Trackless online algorithms for the server problem

Transcription

1 Iformatio Processig Letters 74 (2000) Trackless olie algorithms for the server problem Wolfgag W. Bei,LawreceL.Larmore 1 Departmet of Computer Sciece, Uiversity of Nevada, Las Vegas, NV 89154, USA Received 22 Jue 1999; received i revised form 1 November 1999 Commuicated by S.E. Hambrusch Abstract A class of simple olie algorithms for the k-server problem is idetified. This class, for which the term trackless is itroduced, icludes may kow server algorithms. The k-server cojecture fails for trackless algorithms. A lower boud of 23/11 o the competitiveess of ay determiistic trackless 2-server algorithm ad a lower boud of 1 + 2/2 othe competitiveess of ay radomized trackless 2-server problem are give Published by Elsevier Sciece B.V. All rights reserved. Keywords: Desig of algorithms; Olie algorithms; Radomized algorithms; Competitive aalysis; k-server problem; Pagig 1. Itroductio Irai ad Rubifeld [10] give a simple algorithm for the 2-server problem i a arbitrary metric space, which is betwee 6- ad 10-competitive. This algorithm is remarkable i at least two ways. First, it uses oly costat memory ad, for each request, serves the request i costat time. Secodly, the algorithm ca be oblivious to the actual metric space, as log as the distaces from the servers to the request poit are kow. I effect, the algorithm i its implemetatio ever requires a data type poit. By cotrast, the work fuctio algorithm (WFA) [7], which is kow to be 2-competitive for the 2-server problem (ad (2k 1)-competitive for the k-server problem [11]) may eed to retai iformatio about all previously requested poits. Correspodig author. bei@cs.ulv.edu. Research supported by NSF grat CCR larmore@cs.ulv.edu. Research supported by NSF grat CCR Borodi ad El-Yaiv [2] posed as a ope questio to show a lower boud greater tha 2 o the competitive ratio of ay determiistic 2-server algorithm that uses costat time ad space per request. They also metio that for such a lower boud the model of computatio eeded to be formalized. Such a model is difficult to formalize because ulimited iformatio ca be hidde eve i a sigle real umber. I this paper we itroduce the cocept of a trackless olie algorithm. Tracklessess is a property of a olie algorithm for the server problem which is a restrictio o what iput data the algorithm uses i its computatio, ad o what outputs the algorithm gives. Specifically, for each request poit, the algorithm is oly give as iput the distaces of the curret server positios to the request poit. A trackless algorithm may memorize such distace values, but is restricted from storig explicitly ay poits of the metric space. I the algorithm s respose to a request, the algorithm is limited to producig output that idetifies which server moves, ad does ot metio poits i the metric space explicitly. May kow /00/$ see frot matter 2000 Published by Elsevier Sciece B.V. All rights reserved. PII: S (00)00034-X

2 74 W.W. Bei, L.L. Larmore / Iformatio Processig Letters 74 (2000) algorithms are trackless. These iclude BALANCE2, BALANCE_SLACK, RANDOM_SLACK ad HAR- MONIC, whereas, for example, WFA is ot trackless. The cocept of tracklessess ca be geeralized to other olie problems. For example, there is a obvious extesio to the pagig problem. It would be desirable to settle the k-server cojecture by fidig a k-competitive algorithm for the k- server problem, ad it would be desirable if that algorithm were simple, i the sese that it uses few iputs as well as little time ad space. The results i this paper give a idicatio that this may ot be possible. We suspect that if there is a k- competitive olie algorithm for the k-server problem, it caot be simple. I particular, we show that, if oly trackless algorithms are cosidered, the k-server cojecture fails. (I fact, it fails for k = 2.) This is true despite the fact that the trackless model does ot require memoryless computatio. Our mai result is a lower boud of o the competiveess of ay determiistic trackless olie algorithm for the 2-server problem. For radomized trackless olie algorithms for the 2-server problem agaist a oblivious adversary, we show a lower boud of 1 + 2/ o the competitiveess. This is higher tha the best kow lower boud for the same problem without the tracklessess restrictio. 2. The trackless k-server problem I the k-server problem we are give k 2 mobile servers 1,...,k that reside i a metric space M. A sequece of requests ϱ = r 1...r is issued, where each request is specified by a poit r t M. To satisfy this request, oe of the servers must be moved to r t, t = 1,...,, at a cost equal to the distace from its curret locatio to r t. A algorithm A for the k-server problem decides which server should be moved at each step t. A is said to be olie if its decisios are made without the kowledge of future requests. Our goal is to miimize the total service cost. A is C-competitive ifthecosticurredbya to service each request sequece ϱ is at most C times the optimal (offlie) service cost for ϱ, plus possibly a additive costat idepedet of ϱ. (See Chapter 1 of [2] for a comprehesive discussio of competitiveess.) The competitive ratio of A is the smallest C for which A is C-competitive. Assume ow that the servers are iitially located at poits s1 0,...,s0 k M. Letst i M deote the locatio of server i at time t, meaig after the request r t has bee serviced. Note that si t 1 r t is the distace betwee the locatio of s i ad the request r t at the time that request is made, ad before ay server has moved. We say that a poit p M is active at step t if there is either a server or a request at p at the time of the tth request. That is, p is active at step t if ad oly if p {r t,s1 t 1,...,sk t 1 }. We defie a A to be a trackless algorithm for the k-server problem if: (1) A ca oly move a server to a active poit. That is, every output of A at step t is of the form Move s i to r or Moves i to s j. (2) A receives at step t as iputs oly {si t 1 r t } for 1 i k. We assume that A iitially kows the distaces amog all servers. We do ot restrict the computatioal power of a trackless algorithm, ad we allow A to store iputs from previous steps ad use them i its calculatios. 2 However, all trackless algorithms curretly i the literature use oly O(k) time at each step to make their calculatios. We ow review some of these. Irai Rubifeld Algorithm. The Irai Rubifeld Algorithm, also kow as BALANCE2 [10], is a determiistic trackless algorithm for the k-server problem i a arbitrary metric space. It uses O(k) memory ad O(k) time at each step. For k = 2, the competitiveess of BALANCE2 is at most 10, but a 6 poit couter-example shows that the competitiveess caot be less tha 6 for geeral spaces [4]. BALANCE_SLACK. BALANCE_SLACK is a determiistic trackless algorithm for the 2-server problem i a arbitrary metric space [4]. It uses O(1) memory ad O(1) time at each step. It is 4-competitive for a arbitrary metric space. RANDOM_SLACK. RANDOM_SLACK is a radomized trackless algorithm for the 2-server problem 2 Note therefore, that at step t, A may kow the distaces amog all servers before ad after each request.

3 W.W. Bei, L.L. Larmore / Iformatio Processig Letters 74 (2000) i a arbitrary metric space [4]. It is memoryless ad uses O(1) time at each step. It is 2-competitive for a arbitrary metric space. HARMONIC. HARMONIC is a radomized trackless algorithm for the k-server problem i a arbitrary metric space. It uses O(k) time at each step, ad is memoryless. Its competitiveess is cojectured to be ( k+1) 2, see [13]. For k = 2, its competitiveess is kow to be 3, see [6]. LRU. Least Recetly Used ad other markig algorithms are trackless k-competitive algorithms for the pagig problem with cache size k, which is equivalet to the k-server problem i a uiform space [3,14]. They use O(k) memory. (See [2] for a survey.) MARK. MARK is a trackless radomized algorithm for the pagig problem with cache size k [9]. It uses O(k) memory ad is (2H k 1)-competitive [1], where H k is the kth Harmoic umber. 3. The determiistic lower boud I this sectio, we prove that o trackless algorithm for the 2-server problem which works for all metric spaces ca be less tha competitive. First we state, without proof, three easy techical lemmas. We say that A s servers match the servers of a optimal offlie algorithm (referred to as the optimal servers, for short) if they are at the same poits. Without loss of geerality, every trackless olie algorithm is lazy, as follows from Lemma 3 below. Lemma 3. Give ay trackless olie algorithm for the k-server problem, there is a lazy trackless olie algorithm for the k-server problem which does ot have greater cost. We are ow ready to state the lower boud result. Theorem 1. There is o determiistic trackless olie algorithm for the 2-server problem which is C- competitive for ay C< Proof. Let M be the set of vertices of the tiles of a tilig of the plae ito cogruet equilateral triagles (see Fig. 1). We thik of M as a ifiite graph, where the edges are the edges of the triagles. Distace is defied to be the legth (umber of edges) of the shortest path betwee two poits. Now, let A be ay trackless algorithm for the 2- server problem i M. We assume that the two servers begi at poits which are distace 1 apart i M. We show that there exists a request sequece, α, im such that, after A services α, its two servers are agai distace 1 apart, ad the total cost of movig the servers is positive ad is at least times the optimal cost. We call such a sequece a phase. To show that A caot be C-competitive for ay C< 23 11,wesimply request may phases. We ow describe the request sequece α.weleta,b be the locatios of the servers 1 ad 2, respectively, at Lemma 1. If the optimal servers are at two distict poits x ad y, there is a request sequece whose optimal cost is zero, ad which forces A to move its two servers to x ad y. Lemma 2. If the optimal servers match A s servers at distict poits x ad y,wherexy = d 1, ad if there is a poit z such that xz = yz = d 2, the there is a request sequece such that, if A serves this sequece, it pays d 1 + d 2, the optimal cost is d 2, ad at the ed of the service A s servers match the optimal servers, either at x,z or at y,z. We say that A is lazy if it moves at most oe server at each step, ad the oly to the curret request poit. Fig. 1. The metric space M.

4 76 W.W. Bei, L.L. Larmore / Iformatio Processig Letters 74 (2000) the begiig of a phase. Pick poits c,d,e,p,q M such that: (1) cd = de = 1, ad = ae = 4, ad ac = bc = bd = be = 3. (2) pq = 1, aq = 2, ad ap = bp = bq = 3. Because of the geometry of M such poits exist, see Fig. 1. There are six cases depedig o whe A moves server 2 for the first time: we assume that A serves c with 1. Case 1: A serves cd with 1, 2. I this case, defie α to be the cocateatio of four parts. The first part is cd. The secod part is a sequece give by Lemma 1 which forces the servers to a ad d, which are 4 apart. The third part is give by Lemma 2, ad forces the servers to poits which are 2 apart. The fourth part is give by Lemma 2 agai, ad forces the servers to poits which are 1 apart. The algorithm pays 6 to service cd, paysatleast 3 to servicethe secodpart, at least 6 to service the third part, ad at least 3 to service the fourth part. Thus, the algorithm pays at least 18 to service α. The optimal service serves cd with servers 2, 2, at a cost of 4. The cost of servicig the secod part is 0, as the optimal servers are already at a,d.the optimal cost to service the third part is 2 ad to service the fourth part is 1, by Lemma 2. Thus, the optimal cost of servicig α is 7. Case 2: A serves cde with 1, 1, 2. I this case, defie α to be cde, followed by a sequece, as give by Lemma 1, that forces the servers to a ad e, followed by two sequeces that force the servers to poits which are 1 apart, due to Lemma 2 as described i Case 1. The total cost for A for the phase is at least 20, while the optimal cost is 8. Case 3: A serves cded with 1, 1, 1, 2. I this case, defie α to be cded, followed by a sequece, as give by Lemma 1, that forces the servers to a ad d, followed by two sequeces that force the servers to poits which are 1 apart, as give by Lemma 2. The total cost for A for the phase is at least 21, while the optimal cost is 9. Case 4: A serves cdede with 1, 1, 1, 1, 2. I this case, defie α to be cdede, followed by a sequece, as give by Lemma 1, that forces the servers to a ad e, followed by two sequeces that force the servers to poits which are 1 apart, as give by Lemma 2. The total cost for A for the phase is at least 22, while the optimal cost is 10. Case 5: A serves cdeded with 1, 1, 1, 1, 1, 2. I this case, defie α to be cdeded, followed by a sequece, as give by Lemma 1, that forces the servers to a ad d, followed by two sequeces that force the servers to poits which are 1 apart, as give by Lemma 2. The total cost for A for the phase is at least 23, while the optimal cost is 11. Case 6: A serves cdeded with 1, 1, 1, 1, 1, 1. I this case, defie α to be pqpqpq, followed by a sequece, as give by Lemma 1, that forces the servers to p ad q, which are 1 apart. It is here that the trackless property of A plays a role. Sice A is trackless, it must respod to p the same way it would respod to c, amely by movig server 1, because the oly iput A uses is the pair of distaces (s 1 r, s 2 r),whichis(3, 3) whether r is c or p. Similarly, oce server 1 is at p, A must respod to the request at q i the same maer as it would to the request d if server 1 were at c, because i each case, the iput is the pair (1, 3). Extedig this argumet for the etire sequece, we see that A must serve α = pqpqpq i the same way as it would serve cdeded, amely with 1, 1, 1, 1, 1, 1. Thus, after six steps, server 1 is at q ad server 2 is at b. The total cost for A for the phase is at least 11, while the optimal cost is 5. Sice mi { 18 7, 20 8, 21 9, 22 10, 23 11, 11 5 we have cost A cost opt } = 23 11, for the phase. We retur to the situatio where if c is the first request, A serves that request with server 2. By the property of tracklessess, A must also serve with 2 if the first request is p. There is a symmetry of M which iterchages a ad b, ad which iterchages c ad p (see Fig. 1). Thus, by symmetry, a request sequece ca always be chose so that cost A cost opt for the phase. This completes the proof of the lower boud. Fially we ote that the metric space M ca be mapped oto a torus as i Fig. 2. A momet s thought shows that with such a mappig the couterexample ca be produced usig this fiite space of oly 64 poits.

5 W.W. Bei, L.L. Larmore / Iformatio Processig Letters 74 (2000) Fig. 2. A mappig of M oto a torus. 4. The radomized lower boud I this sectio, we show that o trackless radomized olie algorithm for the 2-server problem ca have competitiveess less tha / Let be a large iteger, ad let ε = 2 2/. Let M be a metric space cosistig of a poit a ad a ulimited umber of poits b 1,b 1,b 2,... where the distace from a to each b i is 1, ad the distace from b i to b j is ε for i j. We refer to the set b 1,b 2,b 3,... as the B-cluster. Let the iitial positios of the servers be a ad b 1. We defie a radomized oblivious adversary agaist which every determiistic algorithm is ot better tha (1+ 2/2)-competitive. By [15], this proves the lower boud. The adversary request sequece cosists of a sufficietly large umber of phases. At the begiig of each phase, the optimal servers are located at a ad at some b k, ad the algorithm s two servers are located at a ad b l for some l. We ow describe oe phase. Poits i the B-cluster will ot be re-used from phase to phase; b 1 will be cosidered to be used before the first phase, hece will ever be requested. Let m be the umber of poits i the B-cluster that have bee used up to the begiig of this phase. (Those poits will ecessarily be b 1...b m.) For each phase, idepedetly, the adversary chooses oe of two strategies, which we call buzz ad wader. With probability 2 1, the adversary chooses to buzz durig that phase, ad with probability 2 2 the adversary chooses to wader durig that phase. If the adversary chooses to buzz durig the curret phase, it requests (b m+1 b m+2 ) a. I this case, the adversary moves its servers to b m+1 ad b m+2 ad the at the ed of the phase moves oe server to a. The optimal cost for this phase is thus 2 + ε. With probability 2 2, the adversary chooses to wader durig the curret phase. I this case, the adversary radomly chooses a iteger i i the rage 1..., with equal probability for each i. The adversary the requests the sequece b m+1 b m+2 b m+3...b m+i a. The adversary serves by movig oe server aroud iside the B-cluster ad leavig the other server at a. Thus, the optimal cost for the phase is iε. I either case, the phase eds with the request a. The expected optimal cost for oe phase is E cost opt = + 1 ( ) ( ) 2 2 ε + (2 + ε) = 4 ( ) O. We ow compute the expected cost E cost A durig oe phase. For ay iteger t 0, let A t be the determiistic algorithm which serves the first t requests i the B-cluster with the same oe server that starts i the B-cluster, ad the, at step t + 1, if the request poit is i the B-cluster, moves its server from a. After that, A t uses its two servers to serve all further requests i the B-cluster, fially movig oe of them back to a whe a is requested at the ed of the phase. Let A be the stubbor algorithm, which serves all requests i the B-cluster with oe server, ever movig the server from a durig that phase. It is clear that ay lazy radomized algorithm A must behave as either A,or A t for some t, durig each phase. Lemma 4. Durig each phase, E cost A 2 ( ) 1 2 O. Proof. Case I: A = A. I this case, A expeds 2ε with probability 2 1, ad with probability 2 2, expeds εi where the average value of i is ( + 1)/2. Thus E cost A = + 1 ( ) ( ) 2 2 ε + 2ε > 2 2.

6 78 W.W. Bei, L.L. Larmore / Iformatio Processig Letters 74 (2000) Case II: A = A t for t<. If the adversary buzzes, the A expeds tε before it moves a over to the B-cluster, after which further service is free, other tha the fial move back to a. Itmustalsopay2to jump from a to the B-cluster ad back. If the adversary waders, the either i t,iwhich case A ever moves its server from a, but pays iε to move its server aroud iside the B-cluster, or i>t,i which case A pays tε to move its server aroud iside the B-cluster before it moves a server from a,butthe must pay (i t 1)ε more to move servers aroud iside the B cluster after that. It must also pay 2 to jump from a to the B-cluster ad back. Summig up the terms, we have E cost At = t i=1 + 1 ( ) 2 2 iε i=t+1 1 ( )( ) (i 1)ε + ( 2 1 ) (2 + tε) = 2 ( ) 1 2 ± O. Case III: A = A t for t. Ift 2, thea t behaves exactly as A, ad we are doe. Thus, we ca assume that t<2. If the adversary buzzes, A expeds tε + 2, while if the adversary waders, A expeds iε. Summig up the terms, we have E cost A = i=1 1 ( ) ( ) 2 2 iε (2 + tε) (t )ε 2 2. Theorem 2. There is o radomized trackless olie algorithm for the 2-server problem which is C-competitive for ay C<1+2 1/ agaist the oblivious adversary ± O ( 1 ad we are doe. 5. Fial commets ) = /2 ± O ( ) 1 Thetreealgorithmisak-competitive algorithm for the k-server problem i trees [5]. We ote that the tree algorithm, although ot explicitly trackless, ca be rewritte as a trackless algorithm. The trackless versio uses a virtual tree, which the algorithm builds from the trackless iput. The tree may differ from the actual tree, but it is a close eough approximatio to esure k-competitiveess. The costructio ad proof are somewhat tedious ad will be published elsewhere. The radomized competitiveess of the 2-server problem is ot kow. The best kow upper boud is 2, ad the best kow lower boud is 1 + e 1/ [8], while we ca show a lower boud of / for the competitiveess of ay radomized trackless algorithm for the 2-server problem i geeral metric spaces. This is a idicator, but ot a proof, that the radomized competitiveess ad the radomized trackless competitiveess of the k-server problem i geeral are differet. For uiform spaces, the determiistic competitiveess of the k-server problem is k, ad sice a uiform space ca be embedded i a tree, there is also a k- competitive trackless algorithm. But the situatio for radomized algorithms is differet. I particular, there is a 3 2 -competitive radomized olie algorithm for the 2-server problem i a uiform space ad this is kow to be optimal agaist a oblivious adversary [12]. Prelimiary calculatios idicate that there is o radomized trackless algorithm for that problem with competitiveess less tha Refereces Proof. Suppose that C< /2.Letϱ be the request sequece i M cosistig of the cocateatio of a large umber of idepedet radomly chose phases. The, by Lemma 4, for ay determiistic trackless olie algorithm A, the ratio cost A / cost opt coverges to [1] D. Achlioptas, M. Chrobak, J. Noga, Competitive aalysis of radomized pagig algorithms, i: Proc. 4th Europea Symp. o Algorithms, Lecture Notes i Comput. Sci., Vol. 1136, Spriger, Berli, 1996, pp [2] A. Borodi, R. El-Yaiv, Olie computatio ad competitive aalysis, Cambridge Uiversity Press, cup.org/titles/56/ html.

7 W.W. Bei, L.L. Larmore / Iformatio Processig Letters 74 (2000) [3] A. Borodi, S. Irai, P. Raghava, B. Schieber, Competitive pagig with locality of referece, J. Comput. Systems Sci. 50 (1995) [4] M. Chrobak, L.L. Larmore, O fast algorithms for two servers, J. Algorithms 12 (1991) [5] M. Chrobak, L.L. Larmore, A optimal olie algorithm for k servers o trees, SIAM J. Comput. 20 (1991) [6] M. Chrobak, L.L. Larmore, HARMONIC is three-competitive for two servers, Theoret. Comput. Sci. 98 (1992) [7] M. Chrobak, L.L. Larmore, The server problem ad olie games, i: DIMACS Series i Discrete Mathematics ad Theoretical Computer Sciece, Vol. 7, 1992, pp [8] M. Chrobak, L.L. Larmore, C. Lud, N. Reigold, A better lower boud o the competitive ratio of the radomized 2- server problem, Iform. Process. Lett. 63 (2) (1997) [9] A. Fiat, R. Karp, M. Luby, L.A. McGeoch, D. Sleator, N.E. Youg, Competitive pagig algorithms, J. Algorithms 12 (1991) [10] S. Irai, R. Rubifeld, A competitive 2-server algorithm, Iform. Process. Lett. 39 (1991) [11] E. Koutsoupias, C. Papadimitriou, O the k-server cojecture, J. ACM 42 (1995) [12] L. McGeoch, D. Sleator, A strogly competitive radomized pagig algorithm, J. Algorithms 6 (1991) [13] P. Raghava, M. Sir, Memory versus radomizatio i o-lie algorithms, IBM J. Res. Developmet 38 (1994) [14] D. Sleator, R.E. Tarja, Amortized efficiecy of list update ad pagig rules, Comm. ACM 28 (1985) [15] A.C.C. Yao, Probabilistic computatios: Towards a uified measure of complexity, i: Proc. 11th Symp. Theory of Computig, 1980.