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lq wklv duwlfoh olhv lq wkh ehkdylru ri wkh sgi ri wkh krsfrxqw k ri wkh vkruwhvw sdwk dv Q lqfuhdvhv1
Iru wkdw sxusrvh/ wkh phdq H ^k ` dqg yduldqfh ydu^k ` duh sorwwhg lq Iljxuh 4/ iru d vdph ydoxh ri
s @ 3=;/ dv d ixqfwlrq ri Q 1
615
Lqvhqvlwlylw| ri wkh olqn ghqvlw| R iru odujh 1
Wkh ghshqghqf| rq wkh olqn ghqvlw| s ehfrphv ydqlvklqjo| vpdoo dv Q jurzv dv looxvwudwhg lq Iljxuh
51 Zh revhuyh wkdw wkh fxuyhv iru s @ 3=5 dqg s @ 3=; frpsxwhg iru d vdph qxpehu ri qrghv Q whqg
wr hdfk rwkhu zkhq Q lqfuhdvhv1 Wklv phdqv wkdw/ iru vx!flhqwo| odujh judskv +lq sudfwlfh iru Q A 83
lq JR +Q ,,/ wkh ehkdylrxu ri wkh krsfrxqw ri wkh vkruwhvw sdwk rqo| ghshqgv rq wkh qxpehu ri qrghv
Q lq wkh wrsrorj| ri wkh fodvv JR +Q ,1 Lq rwkhu zrugv/ dv orqj dv s A 3 lv frqvwdqw/ wkh suhflvh ydoxh
ri s lv qrw lpsruwdqw lq wkh olplw Q $ 4 1 Wklv skhqrphqrq lv h{sodlqhg lq vhfwlrq 715141
suhvhqwhg e| Iruw} dqg Wkruxs ^:`1
8
10-1
N= 10
N= 15
N= 20
N= 50
Pr [hN = k]
10-2
10-3
10-4
10-5
3
6
9
12
k
Iljxuh 5= Iru ydulrxv ydoxhv ri Q / wkh sgi ri wkh krs frxqw ri wkh vkruwhvw sdwk lv gudzq iru s @ 3=5
+xsshu fxuyh, dqg s @ 3=; +orzhu fxuyh,1 Wkh olqn phwulf z lv xqlirupo| glvwulexwhg lq ^3> 4`1
616
Wkh suhflvh ghwdlov ri wkh wrsrorj| ri wkh judsk duh qrw lpsruwdqw1
Zd{pdq judskv duh eholhyhg wr eh ehwwhu uhsuhvhqwdwlyhv ri whohfrppxqlfdwlrq qhwzrunv wkdq u1j1*v
ri wkh fodvv JR +Q ,1 Lq Iljxuh 6/ wkh sgi ri wkh krsfrxqw ri wkh vkruwhvw sdwk dv ixqfwlrq ri wkh
qxpehu ri krsv lv gudzq wrjhwkhu zlwk wkh fruuhvsrqglqj sgiv lq JR +43,1 Vxusulvlqjo|/ shukdsv/ wkhuh
grhv qrw vhhp wr eh dq| vljqlfdqw glhuhqfh ehwzhhq wkh vkruwhvw sdwk ehkdylru lq JR +Q , dqg wkh
Zd{pdq judsk5 / lq vslwh ri wkhlu ghqlwho| glhuhqw wrsrorjlhv dv looxvwudwhg lq Ilj1 7 dqg Ilj1 81
Wklv uhvxow lqglfdwhv wkdw wkh ghwdlov ri wkh wrsrorjlfdo vwuxfwxuh ri wkh judsk ehfrph revrohwh zkhq
xvlqj udqgrp olqn phdvxuhv1
617
Wkh suhflvh ghwdlov ri 8 E% duh lpsruwdqw1
Dsduw iurp xqlirupo| glvwulexwhg olqn phwulfv z/ rwkhu w|shv vxfk dv wkh h{srqhqwldo I +{, @ 4 h3% /
zlwk { 3 dqg sro|qrpldo glvwulexwlrq ixqfwlrqv I +{, @ {k zlwk { 5 ^3> 4` dqg 3 ? ? 4/ kdyh
Wkh Zd{pdq judsk ehorqjv wr wkh idplo| CRlm E zlwk R ' i TE@oO 3 oO / zkhuh wkh yhfwru oO uhsuhvhqwv wkh
srvlwlrq ri qrgh 1 Wkhvh srvlwlrqv duh xqlirupo| glvwulexwhg lq d vtxduh1 Wkh ghfd| udwh @ ri wkh h{lvwhqfh ri wkh olqn
fdq eh h{suhvvhg xqltxho| lq whupv ri wkh olqn ghqvlw| R1
5
9
10-1
Pr[hN =k]
10-2
p=0.2
10-3
p= 0.4
10-4
p= 0.6-0.8
10-5
10-6
2
4
6
8
k
Iljxuh 6= Wkh sgi ri wkh vkruwhvw sdwk lq d Zd{pdq judskv zlwk Q @ 43 qrghv dqg ydulrxv olqn
ghqvlwlhv s @ i3=5> 3=7> 3=9> 3=;j1 Lq grwwhg olqh/ zh kdyh dovr dgghg wkh fruuhvsrqglqj sgi*v ri JR+43,1
ehhq frqvlghuhg1 Qrwh wkdw iru 9@ 4/ olp%f I +{, @ 3 iru A 4/ zkloh olp%f I +{, @ 4 iru ? 41
Wkh uhvxowlqj phdq dqg yduldqfh wrjhwkhu zlwk wkhlu udwlr lv sorwwhg lq Iljxuh 91 Iljxuh 9 looxvwudwhv
wkdw d xqlirup + @ 4, dqg h{srqhqwldo glvwulexwlrq iru z ohdg wr qhduo| lghqwlfdo uhvxowv/ hvshfldoo|
iru odujh Q 1 Doo rwkhu sro|qrpldo glvwulexwlrqv zlwk srvlwlyh h{srqhqwv 9@ 4/ ohdg wr glvwlqfwo|
glhuhqw ehkdylru iru wkh krsfrxqw k = Wkh sdudphwhu lv nqrzq dv wkh h{wuhph ydoxh lqgh{ ri wkh
. d o
suredelolw| glvwulexwlrq ri z1 Iljxuh : vkrzv wkdw wkh udwlr @o
$ iru Q $ 4/ zklfk zloo eh
d o
h{sodlqhg lq vhfwlrq 715151
Wkh ghshqghqfh ri wkh krsfrxqw rq lv h{sodlqhg lqwxlwlyho| dv iroorzv1 Wkh vkruwhvw sdwk lv
prvw vhqvlwlyh wr vpdoo ydoxhv ri z1 Wkh pruh pdvv wkh sgi ri wkh glvwulexwlrq ri z kdv durxqg }hur
+l1h1 wkh vpdoohu ,/ wkh odujhu zh pd| h{shfw wkh krsfrxqw ri wkh vkruwhvw sdwk ehfdxvh wkh vkruwhvw
ohqjwk lv kdugo| fkdqjhg e| lqfoxglqj dq dgglwlrqdo krs1 Rq wkh rwkhu kdqg/ iru odujh / prvw ri wkh
olqn phwulfv kdyh d ydoxh durxqg 4 dqg wkh vkruwhvw sdwk fkdqjhv urxjko| e| xqlw| li dq dgglwlrqdo
krs lv lqfoxghg1 Khqfh/ wkh d!qlw| wr lqfoxgh pdq| krsv lv orz1
Ilqdoo|/ zh kdyh frqvlghuhg udqgrp judskv JR +Q , zlwk wkh olqn phwulf rq doo olqnv htxdo wr rqh
ehfdxvh lw lv h{shfwhg wkdw vwloo d qxpehu ri urxwhuv lq wkh Lqwhuqhw lv frqjxuhg zlwk z @ 41 Li
. d o
@ :=831 Wklv
s @ 3=;/ zh irxqg/ iru doo Q 43/ wkdw H ^k ` @ 4=53/ ydu^k ` @ 3=493 dqg wkdw @o
d o
d!upv wkh lqwxlwlyh uhvxow wkdw/ li wkhuh duh hqrxjk olqnv/ wkh krs frxqw ri wkh vkruwhvw sdwk lv kdugo|
:
Iljxuh 7= D uhsuhvhqwdwlrq ri wkh Zd{pdq judsk zlwk Q
dssur{lpdwho| 533 olqnv1
@ 433
dqg s @ 3=37/ phdqlqj wkdw wkhuh duh
hyhu odujhu wkdq 51
Dw odvw/ zh rxjkw wr phqwlrq wkdw vlpxodwlrqv duh qrw hhfwlyh dw doo wr lqyhvwljdwh wkh fdvh
s @ s @ i +Q , vxfk wkdw wkh dyhudjh qxpehu ri olqnv shu qrgh sQ iru odujh Q / rqo| whqgv wr lqqlw|
yhu| vorzo|1 Wkh dyhudjh qxpehu ri olqnv shu urxwhu fdqqrw eh khog frqvwdqw/ l1h1 s @ li Q $ 4/
rwkhuzlvh wkh judsk lv doprvw vxuho| glvfrqqhfwhg lq wkdw olplw1
7
Wkh vkruwhvw sdwk ehwzhhq D dqg E= wkhru|1
Wkh dqdo|vlv wkdw ohdg wr wkh nh| uhvxowv ghulyhg iurp +4, lv rxwolqhg dqg/ dv phqwlrqhg ehiruh/ wkh
uljrurxv pdwkhpdwlfdo surriv duh irxqg lq ^47`1 Zh uvw frqfhqwudwh rq wkh sduwlfxodu dqg prvw hdv|
fdvh ri wkh frpsohwh judsk N / zkhuh s @ 4 dqg zkhuh doo zhljkwv ru olqn phwulfv duh h{srqhqwldoo|
glvwulexwhg zlwk sdudphwhu 41 Qh{w/ wkh revhuyhg lqghshqghqfh rq s lv glvfxvvhg dqg/ qdoo|/ wkh
jhqhudo fdvh iru wkh olqn ghqvlw| s @ s @ i +Q , lv wuhdwhg1
714
Wkh frpsohwh judsk g
Wkh sureohp ri qglqj wkh vkruwhvw sdwk ehwzhhq wzr qrghv D dqg E lq d judsk zlwk h{srqhqwldoo|
glvwulexwhg olqn phwulfv fdq eh uhskudvhg lq whupv ri d Pdunry glvfryhu| surfhvv1 Wkh glvfryhu|
surfhvv hyroyhv dv ixqfwlrq ri wlph dqg vwrsv dw wlph W zkhq qrgh E lv irxqg1 Wkh surfhvv lv vkrzq
lq Iljxuh ;1
Wkh hyroxwlrq ri wkh glvfryhu| surfhvv fdq eh ghvfulehg e| d frqwlqxrxv wlph Pdunry fkdlq [ +w,/
zkhuh [ +w, ghqrwhv wkh qxpehu ri glvfryhuhg qrghv dw wlph w ehfdxvh wkh fkdudfwhulvwlfv ri d Pdunry
fkdlq ^9` duh edvhg rq wkh h{srqhqwldo glvwulexwlrq dqg wkh phpru|ohvv surshuw|1 Ri sduwlfxodu lqwhuhvw
;
Iljxuh 8= Wkh judsk Jffe+433, zlwk vdph qxpehu ri qrghv dqg d fruuhvsrqglj olqn ghqvlw| s wr wkdw
ri suhylrxv Zd{pdq judsk1
khuh lv wkh surshuw| wkdw wkh plqlpxp ri q lqghshqghqw h{srqhqwldo yduldeohv zlwk sdudphwhu lv
S
djdlq dq h{srqhqwldo yduldeoh zlwk sdudphwhu ?' 1
Wkh glvfryhu| surfhvv vwduwv dw wlph w @ 3 zlwk wkh vrxufh qrgh D dqg iru wkh lqlwldo glvwulexwlrq
ri wkh Pdunry fkdlq/ zh kdyh Su^[ +3, @ 4` @ 41 Wkh vwdwh vsdfh ri wkh frqwlqxrxv Pdunry fkdlq lv
wkh vhw V frqvlvwlqj ri doo srvlwlyh lqwhjhuv q zlwk q Q 1 Iru wkh frpsohwh judsk N / wkh wudqvlwlrq
udwhv duh jlyhq e|
+8,
?c?n @ q+Q q,
Lqghhg/ lqlwldoo| wkhuh lv rqo| wkh vrxufh qrgh D/ khqfh q @ 41 Iurp wklv uvw qrgh D suhflvho| Q 4
qhz qrghv fdq eh uhdfkhg lq wkh frpsohwh judsk N 1 Dowhuqdwlyho| rqh fdq vd| wkdw Q 4 qrghv
duh frpshwlqj zlwk hdfk rwkhu hdfk zlwk h{srqhqwldoo| glvwulexwhg vwuhqjwk wr eh glvfryhuhg dqg wkh
zlqqhu dprqjvw wkhp/ vd| F / lv wkh rqh uhdfkhg lq vkruwhvw wlph zklfk fruuhvsrqgv wr dq h{srqhqwldo
yduldeoh zlwk udwh Q 41 Diwhu kdylqj uhdfkhg F iurp D dw klwwlqj wlph W2/ q @ 5 dqg wkh glvfryhu|
surfhvv uhvwduwv iurp erwk D dqg F 1 Dowkrxjk dw wlph W2 zh zhuh douhdg| surjuhvvhg d fhuwdlq
glvwdqfh wrzdugv hdfk ri wkh Q 5 rwkhu/ qrw |hw glvfryhuhg qrghv/ wkh phpru|ohvv surshuw| ri wkh
h{srqhqwldo glvwulexwlrq whoov xv wkdw wkh uhpdlqlqj glvwdqfh wr wkhvh Q 5 qrgh lv djdlq h{srqhqwldoo|
glvwulexwhg zlwk wkh vdph sdudphwhu 41 Khqfh/ wklv doorzv xv wr uhvwduw wkh surfhvv iurp D dqg F
e| hudvlqj wkh suhylrxvo| sduwldo glvwdqfh dv li zh ljqruh wkdw lw zhuh hyhu wudyhoohg1 Iurp W2 rq/ wkh
glvfryhu| surfhvv kdv grxeoh vwuhqjwk wr uhdfk suhflvho| Q 5 qhz qrghv1 Khqfh/ wkh qh{w zlqqhu/
vd| G/ lv uhdfkhg lq wkh plqlpxp wlph rxw ri 5+Q 5, +grxeo| uhdfkhg, qrghv1 Wklv qrgh G kdv
htxdo suredelolw| wr eh dwwdfkhg wr D ru F ehfdxvh ri v|pphwu|1 Zkhq G lv dwwdfkhg wr F +wkh
dujxphqw ehorz krogv vlploduo| iru dwwdfkphqw wr D,/ v|pphwu| dsshduv wr eh eurnhq/ ehfdxvh G dqg
D kdyh rqo| rqh olqn xvhg zkhuhdv F kdv douhdg| wzr olqnv xvhg1 Krzhyhu/ vlqfh zh duh lqwhuhvwhg
lq wkh vkruwhvw sdwk sureohp/ wkh gluhfw olqn iurp D wr G lv orqjhu wkdq wkh sdwk D $ F $ G dqg/
wkxv/ zh h{foxgh wklv olqn lq wkh glvfryhu| surfhvv khuhe| hvwdeolvklqj djdlq wkh ixoo v|pphwu| lq wkh
<
18
exponential
uniform α = 1
α = 0.2
α = 0.5
α= 2
α= 5
16
14
E[hN] and var[hN]
12
10
8
6
4
2
0
10
100
Number of Nodes N
Iljxuh 9= H ^k ` dqg ydu^k ` +orzhu vwudljkw olqh, yhuvxv wkh qxpehu ri qrghv Q rq d orjdulwkplf
vfdoh1 Glhuhqw glvwulexwlrqv iru wkh olqn phwulfv z kdyh ehhq frqvlghuhg= wkh h{srqhqwldo glvwulexwlrq
dqg wkh sro|qrpldo zlwk h{srqhqw +iru @ 3=5> 3=8> 4> 5> 8,1
Pdunry fkdlq1 Wklv h{foxvlrq dovr phdqv wkdw wkh Pdunry fkdlq pdlqwdlqv vlqjoh sdwkv iurp D wr hdfk
qhzo| glvfryhuhg qrgh dqg wklv sdwk lv dovr wkh vkruwhvw sdwk1 Khqfh/ wkhuh duh qr f|fohv srvvleoh1
Ixuwkhupruh/ vlplodu wr Glmnvwud*v vkruwhvw sdwk dojrulwkp/ hdfk qhzo| uhdfkhg qrgh lv zlwkgudzq
iurp wkh qh{w frpshwlwlrq urxqg jxdudqwhhlqj wkdw wkh Pdunry fkdlq hyhqwxdoo| whuplqdwhv1 Ehvlghv
whuplqdwlqj e| h{wlqfwlrq ri doo dydlodeoh qrghv/ diwhu hdfk wudqvlwlrq +ru qhzo| glvfryhuhg qrgh,/ wkh
Pdunry fkdlq vwrsv zlwk suredelolw| htxdo wr 3? vlqfh hdfk ri wkh q douhdg| glvfryhuhg qrghv kdv
suhflvho| 4 srvvlelolw| rxw ri wkh uhpdlqlqj Q q wr uhdfk E dqg rqo| rqh ri wkhp lv wkh glvfryhuhu1
Wkh vwrsslqj wlph W lv ghqhg dv wkh lqpxp iru w 3 dw zklfk ghvwlqdwlrq qrgh E lv glvfryhuhg1
Lq vxppdu|/ wkh ghvfulehg Pdunry glvfryhu| surfhvv prghov h{dfwo| wkh vkruwhvw sdwk iru doo ydoxhv
ri Q 1
Qh{w/ vlqfh zh duh phuho| lqwhuhvwhg lq wkh krsfrxqw ri wkh vkruwhvw sdwk/ zh qhhg wr ghwhuplqh iru
hdfk qrgh wkh vpdoohvw qxpehu ri krsv wr D1 Wkhuhiruh/ zh frqvlghu dq dvvrfldwhg wuhh wr wkh Pdunry
surfhvv zlwk htxdo qxpehu ri qrghv [ +w, dw wlph w frqvwuxfwhg dv iroorzv1 Mxvw dv wkh glvfryhu| surfhvv/
wkh dvvrfldwhg wuhh vwduwv dw wkh urrw D1 Diwhu hdfk wudqvlwlrq lq wkh frqwlqxrxv wlph Pdunry fkdlq/
[ +w, $ [ +w,.4/ dq hgjh ri xqlw ohqjwk lv dwwdfkhg udqgrpo| wr rqh ri wkh q douhdg| glvfryhuhg qrghv
43
10
α= 5
E[hN]/var[hN]
α= 2
α= 1
1
α = 0.5
α = 0.2
0.1
10
100
Number of Nodes N
Iljxuh := Wkh udwlr
. d o
@od o
yhuvxv qxpehu ri qrghv Q +orjdulwkplf vfdoh, iru ydulrxv lq wkh dvvrfldwhg wuhh ehfdxvh d qhz hgjh lv htxdoo| olnho| wr eh dwwdfkhg wr dq| ri wkh q glvfryhulqj
qrghv1 Khqfh/ e| frqvwuxfwlrq ri wkh dvvrfldwhg wuhh/ doo frqjxudwlrqv duh htxdoo| olnho| dqg wkh
3& E&
suredelolw| sE& wkdw d xqlirupo| fkrvhq qrgh lq wkh wuhh kdv krsfrxqw n/ lv sE& @ E3 - 7 / zkhuh
VE& lv wkh Vwluolqj qxpehu ri wkh uvw nlqg ^4/ 571416`1 Rq wkh h{dfw uhirupxodwlrq ri wkh vkruwhvw
sdwk sureohp lqwr wkh Pdunry glvfryhu| surfhvv dqg dv vkrzq lq wkh dsshqgl{ D14/ wkh jhqhudwlqj
ixqfwlrq iru wkh krsfrxqw ri wkh vkruwhvw sdwk lq wkh frpsohwh judsk N lv
K +{, @
+Q . {,
4 =
Q 4 +Q . 4,+{ . 4, Q
Q
+9,
Dowkrxjk h{dfw iru doo Q / wkh qdo vwhs lqyroyhv wkh dv|pswrwlf ehkdylru iru odujh Q zklfk lv ehvw
rewdlqhg iurp wkh Wd|oru vhulhv h{sdqvlrq durxqg { @ 3 ri wkh jhqhudwlqj ixqfwlrq ri wkh krsfrxqw
ri wkh vkruwhvw sdwk K +{, iru odujh Q 1 Lq wkdw dv|pswrwlf uhjlph K +{, lv jlyhq e| +56, dqg wkh
ghulydwlrq ohdglqj wr +7, lv suhvhqwhg lq wkh dsshqgl{ D151
44
node A
T0
T1
T2
B
T3
time
T
X(T) = n
Iljxuh ;= Wkh glvfryhu| surfhvv dv ixqfwlrq ri wlph1 Wkh wlph krul}rq lv gudzq dv d grwwhg vhpl0flufoh
fhqwhuhg dw hdfk vwduwlqj qrgh D1 Dw wlph W wkh surfhvv vwrsv zkhq E lv irxqg1
715
71514
Rughu fdofxoxv
Rq wkh lqghshqghqfh ri
s1
Lq wklv vhfwlrq/ zh vkrz wkdw wkh revhuyhg lqghshqghqfh rq wkh olqn ghqvlw| s lv txlwh qdwxudo iru odujh
Q 1 Zh zloo hvwlpdwh wkh suredelolw| lq JR +Q , wkdw wkhuh h{lvwv d sdwk zlwk n krsv iurp D wr E dqg
wrwdo ohqjwk Z vpdoohu wkdq O1 Wklv suredelolw| vkrxog eh urxjko| wkh vdph dv wkh suredelolw| wkdw
wkh krsfrxqw lv htxdo wr n zkhq O lv fkrvhq dssursuldwho|1
Su^< sdwk =
n krsv dqg Z O` @ Su^^Mtsdwk= & krsv iZ Oj`
[
[
Su^< n olqnv/Z O` @
H ^4& olqnv ` Su^Z
Mtsdwk= & krsv
@ H ^&sdwkv zlwk n krsv` Su^Z
O`
Mtsdwk= & krsv
O`
zkhuh zh kdyh xvhg Erroh*v lqhtxdolw| dqg 4% lv wkh lqglfdwru ixqfwlrq1 Vlqfh Z lv d vxp ri n
S 3 u
h{srqhqwldoo| glvwulexwhg lqghshqghqw yduldeohv zlwk sdudphwhu 4/ Su^Z O` @ 4 h3u &'f
- @
U u %&3 3%
&
h g{ ? u&- dqg wklv odvw lqhtxdolw| lv vkdus6 iru vpdoo O1 Rq wkh rwkhu kdqg/ lw kdv ehhq
f E&3
vkrzq ^48` wkdw H ^qxpehu ri sdwkv zlwk n krsv` @ 33&32 s& * Q &3s& 1 Khqfh/
Su^< sdwk= n krsv dqg Z
6
Dv|pswrwlfdoo| vkdus phdqv wkdw
*4
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'
,
O` * Q &3 s& On$ @ Q4 +sOQ
n$
dqg lv ghqrwhg dv
45
&
s E% s }E%
&
+:,
Vlqfh zh duh lqwhuhvwhg lq wkh vkruwhvw sdwk/ O pxvw uh hfw d w|slfdo ydoxh iru d vkruwhvw sdwk1 Rq
+l1h1 wkh dyhudjh
dyhudjh/ hdfk qrgh kdv sQ olqnv vxfk wkdw wkh dyhudjh plqlpxp ohqjwk htxdov R
ri wkh plqlpxp ri Qs h{srqhqwldo yduldeohv zlwk sdudphwhu 4, dqg wkhuh duh rq dyhudjh H ^k ` krsv1
d o
/ zh rewdlq
Wkxv/ li O @ .R
4 +H ^k `,&
H ^k `
+;,
Su^k @ n` Su < sdwk= n krsv dqg Z *
sQ
Q n$
Lw uhpdlqv wr ghprqvwudwh wkdw H ^k ` lv lqghshqghqw ri s1 Dqdorjrxvo| wr wkh frpsohwh judsk/ wkh
wudqvlwlrq udwh lv ?c?n @ q=s+Q q, . y+q,/ zkhuh y+q, lv d vwrfkdvwlf yduldwlrq durxqg wkh phdq
qxpehu ri olqnv dw hdfk qrgh gxh wr wkh udqgrpqhvv ri wkh judsk1 Iru odujh Q / qs+Q q, AA y+q,
vxfk wkdw zh fdq qhjohfw wklv vwrfkdvwlf hhfw lq d uvw rughu frpsxwdwlrq1 Lw zloo eh dq lpsruwdqw
S
&
frqfhuq glvfxvvhg lq wkh qh{w vhfwlrq1 Wkh glvfryhu| wlph W& wkhq htxdov7 &?' H{s+qs+Q q,, @
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