0.

)

These d~5 I O.WI s~end;n~ ~OW\e or W\~ 5F-V"e

tlwae on the deve\opMeV'If ot o.. so- oo.l\ed c\; ~k-co..che pro~v-a.WI ~r md per-sona.l CDW\p\A...\.er (, \-Uh:ch tor-fu"o.k~ i5 V\of o.. --pc ) . The pu.rpo5e of +hi.s pro~ra.wt \~ to speed lA.f ru:ce!;)s.es -\-o the. t\ \;\'\~ s~~:JeW) b~ keep; V\~ c.cp ies o\ +he Most Y"ecevJlj ~ccessed . disk sec~ov-s in a. b\A.~~ \n pv-iWta.("'~ stor-e. . T\,e 0..6o'-t.Mpfion \6 +ha.t t~e&e. .e,ec~or6 o.re ~lke\~ to be. o..ccessed ~Q\Vl \n +he Vle£A..- flN-\-I.U'"e _; b~ keep~n~ theW\ i~ pr:W\(A.V'~ 6..\-ofe ~CtV\~ CA.Ccesses \-o the r"eL:t~ive\~ s\o\.U d~sk co..n be o..voidecl.

The pr\ce .\o be p"-id Is the ne.ec\ to a,l\ocCA.te CL r.'xeJ pC\.rt o~ pri Mo..r~ skve. {or -\-he. bu..~r.;, \r pri W\a.rd ~+ore is

so lcur3e -\~o.,~ +his p~(-l \..Vou.lc\ o-therwise V"eMe>.-Lit) LA-n\Aoed.)

+hi~ modu.s opera..VId: \s vi a..b\e . A disk co..che rese\'V\b\es a, v\r-\-UQ.\ ~tor-~e s~~feW)

ver-~ INlu.ch, b"'~ -l-hev-e i.s C\Y\ lVY\porb~ di~tev-eVlc.e. . In Q v~ r~IAO.\ 6tov-o.se 5~~JeW\ the 5VV\a..\ \e s-\. a.c-\-ioVl Is

o.n Q..ccess to u.. 5;n~\e W'ord in pv-iiNletr~ ~tot'€.) whic~ re<j_lAi ve& ve.r~ \ ~-\-~\e t\W\e. - t\'l ~ CoW\p~te.r ; 0.5 )AS- .

In a.. diok c.a.c~e. -\-\,e SmC\.lle~J o.cilon \nvo\ves cop21in3 Q... \Z>ho\e. .sector.) \.V~;c~ req_u.~r-es m!A.Ch more .\-\VYie

- in W\!:\ c.oMp\A.te~ : Zllio }A6 , \?ov- a. 512. b~rte.s 5ec+ov-- .

As o.. re.su..H:. ..> ~e. co..n o..l\ow fhe c\if=»k CQ.che -\o spend VYl u.c~ mot"e t\W\e OVl bookkeej:> iV\~ opev--CttioVIS thCA.Vl wouJd be o.ccepta.b\e in . Q. viv-liAO\-\ ~tov-Of,e ~jo~eW\ .

I ha.ve decLJe.J thC~..~ the rep\a.ce~eVl~ ~Jv-a.te~!a V\e.eded ~v- VlcJi !IV\ se \e.choV'I wd \ be leo..sl 'ReceVItl ~

)

Uoe.cl. l {s I vnp\eW\e~JC1.-~ion t"e.!iu.i re5 5oW\e O..WloiA.nf o\ bookk.eeriVl~ per l< SMo..\leeJ u.ct Ion)> ., \A>~ich \::, the f'eA50Vl

wh~ the U.&e or L 'R u. iV) v; rtLW.-\ €>~ov-~e S<'j6kW\s l5

OV\\~ teo..sib\e ~ifh the. a.id or decLcc.Jed ~o..v-dwa.re. ( See.. .) for exo...mp\e > E\.JD L.f~S in [o J . ) ln ~ CAse .~ however l +he. overhe~ due +o the bookkeepin~ Co.vl be kep-b \.\?;thin re.o.5ono.b \e. \ ;~~+s .

In this no+e I wish -\.o \.0ork O\A.~ in de.\.a.i \ two

a.opect5 or 4-he a.c\W\~Vli:Stv-a.tioV\ V"'eeded -\or t~e IW\p\eW\en+CA.~Ion ot v\chW\ se\ec-hon cl la.. L'R U. The

~o\u..~ioVlS pre.se~Je.cl here a.v-e e.\a.bor-a..hons o~ Q

s~~e~.Jioh b~ L.A.M. Sc~oe\'\W\o..kev-5 l ~r w~lch he Is

k;Yld l ~ acl\:now \e..d:jed . -,he -hvo 0-5pec\.s a.re c\ose\~ covH'lectecl btt.t co.n be discl.L:::.~ed In re\o..-l-ive i5o\a.tlon .

The ~ir.!l.\. problem is to Wla.in l-Q.i n a.. se.q_~.tenc.e oc ot va,\uee, o..nd k lmp\~me~·J t~e .fo\lov:>in~ opera.-ho\1\S OV\ x.:

(:i..)

(-liJ (U:i)

extend x a..t Its \,e.cJ wit~ a. ~\ve~ va.\~e reWioVe O.n cc ideY)ti nec1l) ele.mevJ tV"o~ X.

~ie\d +~e va.\u.e at JC.\5 \a.~J eleW\etf)t

LlVl.\-d Sc~oe.nwta..kev-s to\d Wle. hi~ s~3e.~Jion l +\.te on\~ \Wlp\eW\en+aJio\1 I knew 0~ Wo..5 -\-~e OVIe. lV\ w~ic~ ..:X:

\o repv-ese\'Jed u.s o.. doub~ \ i~\,ed \ist ~ -\-his l5

te.o..5 \ b \ e in tha.t t~e a.bove .2> opei'oJ.-loVI5 CQ..VI tl,en be. pev-G, v-W\ed IV\ CoV\SkVI+ time , bu.~ I never ho..Ve been V€V"~ tond or doL.Lb\~ \inked \\s~s (o.V\d I pV"o~et.bl~ never- wit\ be ) .

l

A 6~Mp\er v--epr-eseV\-kJtoV\ \s the oVte in w~ich the

e\eW\evrL:, o~ ~ a.v-e SIW\p\d enu.W\en:xted J in ov-der-> In O.V\ u.v-vo.~ ~ (sa.~) . T~ ~ ~a5 a. fixed \eVJsth fhen """is \5 o\'\\~ po55i ble u.ndev- o..dd;\;OV!Cl-~ a.ssu.W\ptlon o~ the invo.v- la. VICe ot .> ~Y :SoWI.e ~I VeVJ eo V\ csbnt 1'1 .)

( IV\ ~ a.pp \;cu,tloVI {hi 5 us:su.W~.p-lioVJ ~~ va.\id . ) In orcler +o O..Mor~i6e +~e cost of -\-he .SoVV'Ie~iW\eS necessu.r~ shl(l.iVl~ open:~J;on5 In Q.Y'ro..~ ~, \.Je decide to let ~ h<:A.Ve a_

te\1\sth -\-ha.t \5 5\A.bs+C\.VI~Ia.\1~ la.r~er tho..\'l M .

With N tov- the \eh~+h of ~ ~V\d. LJi~~ 1 tor- a.. vo...\u.e nev~r cccu.ry-~V\~ in x J we need un

o.o\J;~io\1\a.\ vo.x-io.b\e.. t o.nd 1.0e u.se t~e \?ollotDIVt~ ~ V\VQV' iCAV\~5 .

""Po :

1>1: 1'2.;

le .X. io the Su.b&e<j_U.eV\Ce Or ~)5 ho\'\-1 e\eV!flev-Js ))

o ~ r ~ N

(\i-t.~ 0 ~~<t; ~·i.= 1)

-, ~e. thv-ee oper-a.\.ioV\5 OVl oc CO.Vl +~eV"I be LWlp\eW\ev-Jed CAS ro \ \ow5 .

(i) J t= 0 ·~ c \ eCA..\'\ LLf { t > 0 1 0 ~ > 0 -7 sk;p

fl { t > 0 1 . + == r- i .)

Vtew va.\lAe -b be added ~·t := ut~e +o.x. )) . ~

' t5 :

)

(ii)

t '"Po · A #:x-< M } c\eu.nu.p { '?o A 'PiA '1>2 /\ (Vi~ f~ i-< N : ~·1. I..L) 3 j

whev-e it Is \e.\+ u.nden>tood +~et+ c\eo.~up s~o~.<Jd Vlot c:.ha.Vl3e x. . :Deve\opiVI;j o.. pv-o~ro.Wl ~v-c \ eC\.Vl ur j 5 0.. s+ Y'O.l~ ~J ~ Y' WO..V'd e Xe f'Ct5€ _; {hiS

pr-o~ro..W\ +a.kes l\J .steps . 'Beca.u.se

#..x < M ~ N O..V\d #;x.. = (f.;.; o ~ i< N ; ~·i.J.l) ~ We. ho..ve .\.hQ.l N-M < t i6 a.. pos+coV'Ic\i~on of c\eo.nu.f ets \...Ve \\ . 'Beca.u.se t 15 decv-ea.sed bj i du.v-i n3 eveY ~ ex{end-::x:. opev-a.ho n ) deo.V~u.p \5 i V\ vo ked a.t most OV'Ice. evere1 ~-M ex~end -x.

o~ero..hoV~6 . So, on {he u.ver~e fhe lV'IVoca .. ..\-lons

ot' c\ea.hul:> f'e<J. u..\re N / (N- M) 5~e ps per ex\ehd-x operO\.~toVI . For N = 2 *M {t,io

express;o., ~le\ds 2 .

lt is 6u.Wcaen+ (~v- ~~ pu.rpose~) -\-o \d· -\he e\eMenf \..o be r-eMoved be \den~;~ed b~ i{s

iV\dex \Vl ~ .; -\-~u.s J l.Je StW\pl~ ha.ve:

t o ~ p < N A ~·p ! _L }

zj·p ::: j_

In order +o obb...in Q.. solu.-l-ion wifh :> On the constant co~J) We. IV'I.\-rod\A.Ce a.. va.riCLb\e

with u.cldi-ho V\a.\ lV'IVC\.V'ia.V\~6:

rp.3: t ~ "'" ~ N "PCi : (V -1. : ~ ~ _;,_ < N ~ . .f.. = 1 )

)

?s:

In order- ..\-o W\~iV\to..in 'Plf clmV\~p mu.6~ be O..CCOW\fOc.Vlied

b~ '\J' := N . Under- the precoV\dition 1-ha.l ~ i!>

YlOVleW\p+~ \ +~ lo.~J e\eWlevr\. ~n be COWlfLA~ed b~ :

_; do ~·('\>-i) :.1. ~ 'U ~= '\>"-.i od { ~-l'\>-1) I .l 1\ (Vi~ " .. ~ i< N : ~-..: = 1) }

"PS i {) Wla.i vJQ,; V\ ed \o ~ O..CCO \M po.\'\ j i Vl~ C \e.o..\'lu.p LcJ; fh h == 0 o..nd b~ U.CCoWlpo.Vl~JVl~ ~·p !: l_ in (ii)

b~ \, !:: h + 1.. . C.no~ice tho.+ b~ ~oLD ~·p j. l_ ~ t ~ p < '-'". ) In t~e o..bove pr'"o~ra.W> e.<U:~ sb.teWtevJ

"'"~= V-1. WlLA.-5t be reFlo..ced b~ vJt ~= v-1),-i . This shows tho.t -\-he number of okp6 ta.ken b~ the o..bove pr-o~ro..m is a..t h'\o~J h _; a.Vld, b~ 'Ps ) h is

a.tmosf thehu.W~ber o~ -\-imes operuJ~oVl (i.i) ha.s

been lV\Vo ked s \nee the VY'losf r-ecen~ \nvoca...\-ion of c\e<AVI\..\.P· T~u.s, -\-\,e execl.(..~ioV'I -\-aW\e o~ oper-aJ~on (-i.i.i) Co.Vl be Q.W\or-hsed b~ o..-\lri bu.h ~ ~~ p\ece ~ise to ~~e pre.ced;Vl21 reW\ov<N\ open:J--io~5. (see 'Y.S. on p.io~)

Thi6 COV\clude~ +~e ~~u.~oV\ to ou..r -hr6~ prob\eM. T\,e

D..W\ov-~; oa.~ion c:Lscu.5s;oV'I \n pa.vl (ii.i) CCc.-n be ~ v-vno...\ i sed bj the. ivJ-rod\Ac~'on or Q.. SO-CCLlled Credit }u.V\cko~ _, bu.t l-he sit\Ao..~on here i6 5o &; W\p\~ t~CLt cloiV\2) so is hctrd~~ worth +he tro u.b\e. .

I O-Wl ver~ r\~sed \.0it~ thi5 sollA-~lOVl Q.V\d I nV\d it deGnJel ~ mov-e e\esa.~J -\-hQ.n +he vero;OV1 w~fh dou.bt2\ \ ;nked \ i s+5. moreover;, i} is O..V\Ot~er j ll \A!:Jv-a..hon of wh~f I +evJa..\. i vel~ ha.ve dubbed +he 5o% V't.t..le . CoV\5\der the ex tr"eVVte C.CA.Se5 N = M o.nd t\J = oo . In """e eo..se N = M storVl5e \A..\.disa.~loV'l 15 \'no..XiW\o.\ CA.V\d perk,riMa.nce ,·<E.

o11Mo<5t ~ero .> beco.\A5€ c\eo..Vlu.p VV\lA~ be \V)vokec\ lVl

everd executioVl o~ opera.~on (-l.) . In -\1-,e case N::. oo

5tor-"5e uti\iQo..+ion ~~ ~ero o.V\d per{~rW\ctV!Ce ico W\etXiWlct.l .)

beca.u&e c\eetnup is never invoked (, ~r we c~oose ~ to be G.VI o.rv-u~ 1,0it~ do~a.i Vl ~ ~ 0 { -L O...V\J v-evevse

+he order- or X )5 e\eMeV\~5 In ~ ) . So) N =-M v-epreseVltS t~e opti W\ct\ 5 I t~Aa.tioVl t;-OW\ -\-he v;ewpol V)t of

stor-~e uJ\ \i salaoV'l , L0~e.r-eets N = oo represento the

optiW\a.\ s\iu.Q..\.;OV\ rV"oiM +he viewpoinf of 5xeed. The

in betweeV\ o\tu.o.Jion t\1 = 2Jt- M jle\ds 0-. ~\r COV\1\pr-oW\;Se

between +he se two ex~v-eW~eS ~ s+-ov-~e. lll.-\-; li sa..hon \~ 50 °/o -\.J.r--,t. t~e extv-eW\e. N:::. M- Q.ll)d perR>v-ll!llaV\ce

wi I\ be ~~pv-oxi W\a.k\~ So o/o - \.V,v-,-\:.. +~e ex.~reW\e N :oo­

The so% v-Je sta..-\.e5 fho.l ~ if we are o.b\e .\-o obki~ su.ch ~ ColMpv-ow\ISe :> we ha.ve ever~ v-ea.SOV\ +o 6e sa..-\-i&~e.d .

A~ hu.& beevt oboer--ved b~ E. \J. J)jks.\-r-a. ( EW'Dltb2 0\.Vlcl E\.vl)Lt 6s .:> \n [o]) > L 'R U.. io W~onoto\t} ic. ill) thca ~ llowi n~ 5eV\ oe _ Fv-oWa the v;ewpo iV\t of v\r~v..ct.l ~torOfle W\O..ho..~eV't'\ent CL C.o\1\1\pu~.\-ioVl is Q... Se<J_uence or p~e V)a.VYt€6 - ea.\\e.d Q.. rererence &tr'";~- . Cons;der ~e eWect of o.... 5~Vlgle .. r•xed, v-e~re~ce str'"i~~ u.pon +he Conte\'\tS. or -\-\,e bv.:~r - Le.: ~e set oF p~es prese~r~f

in P"'i mo.r~ 5to r-e - > a..s a.. .{:nc.\-;o n or t~e bu..~r s ~ ~ , rf.) oJ O..V\~ po,vJ \n the re~v-ence .siv-&n~ .) +he sel et

ruses in +~e SWlD.l \er b,AS·+'er- I~ Q.. s~b~e.t 0 ~ t~e ~ i of p~e..s In the \o.r-~er b\A.\-rev- > a.nd t~;s holds tr~Ae ~v­a.\l v-e.~er-ence s~v---i n~.S J ~eVl the v-e.p~cu:.eW~ent o.l~ov- i thm l~ ~<A.\cl +o be vnonoto\'\ic .

We. CO..v'\ de.~ ne > o.t ~n::\ ~ int \n -\-he v-eG.re~ce str;"'a a.nd ~r Q.V)~ r~e .) the .sire ot' the ~Wlcxl\est biA-~r con-\.·D.IIfl\n~ +ha.~ p"'-~e .; with a. Wlonoto~ic replctceW\enf-

o..\~ithWI > +his s\n§\e n\A\JY)ber \dev.JL\]es o..\l poooible 6l-lt1er :si1!-es ~r whic~ a.v-t a.ccess to thD..\. p~e c.o..u.ses

a.- p~e ~l4-\t . l.n ov-c\er to be a.b\e to evaJuu..te ;}s pev-~V"W\0.\'\C€

o..nd -\.o obf-a.iVI LVI tov-m<xtlon o.bou.t \.Vho..-\. wou.\J be Cl.

V'"'eCl.SoVla.b\e b"""-Wer siee , I I.Vish VY\~ dtsk-ca.che pro~raWl to r-ecor-d > ~r- ~eh pooo\ b\e bu .. .-~rer size. (u.pto a... ce.v--l="a~n Wlo..XIW\\A~) .> the nu..Wtber o\? r,~e tCUA.lto Correspond\vt~ to -\-\,a.~ h\.\.~Y ol~e. I In viel.V of +he ctbove thi5 bot\~ clowVl -b fhe ~ llowih~ .

LJe cons ;Jev- O.V\ a.v-rQ..d ~ (\ ~ 0 ~ i < N) whose

e\eMevJs o.v-e in.\-e~ers . E leMevJs a~ ~ wi \\ be modi~ec\ o.rbitv-ar~\~ .> nevevthe\ess .J lt m\A.s.\. be po&s\ b\e +o Co~ptAfe (L i ~ 0 ~ i < 9.: ~-i) ) ~v eveY~

<j_ ~ 0 ~ q_ ~ N . S imp\~ CoVY\pu..-hn~ +he Vet.\ ~.e o\? fh i5

expves5ioV\ take~ (}(q_) .\.iW\e _, w~ich i'5 deemed too ex peV\~ive . T~~ o~~eY ex~feMe lo fo !:J·ure the VQ.\ues

of 1-hlb expY'"essioV'I ~V" oJ\ Cj_ , 6'-'-t +his .§lve& r\se.

-lo p~o~obJ;ve\~ ~'>~0.~2\ v.pd~-\-i~ obli~o..\-ion.s LU~el'lev~.­Dl.V\ e\eWievJ o\-' ~ tS W\ocL\1e.d- ~ So> a.~ m t~e preVtou.s

.Sec~ioV) J We Q.re \oo\,;Vl~ ~ov- o.. C.OWlf.V""OWl.ise. ~, How ohoJ:. o... \o~u..v-;H,WliC. ~o\LA..tioVl ~ LJe de.~ne:

S·f·~- (~i:p~i<p+i~~·i)

-1 heVl:

(sO) S·p·O - 0

(oi) S·p·1. - c:ti. p (52) .S·p·c\1\:+.Q.) = .S·p·\( + S·(p+k) . .t

I+ t0; \1 be vtice if we~Vlslruc~ o.. set V or po.iv-s

(p "1) _, 0 ( p ~ p+~ ~ N ;) \,.);-\-~ ~e. ~ Uowl VI <a pv-o per-lies ;

• Ever~ jV\+ev-vo.,\ [o ~ q) ;5 1-he c\ ;~0\Vlf U..V\ lOV\ or 6( \o~.N) W\O..Vl'a iV1-\-er-vQ..\s [p-> p+i) ) w~.\.\-, (p).) 6 V.

•

Ir the Va..\1Ae5 S·p· ~ ~CA.Ve beeV\ pv-ecoW\pu.tecl O.Vld obved > -\-~\5 o..\\owS CoW\plA.+Q,ttOVl or s.Q,i \Vl

\o~o.xi +~ \1\1\ ic tiMe. .

l:.ver~ po\V\t g_: O~q_< N oc.cur5 lVl (}(\o~r\J) MeA-n~ \vJervQ..\5 [ p .. p +l) _, 1.0ith (p ~ ~) ~V. Th\s V"'eeJcict6 -\-he nu.Wlber of lA.pdll--h\1\~ cbl~oJ-iovt5 o~ va.\u.e5 s-p·t -b C9{\o~1'\\1) ,. . t~~ ever~ ch~h~f- or et. (oin~\e) e\e.W\evJ 0~ 2\

S·O·(q_+i)

{ (52) ~

S·O·<j + S·<j:i

{ (s.i) ~

S·O·'.l + ~·c_t ~

\A)~ic~ \s 5\mp\e e.V\OlAjh. IV) view or ou.r e.WicteVlC!:\

re.'iu..ireme\1\t :> we a15o c.ons\der :

5 .Q. (2~t-q)

{ (52) j

S·O ·<j_ + ~·<i·'l )

w~ic~ t5 V\ot s\mf\e e.Vlo~~ t~ere o..~e too W\a.~ of

5uc~ 6U.W\mo..~d5 . T~e.r-e~v-e, We ob5er-ve t~a.t s.Q.9.n

0..V)cl 5·0·(2*<.1.) Co.V\ loe. co~sider-ed 0..5 iV\5kVlce..5 or -\-he Wlore. ~eV\e~oJ s.Q.(o.*q) ..) \.Ve IV"l.\-rod\..lce a..

tu..V)c~ovt t which we. Y'e<j_u.i r-e -b 5o..\.isrt :

f.a..(2ifCJ. + 1..)

[ (to) )

s . 0 . (~ ~ (2 *q_ + i))

{ (52) 1 5·0· (cd-2 ~g) + s-(c.Hf·2*-9)· a..

{ (to) .> \ntr-od\A.ChoVI o~ ~U.Y\choYl ~ )

fco.*2). c:t. + ~.a..·< 2.c3) ,

lJ~ sf.orin~ -\-he va.llle5 or +U.V\Cl;OV\ 2 J 1Je.. obfo.,;"' ~a.t S·0·'j_

CQ.n be Co""f"J-ec\ in 2\o~ ·<J_ 5kp5 ) beC(A.u.5e we ho_.ve.:

S·O·s_ - r. i·q_ f.~.o - 0

f. 0..·(2*q) - r.(0..*2)·q_ t~o.·(2.•q_+ i) - .f. (O..f-2.) ''i + ~·et.·C2~q)

Hence l -\-l.,e va.\u.es ~ · i·q_ Vleed \'lot be s-bv-ec\ . T~e On\~ VCA.\t..teS. or po..r-a.W\e..kr Q t~ct.t Occur- o.re powers

o~ 2 ) ~r f\xed k _, +~e ro..V\~e.. of q_ In ~·2k+-f; ('2 -.q_)

w~;c~ is saJi s0ed b~ a.t W\o5t N / 2k+l '\>Q.lM.es q_. fis o.. v-es\4.\ \:. > -\-\,e tok\ VHAW\ bet'" o~ g - Vo.\ u.e:; to be s-\-ored I& o.t mo~J N/2.. ( T~i~ bo\-\.l'1d ,,~ s~a.v-p: +he o..c tlAo..l Vlu.W\ber i~ l\1 div 2. . )

We COV\C\lAde thio 5echoV'\ t.);th ClVl O.V'\Q.,~5i5 or +he u.pd<At;vt~ obl i~a..ho~s indu.cecl b~ CA. modi ~i ccJ-lon ot ~ · p l 0 ~ p < N . A sto n~d va.l \.le ~·a.. (2•q) W\\-\.5t

be u.pdCA..\-ed - I.e.: I ne rea.&ed b~ 6 ~· p - lt o..\'\d OV'\ \~ it p \ ies in +~e interva.J [ 0..*2 *'3. .> CdE· 2 *"'3. + a. ) ~ this O.MOlA.Vl\-s to :

This Wlea.Vl5 +haJ ~v- tixed Q we ob.\-CA.iV) t~e ~l\ot.aln~ pt'"o~v-a.Wl. t~IMevJ +ha.-\- mus-\- ().CCOWl\:>etV\~ ~·p ;= ~·p + ~ :

)

rhib&: 10

h_ := r div et.. .; ir even. h -~ 0 odd .l, -::>

t

-,~is prosra.Wt ~~W\e\'J mu.5f be exec~..J-ed ~r- eve.v-<'3 a..

tv-ovn those powers or 2 tha.-\. o.\5o scJ·isr~ 2 ~ u.. /\ ~ < N

tleV\ce) +~e nu.W\ber o\? u.pda.te. cb\i5<l.-\.i'ons ~~ Q.t Wloo+

2 \o~· N .

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