On ine hasing robems orRegular n- onsHiroshi Fujiwara Kazuo Iwama Kouki Yonezawa

Kwansei Gakuin University Kyoto University Hokkaido Universityh-fujiwara @ ksc.kwansei.ac jp iwama @ kuis.kyoto-u.ac jp yonezawa@ mee.hokudai.ac jp

Abstract-We consider a server location problem with only is, GRD is optimal. Our result for a general n is sin_ for oddone server to move. If each request must be served on the exact 2.. . . . r2~~~~~~~~~~~amdc.l for even nz. InterestinglLy, there is a remarkablLeposition, there is no choice for the online player and the problem and 1v-is trivial. In this paper we assume that a reqnuest is given as a difference between odd rn's and even n's. (The reason is givenregion and that the service can be done anywhere inside the at the end of Section I.) (ii) Our analysis is tight, namely, thereregion. Namely, for each request an online algorithm chooses are request sequences for which the competitive ratio of GRDan arbitrary point in the region and moves the server there. coincides th bo vLus (iii) W lso i reliminOur main result shows that if the region is a regular n-gon, the observtio of ewrfun tion algorith a showithaticopttv rai of th greyagrtmi 1 forod observation of the work lFunction algorithmnand show that itcompetitive ratio of the glreedy algorithm iS .n for0 odd n.

1 forSeven1i Especially for a sqnare rgi 2the works well for hard examples against GRD. (iv) As for theand for even n. Especiallyufor a square re on,rlthe greedyuan lower bound, there is no good way of exploiting a specificalgorithm turns ont to be optimal. shape to improve 2 of [2]. However, if we drop the convex

I. INTRODUCTION condition from a simple polygon, then we can get a lowerbound of 2 -.

In the k-server problem the player manages k servers by Related Work. In a broader sense, the CNN problem [3],changing their location so that at least one of them serves the [4] can be regarded as a 1-server problem; One can just set therequest at each time step [1]. In this paper we study a 1-server region {(x, y) x or y =ry for a request (scene) onproblem on the xy-plane. One may think that if the player re s ,lR(r,x,.Atog h pe oudrmlssl ag,[owns only one server, there is no choice in server location and s tha withoa ntr resriton temcompetti ve,raitherefore the competitive ratio is always one. This is obviously dereato 9. Chrobak and Sgall provided a 5-compettitvetrue if te playerh to m t serverto the et position work function algorithm for the weighted 2-server problemof the request. hever, ifaitisenuh t movei somewhr which corresponds to a special case of the CNN problem [6].

nearctherequest,th entegpae dos thav choe. Namel [7] illustrated the application of the work function algorithm toeach request has ae ertainr regio. such that the request can be some problems belonging to metrical service systems (MMS).

It is also famous that the work function algorithm is (2N -

can choose an arbitrary point in the region in order to reduce 1)-competitive and optimal for general metrical task systemsthe total travel distance. Natural applications include allocation (MTS) where the size of the space is N [8]. See [9] for theof a relay broadcasting car to follow up consecutive incidents receut progress of the k-server problem.and that of a taxi in an old city with severe traffic restrictions.

Friedman and Linial first studied this problem and proved II. GREEDY ALGORITHMthat there exists a competitive online algorithm for the 1- In the 1-server problem a request R = (xi, yi) is givenserver problem for any convex region [2]. They began with somewhere on the xy-plane at each time step i 1, 2,a line chasing and extended the analysis to a half plane andgeneral convex bodies. However, they were not interested in

Then ar online algorithm ALG sets the sole server on as

oint

any specific shape of the region or specific values of theA (n region D Rs)cite cot of FLG is ined s

competitive ratio (only suh a result is an upper bound ofRm), the cost of ALG S defied as

28.53 for the competitive ratio of the line chasing problem). AS A>(T1N 141 hf Al * fh 1-f f * fh ~~~~~ALG(cu) = SA-1 + SAi-,Ai, )The problem has not appeared in the literature since then.

Our Contribution. In this paper we focus our attention on i=2

regular n-gons (without rotation) as the shape of the region where S is the initial location of the server and Ai-, Ai denotesand investigate how we can take advantage of the information the Euclidean distance between Ai-, and Ai. The offlineof the specific shape. It is shown that: (i) The greedy algorithm problem, i.e., minimization of SA, + EZm2 Ai-,Ai subject(GRD, which moves the server to the nearest position in the to Ai c Di given in advance for all i, is solved in polynomialrequest regioln fromn the previous positioln) works quite well timle if every regioln Di is colnvex [1L0]. In this paper we letfor a smnall ra For ilnstalnce, it achieves comupetitive ratios of regioln Di be the ulnioln of a regular ri-goln (ni > 3) with2, 2, and 3.24 for a regular triangle, a square, and a regular centroid I{ and its interior, The vi-gon does not rotate andpelntagon, respectively. Especially for a square region, the ratio therefore we caln assumne without loss of gelnerality that itsequals the lower boulnd for aln arbitrary colnvex regioln [2]? that bottolm side is always parallel to the x-axis. We set the lelngth

1-4244-0695-1/07/$25.OO ©2007 IEEE. 36

of sides in the rn-gon as one. It should be noticed that the size........... ........................

of the n-gon does not matter to the online competitiveness. 'V,V',,V'Vl

The greedy algorithm for the 1-server problem is defined asbelow. Ai 2n PiAlgorithm GRD: For each request i, (i) if the server's previousposition Ai1 is not in Di, then move the server to X e Di AGRDi FFisuch that minimizes Ai-,X. (ii) Otherwise, do not move theserver.

For regular polygons one can easily see that there are twotypes for the server's behavior: The server arrives on one ofthe sides after vertical movement against that side, or on one Ai_, Pi-of the vertices. Fig. 1. Worst case for the upper bound. (nr odd)We use the definition of the competitive ratio as in [11],

[12]: The competitive ratio of an online algorithm ALG iS c ifthere exists a constant b such that, for all input sequences a from the same position since x2+ y2 < 4(x, y). Moreover,

ALG(u) - c OPT(u) < b, (2) for a fix (D the server of OFF will stop immediately after

where OPT is an optimal offline algorithm. reaching the region in order to minimize AOFFi. Thus we

Lemma 1: For a regular n-gon (n > 3) the competitive ratio haveof GRD is at most £ < AGRD1+ AOFF2-AGRD,2 0)

{ 1itl :Wt if n is odd, -c AOFF,sin 2

nif n is even. ( ) 2cos2 O2

t sln /1rl = i\~~~~~~~~~~~~GRDi+ i rn0OF2iGRDisnProof. We begin with an odd n. Let us look into the _- AOFF (6)

behavior of GRD and an arbitrary offline algorithm OFF foran input sequence cr. We write the positions of GRD'S server Note thatand OFF'S server as A1,A2, .. ,Am, and P1,P2,.. ,Pm, n-k12 2 IFrespectively. Since it may occur for some of the requests that q(1, 0) =.. sin = .2X.sin = . 2n2 (7)GRD moves the server while OFF does not, a simple piecewise k= n k1 sincomparison does not work. Therefore, we must adopt somepotentialsfun tioesnotforkanamortedore,wea iste can semetha The remainder is to see OFF's destination on the side of the

potetia fuctinfr anamotizd aalyis. ne an ee hatn-goln. The right-hand side of (6) is mnaximlized tothe server of GRD always approaches the server of OFF when -OFF does not change the position. So some kind of distance ( 1between the two servers appears to help, but it turns out 1 - c2+1- sA 2 AGRDi (8)that simple ones are insufficient. Our choice of the potential 2nfunction is when

(X,.. 2kwr 2k7 ( 2 cos 2 X 2,y) yL x sin : :y cos ( 2-)i : 1 - 2n C.

k=OT n si7-\.T (9.

=° V \ Slrl ~~~~~~~~~~~~~~~~nJwhere (x, y) is the displacement from the position of OFF'S Hence, the value of E is always non-positive ifserver to that of GRD's. (x, y) represents the total length of 1all projections of vector (X, y) to the normal of each side of the c > sin (10)n-gon. It is easy to confirm that (x, y) satisfies the distance Sil 2naxiom and (Ax, Ay) A * (x, y). We use the notation of (For c 1 E is maximized to zero when ZAiPPiAiAGRDi and AOFF, as the cost of GRD to serve the i-th __ as shoWnin Figure 12)request and that of OFF, respectively, and bi as the value of For an even we make a similar analysis. We useb(x, y) after the i-th movement. We can obtain a competitiveratio as the minimum value of c such that n/2-1 2kT 2kw

(x, )=z xsin +ycos ()E =AGRDi+ i -iI-cAOFFt (2 < n<m) (5) kcO n n

is non-positive for any possible movement. Figure 1 illustrates as a potential function instead of represeknts the totalthe case where the largest c is lneeded to satisfy (5): Without lelngth of projectiolns of vector (x, y) to the lnorlmal of each sideloss of generality we suppose that for request i GRD moves as well. However, it is sufficient to sum up half of projectionsthe server to olne of the sides. The value of £ becolmes largest since there are n2 pairs of parallel sides inl aln eveln a-goln. Thewheln bi-l is zero, i.e., both two servers of GRD alnd OFF start worst situatioln here is giveln by repla ilng the alngle 2,n with

37

TABLE I

A- .................................. .......ICO)MI_TITIVE RATIO OF THE GREEDY ALGO)RITHM FOR REGcULAR

A - .............................................SSeESEESSeSSeESSESSeeSxx//xx/xx//x//xx/xx//xx/xx/S- OB E,I,I I,I I,I I,I I,I I,I,I I,I,I I,I,I I,I,I I,I,I I,I,I I,I IC o m p etitiv eratioSA ............................................. .... I .....32 _

.. 4.1.414213562373095Ax x x x x x x x............... ............... ... ........... 3 2 3606797749978

7r 1 \ \\ /// s;>X wsss osxwSwI..........e. 3 2

4 1.414213562373095

.......... ....................10 3.236067977499789

7. ...........................in Figur 1 As aresut itis obtained that the competitiveratio is atmost for even Consequently, we derive.n pns t that R i a o G.

onlin alorth from th///xxxEeSSSSSSSSSSSSSSSSSSSSSS/ folwnlem., OPT. __ (15)Le m 2 ([]) ThxxxxEere e x it no onin aloih wh s1S-. ri 2meee./ ./////././////////////

Alhog2R maysnoLt be optimaforothloer values( ofd 9vSiiary we8have44 (

Leimm3:ur ForA a regular i ri-gobane(r t3oat 5the GRDpe1i(16)omeiieratio o Ris at least OP sine l

sin2t ve (12 Thorsemunty1: Forthe1sre rbe h ihopttvIn~~~~ipatcua itr i is even rati of then greed algoith iSnopia

* ' ^ GEZD2s~~~~~~r~ ins odd,PrleloofhrFirstly, whe proveowthe lemmra. fo odd =v e 1f (17)usablllthesiesofth ivgo, egnnig ro te* sinT i-co v isievn

bottmsid parallel) tThee r-xis asn side1,e nexrth sidsein2

counerclckwie assid2,.nd.s.on.ntilsidiv........Then....we2.......... .for the case of a s uare the r al...orih in o. tim..al...........drawinga zigzag between side 1 and side ofen another. onin algorithm.74

U1.iv-gon1 and ie 4. These sides form an angle of2.Alhuhazgasoecnse h cmeiiertoo R o euatAvltocan Drla obefoce balention ofo othesetoreqauests it od 2vgo eqastano eua i-o.Frhroe hispsil ooti ve agrgs ai by th folwn competitiv?erativoe 2v+1-o s tlattieta o

trckOan eaho reqestwhesnlyide thelowera1 is-o slgtytoaiggnhTitifeec.n efra eo R s xlie

theright s at t.....h e dc.fosid. to s.id. as f s We an od p n hs.no p' of p

corueisiveqartio thtfGRom sideat~latosd1(seFgr2.sis,aevn-nhsOP pair ofprlenie.In

maytrainsw hv2lr tahe anglhe oneqenty beomeshrfo1dSoygn n r

foreveplygns Thrfr ifo anisodd,plgn tesreOrDoof: (isty+wose prose th lemm f aapo eorod.Let slowly to suhanotma oito hah

-S '0-gf............... .7. .......... .t-||*|......................X.7. .. 'f7

sfterverqdesnt needgtoni movreuany m-ore. >)EpcilL

(13)~ ~ ~ 8 .13299725

On ~ .' o the aeothehand,ar theoptmaloffin algorithmmhas mereltoII.WOKUCTONAGOITM

move to Wnput sequeucessu cthanbne serveddhwrk algorithm plays a. significant rolefor

* * 1 ,~~~~~~~~~~~~ere locatin prblm [6] [7 as we............llnas formercl task

iOT aigur14)Asasystemt,itiss o(MSinedc8 Fthe ompetitive roble th wor functi

tratioisaeven l (ni defied as them cos of teser ol requests3n8paticularfor=42itisturnstthatfor

1 onslneagrthm fomtl te followinage lemma.l by th5llw*Lemm............a2ra-gon.- Thidflerenexit noline aelgoritmacwose nR iSepane

rcmeitvOn ratioieustwemaller thane- fowr ranygonvereUgiony Whnnitoee ubr uc izgi rw ewethe Rgh siotha andsideenl4cbac These sides for an angeo2Althug* GR+D ., > .............. l maynotl bel optma fo other vaue ofale nies If on'

we can equlhow thattheanlysis inLemmascL 1isetight.Simlalywe............ hve

Loemthat3-oahrpegurn-gondosn mov at orl Suppose theatDcompetitive *rati of GR isatza letwee toP sie sfring assa'T nlL spsil

manylteratlLolifs ishavdndI2n,~~ ~~~~~~~loe(1poygns Thereoremf:orthansrepoddl olyonthetih scmetitver

si ra i n is e-O even. rati r eeoofovegredy agmore.i

uslben lthesiend,tes ofptheaoffline beginnithm hasomerl the III. if UCIN iseven,botetomPshownparall togure x2axsic allsiequ test nextb sierved terqes eini eglrngn(>3.Epcalcounterclockwise~ ~ ~ ~ ~~~ ~~Th asr sidetol 2,olr andy so on until sidetnroleefor

consderan npuseqenc suh tat he srve ofGRDmovsefrvter lcaseiofn squarem6,th[7edaselgoih isfor optrimaltsdrawing~~~~OPa ziza bewe sid I4 anderuside [8].2 For anoheonlineealgoritthhwokm.rL

slrlL~~ AsoneX cansefne,a the competitivelratios of GeRD foral reguestsn-on Tes ids or a nge f lhogha iga od -gneqal tatfr rgua 2-gn.Futeror,8h

II~~~~~~~~~~~s ...IIIII

4~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ P

~~~~~~~~~~~~~~.~~~~~~~~~~~~~Fomaly given, XC DXXX E,Fig 4 RT wok bete o a zizatri

t I / / _ X x x x x x x x x x x/xSxSs,/N I ........ .. ....

w~(X) m * SB -* B1B -F B SS1_X

defined as follows, than

Proof..We dics odIt Lt{ he th fail ofInesI

..... . . . . . . . . . . . .X I/

FiX u 3h CrueliAput for RTR. a- PH A)forod and A(PHcosn siny

///////..,,,,,,.....,1t 2m.. i. -I and finally reachrogposetion X [Di for requestt r s e h s. irf is hpt. Formally, given X c D g, Fig. 4. RTR works better ftvra zigzag trip.

wi(X)of R insat SBl + eBlreBa+ Bi_eX .BjDJ<<_ 1= a

t

_ir _ (1-=2os )~ J/ +-sin-itial location of S'-a')for even as andd 0 < as <

2 2 t 2

that the server is initially located at S~(- tan ~-' 2tan -) RT /TR1RTR.Pleapseiealthatparaequest rerset teenridothe-gkfncinagoihon i hR(ost (atio- of an 2sin- no la(19)

and the lengthof each side i one. This inpt sequence fores OPT(tn2) PT2(siny ,T(i'

ReTtneo as lOlgOWS. than oveProof: We discuss odd n. Letweir be the family of lines

Algorithm WFA,A1 For each request i, (i) if the server's 1i y=-tan . - The location of RTR'S server

RTRQ7i..:.)increase.s hy one......,whE---lecos .F 1 increass h:.._.yho-ld -..sfor RTR(K)............

to 1Dis* serven rS finRemarkm 1, thos raNthattioRofTRwX) approahes

input sequen e o'~~~~~~ given as R (1, 1) for oddH;'0 for dd andd 1 Pif -o Hsin-2tan- ens - 1~~~si2 sin2 - ~ sin n

dono mo ) forserven. s,adtesrei nta oaino ~ fcS2

coffesponlds to the case of a oo. Let us denote this through lines 1k (I < k < i -1) onLthe tour from S2 toalgorithm as R%TR. HE. One can observe that each HIi represenlts the image of Ai

-ofjR-a).als < e ,ratioofRT iS at lLesin-E. each request untilL the (nl-th request. Finalkly, the server

cosr odis -CosandR1(l, 2 ) sinf oreven i's, andtheqserver'sthat the server is ilnitially located at S1(_ slSl n n_I7-732tarn _E 2 taii -. n1

anLd the1length of eah side is one. This ilnput sequelnce forces nP(r) u2 llRTR to chase a sinlglLe vertex of the polygonl as Al, A2,A3 . . ..'.' ' ~~~for alUl odd n. As for even n and ,T and S', RTR'S server(see Figure 3). Onl the other hand, an offline algorithmnOFF 2 2

' . ~~~~~~~~~~~movesa total distance of a n-, whereas the optimal offlinemay mnove the server as PI,P2,P3,.. where each Pi lLS the2' ~~~~~~~~~~costlLS .a after the ()-th request. An upper boulnd of2midpoint of AiAi+l . On each request, except the first onle, cost SllRTR((71) increases by one, while cost OFF((71) increases by holds for op(2) as well. 0sin .As this iteration continues, cost ratio RT(1)converges Remark 3: For input sequence ci and initial position of then ?OFF(cTl WF)ato -.i,.W server SI in Remark 1, the cost ratio of Wop, proachesRemark 2:- Let us colnsider, for odd ra and 0 < a < sinLj,,, ,s, w a,9S ; m ' *;s ff ' ^; { ) ' *' ' i ,', .: 9 i S t1. *- * ! e ; t si~~~~~~~~~~~~~

I

____igigo __sesesxwSxwSesesesxwSw<Xez xwSxwXe......

/ // \ CX/,,X ....... XXSSSSS/..... ........ //XI\|\EEEeeeE

..... .......Ad S S S X X Sx \ X E ESE X, C/CyXyES S S i * I \ I \ ........ESeSeSeSE~~~~~~~~~~~~~

G.. ........

..... ........HeXXSXyXySSSSXyXyXySSSSSXyXyXySA ........

. . . . . . . . . . . . . . .y y \.iE..0 . . . . . . . . . . .

a

Fig ....5.Reio fo There 2. rearedasa -ererprblmwihuc.arein,ha.aloe

bound of.6-. 1 3]. This sectio shows that if we just drop

2 th cove codto fro a bone rein the we caCXXXXXXzwEgetEProof For un - _ ci asiml clulto imle a betteryxxywe low3XerXxSwebound.S3Xe

A2A,..n igre3,drwig zgzg ftr eahig.1.ovs.hi.rgin vrtcalyupan.dwnbewen.w

straigt to he vetex. s theiteraion cntinue, thecost atio ineFEgsuchthatethis iorizfanontllyeapaorithfrm.E(rF)b

foreachrequest covRegesion Totalore2 reandeFV d-s ToIselvect suchle awoitioiscalwayon hsposiloerbcost atio FA+i alsoconveges t thisvalue * coosngaf suvfficientl small angleo ash0w( tFhat if eGI)tro

PT(uu) U is~~~~~~thepoveosditionon froH symmdetri tegon,thV.ecne

Remarkof2 Fsg st 2th W fora some calperforms wellifothe lower bound.

input ine Figure 4b(ehallo tha RTR isWFAh) WithaSom laRgTe freorm2,httin line EGhrinste,hittin fHrwincteh 2andnlncervwe cant actuall obtasing a bigettertresulOthewietan eR(omitted). rtmcno aheeacmeiivaif2-E

Note4that inpu sequrene32dawnd oigzrepresent cruelstathions somon.s (thes greedyn serverjutialmove ban koandotbetweentwfor tRDaist weqelastoei em.Indrdalyapoceed the ostdlra ostioBand F.) ilutrshoud be notuedta ifLth sErver,dsayinsdeS

bestocolmes. paety hpiatasOtraegitbeomes area EFUV,na thenitinomust morverat Leat disane onepoitineacsPtraigtt)h etx steieatootne,tecsai stnep.GAsounot thatith costhoiofntheyaprofliesrver isat(ostd.y1oecfrevuenst Monvrgeovrs codnto Remark ot3,WF anTherFore, the Tonln servert must gopousideoi EFUVy eventuallbytan -~ ~ ~ ~ osV- 2sn27r- in2-ovroestedFfiutAhoni igr ,whrT incursn corderg to aufchievela const antgometitive ratio So suppose

mchs cost. SinTecos ratio coFA,ge(otiJau.i)stepsiino Hsymti oV

RrsgsaFoP ineasesmonoton thatian a shonlinetserv gove outside EeUV foe sth first

winpthirespreto awhialeha0< RiS WFA,one canh seme thatga tiome htingsepk lnaelyG inthsstep, hittimoves fro Atep to Xnsallercan providesbtinbetterpefrmne.sFro thisGR (obstervaion (oon.EG)eoreAkdi torvY (ontGI.oines FVc >ndfrt andt0eiswoe thavet conjuectureneTha Wand attin'bereenttcrer comptiativensml,btA XanA Yaratestd- Thcotf

rati thaGR or TR y setin cprelyUnotaey,hE onl.)Ine server insep kotis thusat lfteastrv(k-s1)y+(nsde-theGRperformance ofnFA for general3 Inpud,thsequecestsemr hlatecstfthiflnosresmn(,d.IteolwmRDuch moreDdificlt'tate FV hni utmv tlatdsac nnec

OPT(U2) tan -11-OPT((k-)-u'd-c 1+

IV~~XXxx.LOWER BOUNS FOR NON-CONVEX REGIONr min(k, d) - dxx

It is alssaenaturlsgoaltteseektatetteroloerobound han forenysk>v1ry settin do2/cd.2a ioeen[]bepotngspreovr ifccshape. HowReverkthi WAppearse h nin evrmutgutieEUeetalharerhanit ook iftheshPe(lishnotconex.rNtettatri VConlnCLUDINGrRoeMARKSeFVotefis

yXsinSXXy (ig.EG. oravo ofaAnLn aUoih

one hallow a conjcavre tand unbonde regaion,betherlowperiivbou bthAndAppretyayrblm aremainto beasttack.Ted icluding

beiomeslhanGRgeor.RRA ly, setheCn proberlem whichtucane a() orde foserveranalpysistofFat lean (ii)thinvesta loferthattheperformanebehviof WFAforgenerainputh saequencs ses wleTheor : Forthe serverismina kshape Itwhiheanyfoll

muchvmrjutkepdifchaigasiglere.t.ews?th nLn talgrth[ (kllo achev a(d netv rati of 2-

srighVotevre.Aste.eailoltne,tecsLOWERBOUNDSFOR NONCOVE REGIO mint,ddshrzntyaatfo o )b

other shapes than regular rn-gons, especially a circle and some [6] M. Chrobak and J. Sgall, "The weighted 2-server problem," in Proc.simple shape which is not convex. STACS '00, 2000, pp. 593-604.

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