Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

Transcription

1 Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr, Boaz Patt-Shamir, Dror Rawitz To cit this vrsion: Zvi Lotkr, Boaz Patt-Shamir, Dror Rawitz. Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal. Susann Albrs, Pascal Wil. STACS 2008, Fb 2008, Bordaux, Franc. IBFI Schloss Dagstuhl, pp , HAL Id: hal Submittd on 3 Fb 2008 HAL is a multi-disciplinary opn accss archiv for th dposit and dissmination of scintific rsarch documnts, whthr thy ar publishd or not. Th documnts may com from taching and rsarch institutions in Franc or abroad, or from public or privat rsarch cntrs. L archiv ouvrt pluridisciplinair HAL, st dstiné au dépôt t à la diffusion d documnts scintifiqus d nivau rchrch, publiés ou non, émanant ds établissmnts d nsignmnt t d rchrch français ou étrangrs, ds laboratoirs publics ou privés.

2 Symposium on Thortical Aspcts of Computr Scinc 2008 (Bordaux), pp RENT, LEASE OR BUY: RANDOMIZED ALGORITHMS FOR MULTISLOPE SKI RENTAL ZVI LOTKER 1, BOAZ PATT-SHAMIR 2, AND DROR RAWITZ 3 1 Dpt. of Communication Systms Enginring, Bn Gurion Univrsity, Br Shva 84105, Isral. 2 School of Elctrical Enginring, Tl Aviv Univrsity, Tl Aviv 69978, Isral. 3 Faculty of Scinc and Scinc Education & C.R.I., Univrsity of Haifa, Haifa 31905, Isral. addrss: zvilo@cs.bgu.ac.il,boaz@ng.tau.ac.il,rawitz@cri.haifa.ac.il hal , vrsion 1-3 Fb 2008 Abstract. In th Multislop Ski Rntal problm, th usr nds a crtain rsourc for som unknown priod of tim. To us th rsourc, th usr must subscrib to on of svral options, ach of which consists of a on-tim stup cost ( buying pric ), and cost proportional to th duration of th usag ( rntal rat ). Th largr th pric, th smallr th rnt. Th actual usag tim is dtrmind by an advrsary, and th goal of an algorithm is to minimiz th cost by choosing th bst option at any point in tim. Multislop Ski Rntal is a natural gnralization of th classical Ski Rntal problm (whr th only options ar pur rnt and pur buy), which is on of th fundamntal problms of onlin computation. Th Multislop Ski Rntal problm is an abstraction of many problms whr onlin dcisions cannot b modld by just two options,.g., powr managmnt in systms which can b shut down in parts. In this papr w study randomizd algorithms for Multislop Ski Rntal. Our rsults includ th bst possibl onlin randomizd stratgy for any additiv instanc, whr th cost of switching from on option to anothr is th diffrnc in thir buying prics; and an algorithm that producs an -comptitiv randomizd stratgy for any (non-additiv) instanc. 1. Introduction Arguably, th rnt or buy dilmma is th fundamntal problm in onlin algorithms: intuitivly, thr is an ongoing gam which may nd at any momnt, and th qustion is to commit or not to commit. Choosing to commit (th buy option) implis paying larg cost immdiatly, but low ovrall cost if th gam lasts for a long tim. Choosing not to commit (th rnt option) mans high spnding rat, but lowr ovrall cost if th gam nds quickly. This problm was first abstractd in th Ski Rntal formulation [10] as follows. In th buy option, a on-tim cost is incurrd, and thraftr usag is fr of charg. In th rnt option, th cost is proportional to usag tim, and thr is no on-tim cost. Th dtrministic solution is straightforward (with comptitiv factor 2). In th randomizd Ky words and phrass: comptitiv analysis; ski rntal; randomizd algorithms. Th scond author was supportd in part by th Isral Scinc Foundation (grant 664/05) and by Isral Ministry of Scinc and Tchnology Foundation. c CC Z. Lotkr, B. Patt-Shamir, and D. Rawitz Crativ Commons Attribution-NoDrivs Licns

3 504 Z. LOTKER, B. PATT-SHAMIR, AND D. RAWITZ modl, th algorithm chooss a random tim to switch from th rnt to th buy option (th advrsary is assumd to know th algorithm but not th actual outcoms of random xprimnts). As is wll known, th bst possibl onlin stratgy for classical ski rntal has comptitiv ratio of In many ralistic cass, thr may b som intrmdiat options btwn th xtrm altrnativs of pur buy and pur rnt: in gnral, it may b possibl to pay only a part of th buying cost and thn pay only partial rnt. Th gnral problm, calld hr th Multislop Ski Rntal problm, can b dscribd as follows. Thr ar svral stats (or slops), whr ach stat i is charactrizd by two numbrs: a buying cost b i and a rntal rat r i (s Fig. 1). Without loss of gnrality, w may assum that for all i, b i < b i+1 and r i > r i+1, namly that aftr ordring th stats in incrasing buying costs, th rntal rats ar dcrasing. Th basic smantics of th multislop problm is natural: to hold th rsourc undr stat i for t tim units, th usr is chargd b i + r i t cost units. An advrsary gts to choos how long th gam will last, and th task is to minimiz total cost until th gam is ovr. Th Multislop Ski Rntal problm introducs ntirly nw difficultis whn compard to th classical Ski Rntal problm. Intuitivly, whras th only qustion in th classical vrsion is whn to buy, in th multislop vrsion w nd also to answr th qustion of what to buy. Anothr way to s th difficulty is that th numbr of potntial transitions from on slop to anothr in a stratgy is on lss than th numbr of slops, and finding a singl point of transition is qualitativly asir than finding mor than on such point. In addition, th possibility of multipl transitions forcs us to dfin th rlation btwn multipl buys. Following [2], w distinguish btwn two natural cass. In th additiv cas, buying costs ar cumulativ, namly to mov from stat i to stat j w only nd to pay th diffrnc in buying prics b j b i. In th non-additiv cas, thr is an arbitrarily dfind transition cost b ij for ach pair of stats i and j. Our rsults. In this papr w analyz randomizd stratgis for Multislop Ski Rntal. (W us th trm stratgy to rfr to th procdur that maks onlin dcisions, and th trm algorithm to rfr to th procdur that computs stratgis.) Our main focus is th additiv cas, and our main rsult is an fficint algorithm that computs th bst possibl randomizd onlin stratgy for any givn instanc of additiv Multislop Ski Rntal problm. W first giv a simplr algorithm which dcomposs a (k + 1)-slop instanc into k two-slop instancs, whos comptitiv factor is 1. For th non-additiv modl, w giv a simpl -comptitiv randomizd stratgy. Rlatd Work. Variants of ski rntal ar implicit in many onlin problms. Th classical (two-slop) ski rntal problm, whr th buying cost of th first slop and th rntal rat of th scond slop ar 0, was introducd in [10], with optimal stratgis achiving comptitiv factors of 2 (dtrministic) and 1 (randomizd). Karlin t al. [9] apply th randomizd stratgy to TCP acknowldgmnt mchanism and othr problms. Th classical ski rntal is somtims calld th lasing problm [5]. Azar t al. [3] considr a problm that can b viwd as non-additiv multislop ski rntal whr slops bcom availabl ovr tim, and obtain an onlin stratgy whos comptitiv ratio is Bjrano t al. [4], motivatd by rrouting in ATM ntworks, study th non-additiv multislop problm. Thy giv a dtrministic 4-comptitiv stratgy, and show that th factor of 4 holds assuming only that th slops ar concav, i.., whn th rnt in a slop may dcras with tim. Damaschk [6] considrs a static vrsion

4 RENT, LEASE OR BUY 505 cost b 4 b 3 b 2 b 1 s 1 s 2 s 3 Figur 1: A multislop ski rntal instanc with 5 slops: Th thick lin indicats th optimal cost as a function of th gam duration tim. s 4 t of th problm from [3], namly non-additiv multislop ski rntal problm whr ach slop is bought from scratch. 1 For dtrministic stratgis, [6] givs an uppr bound of 4 and a lowr bound of ; [6] also prsnts a randomizd stratgy whos comptitiv factor is 2/ln 2 = As far as w know, Damaschk s stratgy is th only randomizd stratgy for multislop ski rntal to appar in th litratur. Irani t al. [8] prsnt a dtrministic 2-comptitiv stratgy for th additiv modl that gnralizs th stratgy for th two slops cas. Thy motivat thir work by nrgy saving: ach slop corrsponds to som partial slp mod of th systm. Augustin t al. [2] prsnt a dynamic program that computs th bst dtrministic stratgy for non-additiv multislop instancs. Th cas whr th lngth of th gam is a stochastic variabl with known distribution is also considrd in both [8, 2]. Myrson [12] dfins th smingly rlatd parking prmit problm, whr thr ar k typs of prmits of diffrnt costs, such that ach prmit allows usag for som duration of tim. Myrson s rsults indicat that th problms ar not vry closly rlatd, at last from th comptitiv analysis point of viw: It is shown in [12] that th comptitiv ratio of th parking prmit problm is Θ(k) and Θ(log k) for dtrministic and randomizd stratgis, rspctivly. Organization. Th rmaindr of this papr is organizd as follows. In Sction 2 w dfin th basic additiv modl and mak a fw prliminary obsrvations. In Sction 3 w giv a simpl algorithm to solv th multislop problm, and in Sction 4 w prsnt our main rsult: an optimal onlin algorithm. An -comptitiv algorithm for th non-additiv cas is prsntd in Sction Problm Statmnt and Prliminary Obsrvations In this sction w formaliz th additiv vrsion of th multislop ski rntal problm. A k-ski rntal instanc is dfind by a st of k + 1 stats, and for ach stat i thr is a buying cost b i and a rnting cost r i. A stat can b rprsntd by a lin: th ith stat corrsponds to th lin y = b i + r i x. Fig. 1 givs a gomtrical intrprtation of a multislop ski rntal instanc with fiv stats. W us th trms stat and slop intrchangably. Th rquirmnt of th problm is to spcify, for all tims t, which slop is chosn at tim t. W assum that stat transitions can b only forward, and that stats cannot b 1 It can b shown that stratgis that work for this cas also work for th gnral non-additiv cas (s Sction 5).

5 506 Z. LOTKER, B. PATT-SHAMIR, AND D. RAWITZ skippd, i.., th only transitions allowd ar of th typ i i + 1. W strss that this assumption holds without loss of gnrality in th additiv modl, whr a transition from stat i j for j > i + 1 is quivalnt to a squnc of transitions i i j (cf. Sction 5). It follows that a dtrministic stratgy for th additiv multislop ski rntal problm is a monoton non-dcrasing squnc (t 1,...,t k ) whr t i [0, ) corrsponds to th transition i 1 i. A randomizd stratgy can b dscribd using a probability distribution ovr th family of dtrministic stratgis. Howvr, in this papr w us anothr way to dscrib randomizd stratgis. W spcify, for all tims t, a probability distribution ovr th st of k + 1 slops. Th intuition is that this distribution dtrmins th actual cost paid by any onlin stratgy. Formally, a randomizd profil (or simply a profil) is spcifid by a vctor p(t) = (p 0 (t),...,p k (t)) of k + 1 functions, whr p i (t) is th probability to b in stat i at tim t. Th corrctnss rquirmnt of a profil is k i=0 p i(t) = 1 for all t 0. Clarly, any stratgy is rlatd to som profil. In th squl w considr a spcific typ of profils for which a randomizd stratgy can b asily obtaind. Th prformanc of a profil is dfind by its total accrud cost, which consists of two parts as follows. Givn a randomizd profil p, th xpctd rntal cost of p at tim t is R p (t) df = i p i(t) r i, and th xpctd total rntal cost up to tim t is t z=0 R p (z)dz. Th scond part of th cost is th buying cost. In this cas it is asir to dfin th cumulativ buying cost. Spcifically, th xpctd total buying cost up to tim t is Th xpctd total cost for p up to tim t is B p (t) df = i p i(t) b i. X p (t) df = B p (t) + t z=0 R p (z)dz. Th goal of th algorithm is to minimiz total cost up to tim t for any givn t 0, with rspct to th bst possibl. Intuitivly, w think of a gam that may nd at any tim. For any possibl nding tim, w compar th total cost of th algorithm with th bst possibl (offlin) cost. To this nd, considr th optimal solution of a givn instanc. If th gams nds at tim t, th optimal solution is to slct th slop with th last cost at tim t (th thick lin in Fig. 1 dnots th optimal cost for any givn t). Mor formally, th optimal offlin cost at tim t is opt(t) = min i (b i + r i t). For i > 0, dnot by s i th tim t instanc whr b i 1 + r i 1 t = b i + r i t, and dfin s 0 = 0. It follows that th optimal slop for a gam nding at tim t is th slop i for which t [s i,s i+1 ] (if t = s i for som i thn both slops i 1 and i ar optimal). Finally, lt us rul out a fw trivial cass. First, not that if thr ar two slops such that b i b j and r i r j thn th cost incurrd by slop j is nvr lss than th cost incurrd slop i, and w may thrfor just ignor slop j from th instanc. Consquntly, w will assum hncforth, without loss of gnrality, that th stats ar ordrd such that r i 1 > r i and b i 1 < b i for 1 i k.

6 RENT, LEASE OR BUY 507 Scond, using similar rasoning, not that w may considr only stratgis that ar monoton ovr tim with rspct to majorization [11], i.., stratgis such that for any two tims t t w hav j j p i (t) p i (t ). (2.1) i=0 Intuitivly, Eq. (2.1) mans that thr is no point is rolling back purchass: if at a givn tim w hav a crtain composition of th slops, thn at any latr tim th composition of slops may only improv. Not that Eq. (2.1) implis that B p is monoton incrasing and R p is monoton dcrasing, i.., ovr tim, th stratgy invsts non-ngativ amounts in buying, rsulting in dcrasd rntal rats. 3. An -Comptitiv Algorithm 1 In this sction w dscrib how to solv th multislop problm by rducing it to th classical two-slop vrsion, rsulting in a randomizd stratgy whos comptitiv factor is 1. This rsult srvs as a warm-up and it also givs us a concrt uppr bound on th comptitivnss of th algorithm prsntd in Sction 4. Th cas of r k = 0. Suppos w ar givn an instanc (b,r) with k + 1 slops, whr r k = 0. W dfin th following k instancs of th classical two-slops ski rntal problm: in instanc i for i {1,...,k}, w st i=0 instanc i: b i 0 = 0 and ri 0 = r i 1 r i ; b i 1 = b i b i 1 and r i 1 = 0. (3.1) Obsrv that b i 1 = ri 0 s i, i.., th two slops of th ith instanc intrsct xactly at s i, thir intrsction point at th original multislop instanc. Now, lt opt(t) dnot th optimal offlin solution to th original multislop instanc, and lt opt i (t) dnot th optimal solution of th ith instanc at tim t, i.., opt i (t) = min{b i 1,ri 0 t}. W hav th following. Lmma 3.1. opt(t) = k opti (t). Proof. Considr a tim t and lt i(t) b th optimal multislop stat at tim t. Thn, k opt i (t) = b i 1 + r0 i t i:s i t i:s i >t = (b i b i 1 ) + (r i 1 r i ) t = b i(t) + r i(t) t = opt(t). i:s i t i:s i >t Givn th dcomposition (3.1), it is asy to obtain a stratgy for any multislop instanc by combining stratgis for k classical instancs. Spcifically, what w do is as follows. Lt p i b th 1-comptitiv profil for th ith (two slop) instanc (s [10]). W dfin a profil ˆp for th multislop instanc as follows: ˆp i (t) = p i 1 (t) pi+1 1 (t) for i {1,...,k 1}, ˆp 0 (t) = p 1 0 (t), and ˆp k(t) = p k 1 (t). W first prov that th profil is wll dfind. Lmma 3.2. (1) p i 1 (t) pi 1 1 (t) for vry i {1,...,k} and tim t. (2) k i=0 ˆp i(t) = 1.

7 508 Z. LOTKER, B. PATT-SHAMIR, AND D. RAWITZ Proof. By th algorithm for classical ski rntal, w hav that th stratgy for th i instanc is p i 1 (t) = (t ri 0 /bi 1 1)/( 1). Claim (1) of th lmma now follows from that fact that b i 1 /ri 0 = s i > s i 1 = b1 i 1 /r0 i 1 for vry i {1,...,k}. Claim (2) follows from th tlscopic sum k k 1 ˆp i (t) = p 1 0(t) + (p i 1(t) p i+1 1 (t)) + p k 1(t) = p 1 0(t) + p 1 1(t) = 1. i=0 Nxt, w show how to convrt th profil ˆp into a stratgy. Not that th stratgy uss a singl random xprimnt, sinc arbitrary dpndnc btwn th diffrnt p i s ar allowd. Lmma 3.3. Givn ˆp on can obtain an onlin stratgy whos profil is ˆp. Proof. Dfin ˆP i (t) df = j i ˆp j(t) and lt U b a random variabl that is chosn uniformly from [0,1]. Th stratgy is as follows: w mov from stat i to stat i + 1 whn U = ˆP i (t) for vry stat i. Namly, th ith transition tim t i is th tim t such that U = ˆP i (t). Thus w obtain th following: Thorm 3.4. Th xpctd cost of th stratgy dfind by ˆp is at most optimal offlin cost. 1 tims th Proof. W first show that by linarity, th xpctd cost to th combind stratgy is th sum of th costs to th two-slop stratgis, i.., that Xˆp (t) = k X pi(t). For xampl, th buying cost is k k 1 k k Bˆp (t) = ˆp i (t) b i = (p i 1(t) p i+1 1 (t)) b i +p k 1(t) b k = p i 1(t) (b i b i 1 ) = B p i(t). i=0 i=0 Similarly, Rˆp (t) = k R pi(t) by linarity, and thrfor, t k ( t k ) Xˆp (t) = Bˆp (t) + Rˆp (z)dz = B p i(t) + R p i(z) dz = z=0 z=0 Finally, by Lmma 3.1 and th fact that th stratgis p 1,...,p k ar conclud that k k Xˆp (t) = X p i(t) 1 opti (t) = which mans that ˆp is 1 -comptitiv. 1 opt(t) k X p i(t). 1-comptitiv w Th cas of r k > 0. W not that if th smallst rntal rat r k is positiv, thn th comptitiv ratio is strictly lss that 1 : this can b sn by considring a nw instanc whr r k is subtractd from all rntal rats, i.., b i = b i and r i = r i r k for all 0 i k. Suppos p is 1 -comptitiv with rspct to (r,k ) (not that r k = 0). Thn th comptitiv ratio of p at tim t w.r.t. th original instanc is: c(t) = X p(t) opt(t) = X p(t) + r k t opt (t) + r k t 1 opt (t) + r k t opt = (t) + r k t opt (t) r k t + 1

8 RENT, LEASE OR BUY 509 d dt opt (t) = r i r k for t [s i 1,s i ). Hnc, th ratio opt (t) r k t is monoton dcrasing, and thus c(t) is monoton dcrasing as wll. It follows that c r 0 r k r k + 1 = r k/r 0 1 Obsrv that c = 1 whn r k = 0, and that c = 1 whn r k = r 0 (i.., whn k = 0). 4. An Optimal Onlin Algorithm In this sction w dvlop an optimal onlin stratgy for any givn additiv multislop ski rntal instanc. W rduc th st of all possibl stratgis to a subst of much simplr stratgis, which on on hand contains an optimal stratgy, and on th othr hand is asir to analyz, and in particular, allows us to ffctivly find such an optimal stratgy. Considr an arbitrary profil. (Rcall that w assum w.l.o.g. that no slop is compltly dominatd by anothr.) As a first simplification, w confin ourslvs to profils whr ach p i has only finitly many discontinuitis. This allows us to avoid masur-thortic pathologis without ruling out any rasonabl solution within th Church-Turing computational modl. It can b shown that w may considr only continuous profils (dtails omittd). So lt such a profil p = (p 0,...,p k ) b givn. W show that it can b transformd into a profil of a crtain structur without incrasing th comptitiv factor. Our chain of transformations is as follows. First, w show that it suffics to considr only simpl profils w call prudnt. Prudnt stratgis buy slops in ordr, on by on, without skipping and without buying mor than on slop at a tim. W thn dfin th concpt of tight profils, which ar prudnt profils that spnd mony at a fixd rat rlativ to th optimal offlin stratgy. W prov that thr xists a tight optimal profil. Furthrmor, th bst tight profil can b ffctivly computd: Givn a constant c, w show how to chck whthr thr xists a tight c-comptitiv stratgy, and this way, using binary sarch on c, w can find th bst tight stratgy. Finally, w xplain how to construct that profil and a corrsponding stratgy Prudnt and Tight Profils Our main simplification stp is to show that it is sufficint to considr only profils that buy slops conscutivly on by on. Formally, prudnt profils ar dfind as follows. Dfinition 4.1 (activ slops, prudnt profils). A slop i is activ at tim t if p i (t) > 0. A profil is calld prudnt if at all tims thr is ithr on or two conscutiv activ slops. At any givn tim t, at last on slop is activ bcaus i p i(t) = 1 by th problm dfinition. Considring Eq. (2.1) as wll, w s that a continuous prudnt profil progrsss from on slop to nxt without skipping any slop in btwn: onc slop i is fully paid for (i.., p i (t) = 1), th algorithm will start buying slop i + 1. W now prov that th st of continuous prudnt profils contains an optimal profil. Intuitivly, th ida is that a non-prudnt profil must hav two non-conscutiv slops with positiv probability at som tim. In this cas w can shift som probability toward a middl slop and only improv th ovrall cost. Thorm 4.2. If thr xists a continuous c-comptitiv profil p for som c 1, thn thr xists a prudnt c-comptitiv profil p.

9 510 Z. LOTKER, B. PATT-SHAMIR, AND D. RAWITZ Proof. Lt p = (p 0,...,p k ) b a profil and suppos that all th p i s ar continuous. It follows that B p is also continuous. Dfin bst(t) = max {i : b i B p (t)} and nxt(t) = min {i : b i B p (t)}. In words, bst(t) is th most xpnsiv slop that is fully within th buying budgt of p at tim t, and nxt(t) is th most xpnsiv slop that is at last partially within th buying budgt of p at tim t. Obviously, bst(t) nxt(t) bst(t) + 1 for all t. Now, w dfin p as follows: b nxt B p (t) i = bst(t) and bst(t) nxt(t), b nxt b bst B p(t) b bst p i (t) = b nxt b bst i = nxt(t) and bst(t) nxt(t), 1 i = bst(t) = nxt(t), 0 othrwis. It is not hard to vrify that i p i(t) = 1 for vry tim t. Furthrmor, obsrv that p is prudnt, bcaus B p is continuous. It rmains to show that p is c-comptitiv. W do so by proving that B p (t) = B p (t) and R p (t) R p (t) for all t. First, dirctly from dfinitions w hav B p (t) = p bst(t) (t) b bst(t) + p nxt(t) (t) b nxt(t) = b nxt(t) B p (t) b nxt(t) b bst(t) b bst(t) + B p(t) b bst(t) b nxt(t) b bst(t) b nxt(t) = B p (t). Considr now rntal paymnts. To prov that R p (t) R p (t) for vry tim t w construct inductivly a squnc of probability distributions p = p 0,...,p l = p. Th first distribution p 0 is dfind to b p. Suppos now that p j is not prudnt. Distribution p j+1 is obtaind from p j as follows. For any t such that thr ar two non-conscutiv slops with positiv probability, lt i 1 (t),i 2 (t),i 3 (t) b any thr slops such that i 1 (t) = argmin{i : p j i (t) > 0}, i 3(t) = argmax{i : p j i (t) > 0}, and i 1(t) < i 2 (t) < i 3 (t) (such i 2 (t) xists bcaus p j is not prudnt). Dfin p j i (t) j (t) i = i 1 (t), p j+1 i (t) = p j i (t) + p j i (t) p j i (t) b i2 (t) b i1 (t) j (t) b i2 (t) b + i1 (t) j (t) b i3 (t) b i2 (t) j (t) b i3 (t) b i2 (t) i = i 2 (t), i = i 3 (t), i {i 1(t),i 2 (t),i 3 (t)} whr j (t) > 0 is maximizd so that p j+1 i (t) 0 for all i. Intuitivly, w shift a maximal amount of probability mass from slops i 1 (t) and i 3 (t) to th middl slop i 2 (t). Th fact that j (t) is maximizd mans that w hav ithr that p j+1 i 1 (t) = 0, or p j+1 i 3 (t) = 0, or both. In any cas, w may alrady conclud that l < k. Also not that by construction, for all t w hav B p j+1(t) = i pj+1 i (t) b i = i pj i (t) b i = B p j(t). Hnc, p l = p. As to th rntal cost, fix a tim t, and considr now th rnt paid by p j and p j+1 : R p j(t) R p j+1(t) = j ( (t) j (t) j ) (t) j (t) = r i1 (t) r b i2 (t) b i2 (t) + r i1 (t) b i2 (t) b i1 (t) b i3 (t) b i3 (t) i2 (t) b i3 (t) b i2 (t) ( = j ri1 (t) r i2 (t) (t) r ) i 2 (t) r i3 (t) > 0 b i2 (t) b i1 (t) b i3 (t) b i2 (t)

10 RENT, LEASE OR BUY 511 whr th last inquality follows from th fact that if i < j, thn b j b i r i r j is th x coordinat of th intrsction point btwn th slops i and j. Our nxt stp is to considr profils that invst in buying as much as possibl undr som spnding rat cap. Our approach is motivatd by th following intuitiv obsrvation. Obsrvation 4.3. Lt p 1 and p 2 b two randomizd prudnt profils. If B p 1(t) B p 2(t) for vry t, thn R p 1(t) R p 2(t) for vry t. In othr words, invsting availabl funds in buying as soon as possibl rsults in lowr rnt, and thrfor in mor availabl funds. Hnc, w dfin a class of profils which spnd mony as soon as possibl in buying, as long as thr is a bttr slop to buy, namly as long as p k (t) < 1. Dfinition 4.4. Lt c 1. A prudnt c-comptitiv profil p is calld tight if X p (t) = c opt(t) for all t with p k (t) < 1. Clarly, if th last slop is flat, i.., r k = 0, thn it must b th cas that p k (s k ) = 1 for any profil with finit comptitiv factor: othrwis, th cost to th profil will grow without bound whil th optimal cost rmains constant. Howvr, it is important to not that if r k > 0, thr may xist an optimal profil p that nvr buys th last slop, but still its xpctd spnding rat as t tnds to infinity is c r k. It is asy to s that a tight profil can achiv any achivabl comptitiv factor. Lmma 4.5. If thr xists a c-comptitiv prudnt profil p for som c 1, thn thr xists a c-comptitiv tight profil p. Proof. Lt p b th prudnt profil satisfying X p (t) = c opt(t) for all t for which p k (t) < 1. W nd to show that p is fasibl. Sinc by dfinition, p buys with any amount lft, it suffics to show that for all t, th rnt paid by p is at most c d dtopt(t). Indd, R p (t) R p (t) for vry t du to Obsrvation 4.3, and sinc p is c-comptitiv it follows that R p (t) c d dtopt(t) and w ar don Constructing Optimal Onlin Stratgis W now us th rsults abov to construct an algorithm that producs th bst possibl onlin stratgy for th multislop problm. Th ida is to guss a comptitiv factor c, and thn try to construct a c-comptitiv tight profil. Givn a way to tst for succss, w can apply binary sarch to find th optimal comptitiv ratio c to any dsird prcision. Th main qustions ar how to tst whthr a givn c is fasibl, and how to construct th profils. W answr ths qustions togthr: givn c, w construct a tight c-comptitiv profil until ithr w fail (bcaus c was too small) or until w can guarant succss. In th rmaindr of this sction w dscrib how to construct a tight profil p for a givn comptitiv factor c. W bgin with analyzing th way a tight profil may spnd mony. Considr th situation at som tim t such that p k (t) < 1. Lt j b th maximum indx such that s j t. Thn d dt opt(t) = r j. Thrfor, th spnding rat of a tight profil at tim t must b c r j. If j < k, th tight profil may spnd at rat c r j until tim s j+1 (or until p k (t) = 1), and if

11 512 Z. LOTKER, B. PATT-SHAMIR, AND D. RAWITZ j = k th tight profil may continu spnding at this rat forvr. Hnc, for t (s j,s j+1 ), w hav d dt B p(t) + R p (t) = c d dt opt(t) = c r j. (4.1) Sinc p is tight and thrfor prudnt, w also hav, assuming bst(t) = i and nxt(t) = i+1, that B p (t) = p i (t)b i + p i+1 (t)b i+1, and R p (t) = p i (t)r i + p i+1 (t)r i+1. Plugging th abov quations into Eq. (4.1), w gt d dt p i(t)b i + d dt p i+1(t)b i+1 + p i (t)r i + p i+1 (t)r i+1 = c r j Sinc p is prudnt, p i (t) = 1 p i+1 (t) and hnc d dt p i(t) = d dt p i+1(t). It follows that d dt p i+1(t) + p i+1 (t) ri+1 r i b i+1 b i = c r j r i b i+1 b i (4.2) A solution to a diffrntial quation of th form y (x) + αy(x) = β whr α and β ar constants is y = β α + Γ αx, whr Γ dpnds on th boundary condition. Hnc in our cas w conclud that p i+1 (t) = c r r j r i r i+1 i t b + Γ i+1 b i, (4.3) r i+1 r i and p i (t) = 1 p i+1 (t), whr th constant Γ is dtrmind by th boundary condition. Eq. (4.3) is our tool to construct p in a picwis itrativ fashion. For xampl, w start constructing p from t = 0 using p 1 (t) = c r 0 r 0 r 1 r 0 +Γ r0 r1 t b 1 b 0 and th boundary condition p 1 (0) = 0. W gt that Γ = r 0(c 1) r 0 r 1, i.., p 1 (t) = r 0(c 1) r 0 r 1 ( r0 r1 b 1 b 0 t 1), and this holds for all t min(s 1,t 1 ), whr t 1 is th solution to p 1 (t 1 ) = 1. In gnral, Eq. (4.2) rmains tru so long as thr is no chang in th spnding rat and in th slop th profil p is buying. Th spnding rat changs whn t crosss s j, and th profil starts buying slop i + 2 whn p i+1 (t) = 1. W can now dscrib our algorithm. Givn a ratio c, Algorithm Fasibl is abl to construct th tight profil p or to dtrmin that such a profil dos not xist. It starts with th boundary condition p 1 (0) = 0 and rvals th first part of th profil as shown abov. Thn, ach tim th spnding rat changs or thr is a chang in bst(i) it movs to th nxt diffrntial quation with a nw boundary condition. Aftr at most 2k such itrations it ithr computs a c-comptitiv tight profil p or discovrs that such a profil is infasibl. Sinc w ar abl to tst for succss using Algorithm Fasibl, w can apply binary sarch to find th optimal comptitiv ratio to any dsird prcision. W not that it is asy to construct a stratgy that corrsponds to any givn prudnt profil p, as dscribd in th proof of Lmma 3.3. W conclud with th following thorm. Thorm 4.6. Thr xists an O(k log 1 ε ) tim algorithm that givn an instanc of th additiv multislop ski rntal problm for which th optimal randomizd stratgy has comptitiv ratio c, computs a (c + ε)-comptitiv stratgy.

12 RENT, LEASE OR BUY 513 Algorithm 1 Fasibl(c, M): tru if th k-ski instanc M = (b,r) admits comptitiv factor c 1: Lt s i = bi bi 1 r i 1 r i for ach 1 i k 2: Boundary Condition p 1 (0) = 0 3: j 0; i 1 4: loop 5: Dfin p i (t) = c rj ri 1 r i r i 1 + Γ xp( ri 1 ri b i b i 1 t) 6: Try to solv for Γ using Boundary Condition 7: if no solution thn rturn fals possibl scap if not fasibl 8: y p i (s j ) 9: if y < 1 thn 10: Boundary Condition p i (s j ) = y 11: j j + 1 continu at th nxt intrval [s j, s j1 ] 12: ls 13: Lt x b such that p i (x) = 1 14: Boundary Condition p i+1 (x) = 0 15: i i + 1 mov to nxt slop 16: nd if 17: if i > k or j k thn rturn tru w r don 18: nd loop 5. An -Comptitiv Stratgy for th Non-Additiv Cas In this sction w considr th non-additiv multislop ski rntal problm. W prsnt a simpl randomizd stratgy which improvs th bst known comptitiv ratio from 2/ ln 2 = 2.88 to. Our tchniqu is a simpl application of randomizd rpatd doubling (s,.g., [7]), usd xtnsivly in comptitiv analysis of onlin algorithms. For xampl, dtrministic rpatd doubling appars in [1], and a randomizd vrsion appars in [13]. Bfor prsnting th stratgy lt us considr th dtails of th non-additiv modl. Augustin at l. [2] dfin a gnral non-additiv modl in which a transition cost b ij is associatd with vry two stats i and j, and show that on may assum w.l.o.g. that b ij = 0 if i > j and that b ij b j for vry i < j. Obsrv that w may furthr assum that b ij = b j for vry i and j, sinc th optimal (offlin) stratgy rmains th unchangd. It follows that th stratgis from [3, 4, 6] that wr dsignd for th cas of buying slops from scratch also work for th gnral non-additiv cas. W propos using th following itrativ onlin stratgy, which is similar to th on in [6], xcpt for th choic of th doubling factor. Spcifically, th jth itration is associatd with a bound B j on opt(τ), whr τ dnots th trmination tim of th gam. W dfin df B 1 = opt(s 1 )/α X, whr α > 1 and X is a chosn at random uniformly in [0,1). W also dfin B j+1 = α B j. Lt τ j = opt 1 (B j ) and lt i j b th optimal offlin stat at tim τ j. In cas thr ar two such stats, i.., τ j = s i for som i, w dfin i j = i 1. It follows that i 1 = 0. In th bginning of th jth itration th onlin stratgy buys i j and stays in i j until th this itration nds. Th jth itration nds at tim τ j. Obsrv that th first itration starts with B 1 = opt(s 1 ), namly w us slop 0 until s 1. Thorm 5.1. Th xpctd cost of th stratgy dscribd abov is at most tims th optimum.

13 514 Z. LOTKER, B. PATT-SHAMIR, AND D. RAWITZ Proof. Obsrv that th first itration starts with B 1 = opt(s 1 ), namly w us slop 0 until s 1, and hnc, if th gam nds during th first itration, i.., bfor s 1 /α X, thn th onlin stratgy is optimal. Considr now th cas whr th gam nds at tim τ s 1 /α X, and suppos that τ [τ l,τ l+1 ) for l > 1. In this cas, th xpctd cost of th onlin stratgy is boundd by l l+1 [ ] E opt(τ j ) + opt(τ) E opt(τ j ) α E α 1 opt(τ l+1) j=1 j=1 [ ] α 2 X = E α 1 opt(τ) = α α 1 α By choosing α = th comptitiv ratio is ln α Acknowldgmnt 1 x=0 α x dx opt(τ) = α ln α opt(τ) = as rquird. W thank Sffy Naor and Niv Buchbindr for stimulating discussions. Rfrncs [1] J. Aspns, Y. Azar, A. Fiat, S. A. Plotkin, and O. Waarts. On-lin routing of virtual circuits with applications to load balancing and machin schduling. Journal of th ACM, 44(3): , [2] J. Augustin, S. Irani, and C. Swamy. Optimal powr-down stratgis. In 45th IEEE Symp. on Foundations of Computr Scinc, pags , [3] Y. Azar, Y. Bartal, E. Furstin, A. Fiat, S. Lonardi, and A. Rosén. On capital invstmnt. Algorithmica, 25(1):22 36, [4] Y. Bjrano, I. Cidon, and J. S. Naor. Dynamic sssion managmnt for static and mobil usrs: a comptitiv on-lin algorithmic approach. In 4th Intrnational Workshop on Discrt Algorithms and Mthods for Mobil Computing and Communications, pags ACM, [5] A. Borodin and R. El-Yaniv. Onlin Computation and Comptitiv Analysis. Cambridg Univrsity Prss, [6] P. Damaschk. Narly optimal stratgis for spcial cass of on-lin capital invstmnt. Thortical Computr Scinc, 302(1-3):35 44, [7] S. Gal. Sarch Gams. Acadmic Prss, [8] S. Irani, R. K. Gupta, and S. K. Shukla. Comptitiv analysis of dynamic powr managmnt stratgis for systms with multipl powr savings stats. In Dsign, Automation and Tst in Europ Confrnc and Exhibition, pags , [9] A. R. Karlin, C. Knyon, and D. Randall. Dynamic TCP acknowldgmnt and othr storis about /( 1). Algorithmica, 36(3): , [10] A. R. Karlin, M. S. Manass, L. Rudolph, and D. D. Slator. Comptitiv snoopy caching. Algorithmica, 3(1):77 119, [11] A. W. Marshall and I. Olkin. Inqualitis: Thory of Majorization and Its Applications. Acadmic Prss, [12] A. Myrson. Th parking prmit problm. In 46th IEEE Symp. on Foundations of Computr Scinc, pags , [13] R. Motwani, S. Phillips, and E. Torng. Non-clairvoyant schduling. Thor. Comput. Sci., 130(1):17 47, This work is licnsd undr th Crativ Commons Attribution-NoDrivs Licns. To viw a copy of this licns, visit