no,# s> m ,1,5,2,1 s,1
71 2 s 1 DFS Not Hn St <no,# s> m ,5,2,1 s,1
72 2 s 1 DFS Not Hn St <no,# s> m ,2,1 s,1
73 Forwr E 2 s 1 DFS Not Hn St <no,# s> m ,3,1 s,1
74 2 s 1 DFS Not Hn St <no,# s> m ,1 s,1
75 2 s 1 DFS Not Hn St <no,# s> m ,2 s,1
76 220 s 1 DFS Not Hn St <no,# s> m s,1
77 220 s 1 DFS Not Hn St <no,# s> m ,0 s,2
78 220 s 1 DFS Not Hn St <no,# s> m ,1 s,2
79 220 s 1 DFS Not Hn St <no,# s> m ,2 s,2
80 220 s 1 DFS Not Hn St <no,# s> m ,0,3 s,2
81 220 s 1 DFS Not Hn St <no,# s> m ,1,3 s,2
82 220 s 1 DFS Not Hn St <no,# s> m ,3 s,2
83 220 s 1 DFS Not Hn St <no,# s> m s,2
84 220 s 1 DFS Not Hn St <no,# s> m s,3
85 220 s 1 DFS 25 Not Hn St <no,# s> m ,0 s,4
86 220 s 1 DFS 25 Not Hn St <no,# s> m ,1 s,4
87 220 s 1 DFS 25 Not Hn St <no,# s> m ,2 s,4
88 220 s 1 DFS 25 Not Hn St <no,# s> m ,3 s,4
89 220 s 1 DFS 2526 Not Hn St <no,# s> m s,4
90 Tr Es B Es Forwr Es Cross Es s DFS Fns! m Not Hn St <no,# s>
91 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 s m
92 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 s m
93 93 Unrt Grps In pt-rst sr o n unrt rp, vry s tr tr or. Wy?
94 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 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 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
97 Topoo Sortn (.., puttn tss n nr orr) An ppton o Dpt-Frst Sr
98 98 Lnr Orr unrwr sos pnts sos unrwr pnts sos sos sos unrwr pnts sos
99 99 Lnr Orr unrwr sos pnts sos Too mny vo ms?
100 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 ) W n o ttr!
101 Lnr Orr A: DFS Not Hn St 101..
102 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:..
103 Lnr Orr A: DFS Not Hn St Lnr Orr: 103,
104 Lnr Orr A: DFS Not Hn St Lnr Orr: 104,,
105 Lnr Orr A: DFS Not Hn St Lnr Orr: 105,,,
106 Lnr Orr A: DFS Not Hn St Lnr Orr: 106,,,,
107 Lnr Orr A: DFS Not Hn St Lnr Orr: 107,,,,
108 Lnr Orr A: DFS Not Hn St Lnr Orr: 108,,,,,
109 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,, 109
110 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,, 110
111 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,, 111
112 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,, 112
113 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,, 113
114 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,, 114
115 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,,, 115
116 Lnr Orr A: DFS Not Hn St Lnr Orr:,,,,,,,,,,, Don! 116
117 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
118 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
119 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
120 Lnr Orr A: DFS Anyss: Θ(V+E) Not Hn St Lnr Orr:,,,,,,,,,,, Don! 120
121 Sortst Pts Rvst
122 122 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 v
123 Sn-Sour (Wt) Sortst Pts
124 124 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) = 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).
125 125 Exmp Sn-sour sortst pt sr nus sr tr root t s. Ts tr, n n t pts tmsvs, r not nssry unqu.
126 126 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.
127 127 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.
128 128 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
129 129 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.
130 130 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.
131 131 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
132 132 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
133 133 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.
134 134 Exmp: Sn-sour sortst pts n rt y rp (DAG) Sn rp s DAG, w r urnt no ntv-wt ys.
135 135 Aortm Tm: Θ ( V + E)
136 Exmp 136
137 Exmp 137
138 Exmp 138
139 Exmp 139
140 Exmp 140
141 Exmp 141
142 142 Corrtnss: Pt rxton proprty (Lmm 24.15) Lt p =< v, v,..., v > 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).
143 143 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.
144 144 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.
145 145 Dstr s ortm Dstr s ortm n vw s ry, sn t wys ooss t tst vrtx n V S to to S.
146 146 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)
147 147 Exmp Ky: Wt Gry Not Hnn B Hn
148 Exmp 148
149 Exmp 149
150 Exmp 150
151 Exmp 151
152 Exmp 152
153 153 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.
154 154 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)
155 155 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
156 156 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]
157 157 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.?
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