A ~T~EMATICAL ApPROAC M TO

PROGRAM BEMAV!OUR AHD

LoO K-AME.o.D REPLAC£IIEMf ALGOR lT MMS

OTTO SPA NIO L A 7H6

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1.<.J . DIB."I •• o f 1; to. re glo., ~ o f Int..... I!

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• • ~o rl ~" ) 1. dl • • ctl y .do • •• • • bl., ~ r •• ' .' it Is • •• "~.d

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loto ~j(i) C"" ( ' I • Xj(1) )

,,' b) Re l •• dlng the old <o',e'" of I J(I) boc k Int. S~ .

Th •• elUdlng .'1 b • •• P . . ..... If t h ... , r , un l Iru

P.gu I . ~", o r If th e ,.nto nt< o f XJ(i) hav. not , h • • Qed

du,lng tho Tul dU'y of 11(1) In ~" ( thl' " ' 1 bo ' e co.ni,U

by. r ead_only bit ) .

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SH '"" " " to 90 • • ru~ b, tho p . go ,I t< , tho vt11!rHl.n Of

word. 10. p. g., t • • , •• U . f .001001"9' gr o.".1 "go, . th o

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In p. r t oj . f d. II .I ,lon 1. on l y t n. i •• di ng ( •• t • • r" r e ga rd ed .

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?r><t i « t,. ro? lo co • • nt ,o , to t . n. t o b • • Iho. f 'ICt i on of

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C1o" lo.~ing <.ndiU.n ~AII) I

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h~ .. . ... Ion or on ( Ooe Bloc k loO k-thud ) .nd Hs .odl­

f teo,I •• , Sl ( Sp,"O ' Loc ollty ) ,nd I t ( ', •• or ,1 toe. l lly )

oco d U Ud " 'P'9' 0 9 "gorHll w, ,,n,, ... Jo.Uh . O~l and t h

.. ,.,.10. Sf I 510.10 Predlctl •• 1 or o!t H. , •• • -. hud­

.l gorltll_S.

In ' .W' HY. e • • " os, worts " follow, ; ~.,I~ .. to •• 1"lft9

P'g"! ( . nic~ I , ~llCod Oft t~o top of c ~ " LI U· SU , ' ) Its • •• 1 9 ~ ~our !q. 1 (If not y o t pro , o n' I n ~M ) I, 10'0"0 ( o.

th ~ot.u pI ... o f t~. uac t ) .

Thu s 8'1" 1 Ii r . phno .. t h O'U'".''' o f I ~ .... t p ... h vlt

if tho.o . .. . 0 r oforueo Oft t h i> p.~o I . Uo ..... \ 1~o.

8, U h 0'91.1,.110. "hich i, clo .. ly r ., l.tod to t ~o llU·

~ r ln<lp l • • "" II .0' bur d" . od wi th p. ~., s wh ic h ~ I ' O b"".

pr"p'9"d IopodluOu , ly, but on t~. othor ,10 ' t ho off."t of

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Jo.eph, •• c, I •• Of OR, i , • 100k-,I> •• d- 019 •• II ' _ , bOCoyS '

~M Is '."TVO. to ~ .ld p'g", ~h\ch are 1" .0. I n .d •• n c~ .

It wi ll turn 0. ' th.t DOt' ORt .nd SP Ho ,poe l ,1 c"", o f

LH w,ien will b. c.nst , .c t e. In 'H tio n l by • •••• o f •

•• t h ... t l ,, 1 • • de1. I t .hould bo not e d I h " 08 L. $P , 5l

ond H '.H b ... d.,he. ~.( "11 by he u,h Ue.) •• g • • on t<.

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wit" .... 11." 10 ' ''''. dl st .. , o , I. e. 1> 1 U • • 1"lnq

p. ~. ( 01 90rHI>. 8. of Be l . d1 !'1 ).

Co " o.pondlng Jy o . opt h, 1 d ,~,"d p,.p.gln~ . loo r IU. whic h 10

.bl. to load. po g .. at the sue t\ ". " Ill , . plo n tho •• 'Q" o f .~H "Ith grutes t fO'wor d dhunc. ' I th e \ pog .. .. htc~ ...

not 1 0 ' pre ••• t In ~~ .nd wh i ch h ••••• • 11.'t forwa , d ~I't • • c ••.

t o pr e dict the fulur~ bo~ .. lo u r of tho re f .,"nc .. by t~~ r .Cut

put ond to reph« pa9"' .. ~IC" we upoet to ~ ••• grutnt f o r _

wHd d(Honeo •. Tn" .... applies for I"'_a~ .. d .Igoritn • ••

• • • 1.4 O .. niptl0. of . "ot~.r no •• l of P,"'i,U boll .. ; •• ,.

~''iton . of • Rro9" .

In tho fo ll. w ln~ , .. t nuot l c. ) Md,1 I, ~r • • " t . d wM e l!

I.\ ,.d., .. oropor t l • • of lo " l i t~ whj , ,, H~ ob .oe • • d 'or

... y porto of I F OUR~~-

. od . l wn l," .'Y h • • ,, 11 ., ~" CI . I ' y for l no pl n. proq" ~'.

dulO O PrtPogl ' 9 "" 100 ' -' '''. "~orlt " .' ,dl1 '" d.d,.d.

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Pro9" u"" Iud \0 eo ', ,' lro t . do t . Of th o .... . t yp • •

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Is d. o to prog ..... l'g n." " "d tho ,t,.c t """ of ~ro9r . ~~ I "g

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orog' u , , 'o 'l 'on V of • • ; ~ol. vor i" , ,,, " ' ;0" ~ l' ' l""

of ." ... . . " 91o, C of <o. H • • t> ( (.110 11 • O"'"o t (Ote'"

n. nl " 00 " In [~ . I I] I ,to.

h~. d< . l ro<o l t< o f r. 'b,Q. l ~ ] .. 1I J o,.~" [ 11] ,how t h. ,

for ,,.n, 0'"9" "' •• H, ttl on of ~~ 'n to • or S .. glo'. "i ll

b ••• ffle i , .t. 10 ,.~ •• • t . nt, tnl' ~o l d, I.dooodo.tl , of

th o pogo 'Izo.

Ih. n u~b.' of P09" .. ~I ,h • rooloo ""upi . , '0 .. " • .,i ..

.. I'~ th. " ~. t. ~ •• Io •• " , " to . nu , ~ ", of . O ~H 'n . ,~

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I.,rodu co d

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lure or. othr , oqlo" •• •• q. 'ho 'oqlo. C . f <on'"'''' ( - OU ' ­

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10 I. ~ r oc .... d 10 "o.·du,." .09;'9. t ~ . hltro t . oltl: •

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. I nc o ril i ng 0(') ( h~ propo.1n 9 . dditlo •• l • • q .. It \1 • • t - I I

~' 1 r .. "1t I n h19" t.", It t in" t _ 1 .. . ,11 " In. dHIi ••

of t h o 'Itro to Q(t . l l ( 1>0) , I lo t .. ti n .. . H i , wo l l bo <o n

fro ~ .n • • u .l . In .«lIon 2.2. Thu •• Ino re .. l n ~ 01') .191H

,. .. ,,1 ""OM.

~""' tn . l . .. th. nl\rot • • 111 be. I n Qen" , I • • ~ood or l t . r lon

IF t~ . nu Mb. r of o' ge ' wh l," . r . pr ' . ' 90 d .t the •••• ,I • • I,

. .. or. l y l lM lt , d ( "1 ,or J ).

HI, rest r l,tlon wil l b • • try l~ p" r U"t for t ~ . <o."ruHlon

of nur 100 ' -" " 0 . 190 r lt~ . L'~.

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- 19 -

If I~ . ,.1o tl o .. In quutl .. h.' ~ f or ~R • h' > 1 " "1

t~e. occo,dl'9 to Ih •• r .~ t t he p,~, gu .f 't "d 1',

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to ,0n . .. II .. Ih •• r •• J on h ' > 1.

then •• ,n f or '''~e ... 1.dowSize. 'Ut ,0 .. 1\1 " 1 be Obttl n u

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liS ~ l !>u' not f.r li S • I. Tills .. til be th e CI .e If I

r t ,lo' to <.~po,.d of I wo orray. wh o ... 1 .... " or. r o f.,on -

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Th. be •••• ( on do I. ou r •• ,.pl . I. t. Ilwl, tht 'u"ber

of P' g.' . hle' .' 1 b. r e pl oCtd ., , ,I •• " . Il l /2 or I • • "

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t • • o .. oy . whlc' ar. ,0n UI • •• In S) ,n" ~ tnto I • 11 v '2

w.e .. 1111 • IIZ I. ,.u. fo r tU .ub .. . ~I •• ' $1 ' 52.' h •••

".ot , • • 11 UO .... Ju" p. do.I.&o • • .on for . 1"00<0 ,1 ...

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Th. followln9 r.~ . ,' . bl. conclu,lon. 'hould b. dr •• n f, ...

thl' uo •• lo:

• . Ih "u" ber N o f •• g .. wo1<o .,' b. " ' ph,oo .. t~ . ....

t he . .. to u 11.tted .u. r .ly. A ~o .. all t~to " Ill ~e

th on I 1ft Hde r h gu" .. Ut the ~ r epo.6er .. c. of .... 11

>ddre .. Ju"P"

~ . Horoo •• r 11 K to ... 11 (c ... por.d ,0 th o , I , •• f t~.

'<910n ) ' h. ~ I , .. t • • t I .... tl .." .. III (In ~ "" ... I )

n.t b< rodu'ed h~roPH1, by look-ohud r .pl ••••• ~ ...

- 2 1 _

c. fl •• ll y • I lolt. tIO. of ~ (O~ • • t o •• • l) .Il~.

u. to o.,I.,t th. p • • tltlon of i R •• d "" In t o r' 91 0~ '

.,,' 10.,0 . . ... ~ lth O .t dot.doriM . " dul, to. pu f • • •

• ,.c. o f U ••• Igo . ltn • . Tho. U o u., d lffl c.1t orob l •••

of t ho 4 ..... t.itlon of ' ooioo' II i" S~ •• ~ o f th .

U'pIoCI." .f Iho I • • g ...... 'I I n N" ... 0 . dl"9 to I " ~

<. , ng\ol oc,hlly of • • e gl ••• , ' ~ o • • " ••••.

hOI •••• ,ld ... ' I •• , wl1\ ~ . _0 ', l op •• "n, f or 'h

<00<\'0 .11 •• Of .O.-d •••• d ·p.9 1"9 . 1 ~o r\t" •• In ••• tlo. 1

of tho po pe • .

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•• qt o •• of t .I • ••• 1

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1.3.1 . R'Sl o", " rw cturod lite V

[ j , [ 1i- l [ I1 .1 .[.i'llo - l] I

~ ... U I ~ [

! • '

{ j _ l • ..• ~ · 1 , , • ." , . • • ~ - j.~ • '"' I • ". i-I . .. . o-, , , • ." Je·r-r • .. •

i- .

, . • '" L ~j . ".Hj·"'g"IiI · •• ~. 1"1 • . . • ' - 1 , , • ." ..

Pro o f:

o. QI " - Qj H;g n (il . ,

, (h l )t- ,

- rLPr ' ''j . j~_" , j(lJ · · r Y j "i ~n( j l·I.· · r) } o •

•• I . let . > .!:

• l) .

., " . I

j . ... , . ',., . ) - , '. . 7,. • r ' r " . ,- , I ",. . ' , . , • ) J. 0 . J., " ' • " .

I, ..... ~tI." til" fOl'ow! " q h O~\.lftOd f •• I, :

' , '4' IZ .' ,. , r (I J • '1" J • ) • J. 1 " . , " .

Z • • > . /! · l

" . ... I ", • ",., • ' J.1 " . , ... . • • I (7J , ·'t. J , ) J., " . r

.'0( ".,,· 1) , r (',,',\.,.' J. 1 'r'r

H .. U: ., • dl

• , ~. b, ... ""~tl."

""4 l iI"' f or. Q." . Q\, J J. ••

1 1. lei • • '\

" . " . " .

-, I "I .' . ',., • ) J 0 J.· •• 1 r 'r

'.,'1" 1 . , . . ,-, r (I J ' . 7 J• k • )

J ."r "

- 24 -

1. r > k/2 + 1 :

min(k-r,r-1) e 3 =.L (Y- J. a - Y-k+J· a )

J=l ' r ' r

2. r, k/2 + 1

k -1 k-1 e 3 = .L (Y- J. a -Y-k+J· a )

J=l 'r ' r . L . (Y- J. a -Y-k+J· a ) J=k-r+1 'r . ' r

= min(k-r,r-1)

L (Y- J. a - Y- k+J· a ) j=l ' r ' r

Thus e2 + e 3 = T ~ 0 concluding the proof.

Remarks:

a. Except from the assumption T ~ 0 all the conditions

of theorem 4 are consequences of the relations

b.

y. ~ Y l,a r i+sign(i),a r i + 0

Yi a = Y-i a , r ' r ( 0 ~ i ~ min(a'k-a a-I) ) "" r' r

which have been motivated in 1.4.3.2. for region V.

If Y i a ~ Yi+sign(i),a r for ; E (- k+ 1, k -1]

, r

then only the term d2 (or e2 ) of T might be negative.

In all cases of practical interest this term should be

compensated by the other terms. Moreover, these conside­

rations indicate that our assumptions are satisfied

rather for larger k than for small values of the

page size.

- 2~ -

<. I h • •• 1Idtt1 Of tU rehtlon Q 0, '

d. A •• ' U "it of t hoo,"_ 4 In tddttton to p.~, 80

of

.,. n •• d.d to .'ttblt,h

0t i •• ~ Q~I . , etc . .

2.3 . 2. 109tO", st"uctu,U H~. I

c. 'J.c •• > ~< .•

OI~t,>I , I , I t ·. I

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Proo "

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of the dl ..... lon' ~I d. In I.~.J.I. , they 'hould be

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t •• dltlo., .'1 b. w •• ~ ••• d I. ~'nl c • ••• bu •• On"

of th ••• '1 b ••• tIt ed wlthou, "Ore , do ; Ihl' " n l

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-p ' e "gl.9 or t ne .elghbo"" (P~)' . hlch h" bo •• p,op."d

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<hHHteri,I'9 l oo ' -.hud .e.h, ... ,t.) wh l e" 0" boslng

o. P~ ... l oadl.g r vle .. Il l be ,hdl.d In the fo llowing

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II II .1'1 10 I •• th.t the d •• ,., o.~,.,I". " 90" 10' O' A

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A Hudl .f ,<pTHU •• t .190.tth ~. h. , 1.tu , 1 ~U".,

I t ... ~. '""put o".

18H 51st . J. ~ 1 19U) a . 101.

(s) (_"ou, Eo G. J~ . u d h •• i.g , ,. J.,

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r ~) Ga,. h , r. , It .• ,o.t , J. uo ~otiu, ~.,

Aoo ly" ~'"n . 190,l th •• d. 9 .. 1100 .t_u'ton h

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( 10) Hot.<.td , p, 1. n~ G •• atd . J.,

Pro~ru ,ute.ctu,t n9 f oe .I" u" ~ .,'o'y.

IBM Sy<t. J . 10 n91l) ltiS 191.

( I I] )OHp', ~ . ,

An 0 •• 1,,1. o f p'9;n9 .nd 0' 09" _ boh •• lou r .

Tho C • • put •• Jo"rn, 1 II (IHo) 48 - 5~.

[Il] luJa." J.,

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In: Cooputo, .'(~;trctu"< >nO .~tvo r ~ ,.

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18'1 ~1H . J. , (lHo) 18 - 117.

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FJCC (1912) 591 - ~09.