Randomized algorithm for the k-server problem on decomposable spaces
We study the randomized k-server problem on metric spaces consisting of widely separated subspaces. We give a method which extends existing algorithms to larger spaces with the growth rate of the competitive quotients being at most O(log k). This met…
Authors: Judit Nagy-Gy"orgy
Randomized algorithm for the k-serv er problem on deomp osable spaes ∗ Judit Nagy-Gy örgy † No v em b er 15, 2018 Abstrat W e study the randomized k -serv er problem on metri spaes onsisting of widely separated subspaes. W e giv e a metho d whi h extends existing algorithms to larger spaes with the gro wth rate of the omp etitiv e quotien ts b eing at most O (lo g k ) . This metho d yields o ( k ) -omp etitiv e algorithms solving the randomized k -serv er problem, for some sp eial underlying metri spaes, e.g. HST s of small heigh t (but un b ounded degree). HST s are imp ortan t to ols for probabilisti appro ximation of metri spaes. Keyw ords: k -serv er, on-line, randomized, metri spaes. 1 In tro dution In the theory of designing eien t virtual memory-managemen t algorithms, the w ell studied paging problem pla ys a en tral role. Ev en the earliest op eration systems on- tained some heuristis to minimize the amoun t of op ying memory pages, whi h is an exp ensiv e op eration. A generalization of the paging problem, alled the k -serv er prob- lem w as in tro dued b y Manasse, MGeo h and Sleator in [ 14 ℄, where the rst imp ortan t results w ere also a hiev ed. The problem an b e form ulated as follo ws. Giv en a metri spae with k mobile serv ers that o up y distint p oin ts of the spae and a sequene of requests (p oin ts), ea h of the requests has to b e serv ed, b y mo ving a serv er from its urren t p osition to the requested p oin t. The goal is to minimize the total ost, that is the sum of the distanes o v ered b y the serv ers; the optimal ost for a giv en sequene is denoted opt( ) . An algorithm is online if it serv es ea h request immediately when it arriv es (without an y prior kno wledge ab out the future requests). Denition 1. A n online algorithm A is c - omp etitive if for any initial ongur ation C 0 and r e quest se quen e it holds that cost( A ( )) ≤ c · opt( ) + I ( C 0 ) , wher e I is a non-ne gative onstant dep ending only on C 0 . ∗ Resear h is partially supp orted b y OTKA T049398. † Departmen t of Mathematis, Univ ersit y of Szeged, Aradi v értan úk tere 1, H-6720 Szeged, Hungary , email: Nagy-Gy orgymath.u-szeged.h u 1 The omp etitiv e ratio of a giv en online algorithm A is the inm um of the v alues c with A b eing c -omp etitiv e. The k -serv er onjeture (see [14 ℄) states that there exists an algorithm A that is k -omp etitiv e for an y metri spae. Manasse et al. pro v ed that k is a lo w er b ound [14℄, and K outsoupias and P apadimitriou sho w ed 2 k − 1 is an upp er b ound for an y metri spae [12℄. In the randomized online ase (sometimes this mo del is alled the oblivious adv ersary mo del [5℄) the omp etitiv e ratio an b e dened in terms of the exp eted v alue as follo ws: Denition 2. A r andomize d online algorithm R is c - omp etitive if for any initial on- gur ation C 0 and r e quest se quen e we have E(cost( R ( ))) ≤ c · opt( ) + I ( C 0 ) , wher e I is a non-ne gative onstant dep ending only on C 0 and E(cost( R ( ))) denotes the exp e te d value of cost( R ( )) . The omp etitiv e ratio of the ab o v e randomized algorithm is dened analogously . In the randomized v ersion there are more problems that are still op en. The random- ized k -serv er onjeture states that there exists a randomized algorithm with a omp eti- tiv e ratio Θ(log k ) in an y metri spae. The b est kno wn lo w er b ound is Ω(log k / log 2 log k ) [3℄. A natural upp er b ound is the b ound 2 k + 1 giv en for the deterministi ase. By re- striting our atten tion to metri spaes with a sp eial struture, b etter b ounds an b e a hiev ed: Fiat et al. sho w ed a lo w er b ound H k = P k i =0 i − 1 ≈ log k for uniform metri spaes [10℄, whi h turned out to b e also an upp er b ound, see MGeo h and Sleator, [ 13℄. In this pap er w e also onsider a restrition of the problem, namely w e seek for an eien t randomized online algorithm for metri spaes that are µ -HST spaes [1, 2℄ and dened as follo ws: Denition 3. A µ -hier ar hi al ly wel l-sep ar ate d tr e e ( µ -HST) is a metri sp a e dene d on the le aves of a weighte d, r o ote d tr e e T with the fol lowing pr op erties: 1. The e dge weight fr om any no de to e ah of its hildr en is the same. 2. The e dge weights along any p ath fr om the r o ot to a le af ar e de r e asing by the fator µ fr om one level to the next. The weight of an e dge inident to a le af is one. The µ -HST spaes pla y an imp ortan t role in the so-alled metri spae appro ximation te hnique dev elop ed b y Bartal [ 2 ℄. F ak haro enphol et al [11℄ pro v ed that ev ery w eigh ted graph on n v erties an b e α -probabilistially appro ximated b y a set of µ -HST s, for an arbitrary µ > 1 where α = O ( µ log n/ log µ ) . It has b een sho wn in [15℄ that for an y 2 k -HST with an underlying tree T that has a small depth and maxim um degree there exists a p olylog( k ) -omp etitiv e randomized algorithm for the k -serv er problem. By sligh tly mo difying the approa h of Csaba and Lo dha [7℄ and Bartal and Mendel [4℄ w e sho w that there exists su h an algorithm for an y µ -HST that has a small depth and arbitrary maxim um degree t , giv en µ ≥ min { k , t } . 2 Notation Supp ose the p oin ts of a metri spae an b e partitioned in to t blo ks, B 1 , . . . , B t , su h that the diameter of ea h blo k is at most δ and whenev er x and y are p oin ts of dieren t blo ks, their distane is exatly ∆ . Supp ose also that ∆ /δ = µ ≥ k holds. The ab o v e metri spae is µ -deomp osable [15℄. 2 F or a giv en request sequene w e denote its i th mem b er b y i , and the prex of of length i b y ≤ i . Giv en a blo k B s , a request sequene ha ving only requests from B s and a n um b er ℓ of serv ers inside B s and an algorithm A , let cost( A s ( ℓ, )) denote the ost omputed b y the algorithm A for these inputs and opt s ( ℓ, ) (in the latter ase also the initial p osition of the serv ers an b e hosen). If is nonempt y , opt s (0 , ) is dened to b e innite. Denition 4. The demand of the blo k B s for the r e quest se quen e that ontains only r e quests fr om is D s ( ) := min { ℓ | opt s ( ℓ, ) + ℓ ∆ = min j { opt s ( j, ) + j ∆ }} , if is nonempty, otherwise it is 0 . Visually , D s ( ) denotes the least n um b er of serv ers to b e mo v ed in to the initially empt y blo k B s to a hiev e the optimal ost for the sequene . W e note that the b eha viour of the sequene D s ( 1 ) , D s ( ≤ 2 ) , . . . , D s ( ≤ i ) , . . . , D s ( ) is unlear. 3 Algorithm X In the rest of the pap er w e supp ose that there exists a randomized online algorithm A and a funtion f with f ( ℓ ) log ℓ b eing monotone inreasing (onstan ts are allo w ed), and w e ha v e E[cost( A s ( ℓ, ))] ≤ f ( ℓ ) · opt s ( ℓ, ) + f ( ℓ ) · ℓδ log ℓ (1) for an y ℓ and s . Ha ving Algorithm A , w e an dene our shell algorithm X that uses A as a subroutine inside the blo ks. 3.1 The Algorithm The algorithm uses A as a subroutine and it w orks in phases. Let ( p ) denote the sequene of the p th phase. In this phase the algorithm w orks as follo ws: Initially w e mark the blo ks that on tain no serv ers. When ( p ) i , the i th request of this phase arriv es to blo k B s , w e ompute the demand D s ( ( p ) ≤ i ) and the maximal demand D ∗ s ( ( p ) i ) = max { D s ( ( p ) ≤ j ) | j ≤ i } for this blo k (note that these v alues do not hange in the other blo ks). If D ∗ s ( ( p ) i ) is less than the n um b er of serv ers in B s at that momen t, then the request is serv ed b y Algorithm A , with resp et to the blo k B s . If D ∗ s ( ( p ) i ) b eomes equal to the n um b er of serv ers in B s at that momen t, then the request is serv ed b y Algorithm A , with resp et to the blo k B s and w e mark the blo k B s . If D ∗ s ( ( p ) i ) is greater than the n um b er of serv ers in B s at that momen t, w e mark the blo k B s and p erform the follo wing steps un til w e ha v e D ∗ s ( ( p ) i ) serv ers in that blo k or w e annot exeute the steps (this happ ens when all the blo ks b eome mark ed): 3 • Let us ho ose an unmark ed blo k B s ′ randomly uniformly , and a serv er from this blo k also randomly . W e mo v e this hosen serv er to the blo k B s (su h a mo v e is alled a jump ), either to the requested p oin t, or, if there is already a serv er o up ying that p oin t, to a randomly ho- sen uno upied p oin t of B s . If the n um b er of serv ers in B s ′ b eomes D ∗ s ′ ( ( p ) i ) via this mo v e, w e mark that blo k. In b oth B s and B s ′ w e restart algorithm A from the urren t onguration of the blo k. If w e annot raise the n um b er of serv ers in blo k B s to D ∗ s ( ( p ) i ) b y rep eating the ab o v e steps (all the blo ks b eame mark ed), then Phase p + 1 is starting and the last request is b elonging to this new phase. Our main result is the follo wing: Theorem 5. A lgorithm X is c · log k · f ( k ) - omp etitive for some onstant c . In the follo wing t w o subsetions w e will giv e an upp er b ound for the ost of Algorithm X and sev eral lo w er b ounds for the optimal ost in an arbitrary phase. The ab o v e theorem easily follo ws from these. F or on v eniene w e mo dify the request sequene in a w a y that do es not inrease the optimal ost and do es not derease the ost of an y online algorithm, hene the b ounds w e get for this mo died sequene will hold also in the general ase. The mo diation is dened as follo ws: w e extend the sequene b y rep eatedly requesting the p oin ts of the halting onguration of a (xed) optimal solution. W e do this till P t s =1 D ∗ s ( ( u ) ≤ i ) b eomes k . Observ e that the optimal ost do es not hange via this transformation, and an y online algorithm w orks the same w a y in the original part of the sequene (hene online), so the ost omputed b y an y online algorithm is at least the original omputed ost. 3.2 Upp er b ound In the rst step w e pro v e an auxiliary result. W e reall from [8℄ that an online mat h- ing problem is dened similarly to the online k -serv er problem with the follo wing t w o dierenes: 1. Ea h of the serv ers an mo v ed only one; 2. The n um b er of the requests is at most k , the n um b er of the serv ers. F or an y phase p of Algorithm X w e an asso iate the follo wing mat hing problem MX. The underlying metri spae of the mat hing problem is a nite uniform metri spae that has the blo ks B s as p oin ts and a distane ∆ b et w een an y t w o dieren t p oin ts. Let ˆ D s ( p ) denote the n um b er of serv ers that are in the blo k B s just at the end of phase p . No w in the asso iated mat hing problem w e ha v e ˆ D s ( p − 1) serv ers originally o up ying the p oin t B s . During phase p , if some v alue D ∗ s inreases, w e mak e a n um b er of requests in p oin t B s for the asso iated mat hing problem: w e mak e the same n um b er of requests that the v alue D ∗ s has b een inreased with. W e also asso iate an auxiliary mat hing algorithm (AMA) on this struture as follo ws. Supp ose D ∗ s inreases at some time, ausing jumps. These jumps are orresp onding to requests of the asso iated mat hing problem; AMA satises these requests b y the serv ers that are orresp onding to those in v olv ed in these jumps. 4 Let ˆ D s ( p ) denote the n um b er of serv ers in blo k B s just after phase p . If p is not the last phase, let ( p )+ denote the request sequene w e get b y adding the rst request of phase p + 1 to ( p ) . No w w e ha v e D ∗ s ( ( p ) ) ≤ ˆ D s ( p ) ≤ D ∗ s ( ( p )+ ) (2) and in all blo k but at most one w e ha v e equalities there (this is the blo k that auses termination of the p th phase). Denote m p := t X s =1 max { 0 , ˆ D s ( p ) − ˆ D s ( p − 1) } . (3) Sine the auxiliary metri spae is uniform, the optimal ost is ∆ m p . Lemma 6 (Csaba, Pluhár, [8 ℄) . The exp e te d ost of AMA is at most log k · ∆ m p . Lemma 7. The exp e te d ost of A lgorithm X in the p th phase is at most f ( k ) t X s =1 opt s ( D s ( ( p )+ ) , ( p ) ) + ∆ t X s =1 D s ( ( p )+ ) − k !! + +∆ m p ( f ( k ) log k + f ( k ) + log k ) + ∆ f ( k ) log k . Pr o of . Consider the p th phase of a run of Algorithm X on the request sequene and let τ denote the asso iated run of AMA. Let B s b e a blo k in whi h some request arriv es during this phase. F or the sak e of on v eniene w e omit the index s of the blo k: let opt( ℓ, ′ ) := opt s ( ℓ, ′ ) , ˆ D p − 1 := ˆ D s ( p − 1) and ˆ D p := ˆ D s ( p ) . Denote ( p ) the restition of to B s . While the blo k is unmark ed, only jump-outs an happ en from this blo k; let these jump-outs happ en just b efore the r 1 th, . . . , r d − th request of ( p ) , resp etiv ely . After the blo k has b een mark ed, only jump-ins an happ en; let these happ en when the r d − +1 th, . . . , r d − + d + th request arriv es, resp etiv ely (for an y giv en request there an b e more preeeding jumps). Denote σ i = r i . . . r i +1 − 1 (where r 0 is the rst and r d − + d + +1 − 1 is the last request of the phase in B s ), and let k s,i := k i b e the n um b er of serv ers in B s during σ i . Observ e that the demand at the r i th request is exatly k i . Finally , let ℓ i denote the demand o uring at the r i − 1 th request if this request falls in to the p th phase; otherwise let ℓ i = 0 . A jump-in to the blo k satises the last request, hene there is no serv er mo v emen t inside the blo k during a jump. The exp eted ost of non-jump mo v emen ts in this blo k (this is alled the inner ost ) is, applying (1), at most d − + d + X i =0 E[ A s ( k i , σ i ) | τ ] ≤ d − + d + X i =0 f ( k i )opt( k i , σ i ) + k i · f ( k i ) log k i δ ≤ f ( k ) d − + d + X i =0 opt( k i , σ i ) + δ d − + d + X i =0 k i · f ( k i ) log k i . (4) W e b ound the righ t side of ( 4 ) pieewise. Summing up till the jump-out just b efore the last: d − − 1 X i =0 opt( k i , σ i ) ≤ d − − 1 X i =0 opt( k d − , σ i ) ≤ opt( k d − , ( p ) ≤ r d − ) . (5) 5 F rom the last jump-out till the last jump-in: d − 1 X i = d − opt( k i , σ i ) ≤ d − 1 X i = d − opt( ℓ i +1 , σ i ) = d − 1 X i = d − opt( ℓ i +1 , σ i ) + opt( ℓ i +1 , ( p ) ≤ r i ) − opt( ℓ i +1 , ( p ) ≤ r i ) ≤ d − 1 X i = d − opt( ℓ i +1 , ( p ) 1 m p . Pr o of . Let ( p ) ∗ b e the subsequene of ( p ) whi h w e get b y omitting ea h request that arriv es to a blo k B s after the demand of that giv en blo k rea hes D ∗ s ( ( p )+ ) (note that ( p ) ∗ is not neessarily a prex of ( p ) ). No w w e ha v e t w o ases: rst, if D ∗ s ( ( p )+ ) > C s ( ( p − 1)+ ) holds, then b y Denition 4 opt s ( C ∗ s ( ( p )+ ) , ( p )+ ) + ∆( C ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ opt s ( C ∗ s ( ( p )+ ) , ( p ) ∗ ) + ∆( C ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ (opt s ( D ∗ s ( ( p )+ ) , ( p ) ∗ ) + ∆ 0 , D ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ ∆ max { 0 , ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) } . Otherwise it holds that max { 0 , ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) } = 0 , and also ob viously opt s ( C ∗ s ( ( p )+ ) , ( p )+ ) + ∆( C ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ 0 . F rom b oth ases w e get opt s ( C ∗ s ( ( p )+ ) , ( p )+ ) + ∆( C ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ (21) ∆ max { 0 , ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) } . No w sine t P s =1 ˆ D s ( ( p ) ) = t P s =1 C s ( ( p − 1)+ ) = k , it also holds that t X s =1 ∆ max { 0 , ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) } = t X s =1 1 2 ∆ | ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) | . (22) 8 Summing the ost of the jumps that the optimal solution p erforms w e get ∆ t X s =1 | C s ( ( p )+ ) − C s ( ( p − 1)+ ) | ≤ 2 · opt( k , ( p )+ ) . (23) Note that the fator of 2 omes from the fat that ea h jump app ears t wie on the left hand side. No w summing up (21), (22) and (23 ) w e ha v e 4 · opt( k , ( p )+ ) ≥ ∆ t X s =1 | ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) | + | C s ( ( p )+ ) − C s ( ( p − 1)+ ) | ≥ ∆ t X s =1 | ˆ D s ( ( p ) ) − C s ( ( p )+ )) | . (24) No w summing (21) relativized to phase p and (24) relativized to phase p − 1 w e get that 2 · opt( k , ( p )+ ) + 4 · opt( k , ( p − 1)+ ) ≥ ∆ t X s =1 | ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) | + | ˆ D s ( ( p − 1) ) − C s ( ( p − 1)+ )) | ≥ ∆ t X s =1 | ˆ D s ( ( p ) ) − ˆ D s ( ( p − 1) )) | = ∆ m p , (25) and the statemen t follo ws. Lemma 10. If the p th phase is not the last one, then opt( k , ( p )+ ) ≥ ∆ . Pr o of . Similarly to the pro of of Lemma 9, opt s ( C ∗ s ( ( p )+ ) , ( p )+ ) + ∆( C ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ opt s ( C ∗ s ( ( p )+ ) , ( p ) ∗ ) + ∆( C ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ (opt s ( D ∗ s ( ( p )+ ) , ( p ) ∗ ) + ∆ 0 , D ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) ≥ ∆ max { 0 , ˆ D s ( ( p ) ) − C s ( ( p − 1)+ )) } , so w e get that the optimal ost is at least t X s =1 opt s ( D ∗ s ( ( p )+ ) , ( p ) ∗ ) + ∆( D ∗ s ( ( p )+ ) − C s ( ( p − 1)+ )) . (26) No w sine P t s =1 C s ( ( p − 1)+ ) = k , moreo v er if the p th phase is not the last one, then P t s =1 D ∗ s ( ( p )+ ) > k also holds; applying these w e get the statemen t. 3.4 Pro of of Theorem 5 No w w e are able to pro v e the theorem ab out omp etitiv eness of Algorithm X. Pr o of . [Theorem 5℄ If w e apply (13) to the rst phase and write k instead of m p log k , then Lemmas 7 , 8, 9 and 10 giv e that E(cost( K A ( ))) ≤ f ( k )opt( k , ) + 6 · opt( k , ) f ( k ) log k + f ( k ) + log k + ∆ f ( k ) log k ≤ f ′ ( k ) · opt( k , ) + f ′ ( k ) · k ∆ log k , 9 where f ′ ( k ) = f ( k )(6 log k + 8) . A t this p oin t w e mak e a few remarks: 1. If the starting onguration of the online problem oinides with the starting on- guration of an optimal solution, then it holds that E( K A ( )) ≤ c · f ( k ) · log k · opt( k , ) . 2. It is a bit more natural to require ∆ ≥ δ M to hold, where M is the size of the greatest blo k. If additionally M < k holds, w e get a b etter omp etitiv e ratio. 3.5 Corollaries Starting from the MARKING algorithm [10 ℄ and iterating Theorem 5 w e get the follo wing result: Corollary 11. Ther e exists a ( c 1 log k ) h - omp etitve r andomize d online algorithm on any µ -HST of height h (her e µ ≥ k ), wher e c 1 is a onstant. Conse quently, when h < log k log c 1 +log log k , this algorithm is o ( k ) - omp etitive. Considering only metri spaes ha ving at most t < k blo ks, it is enough to require µ > t . Substituting this to Lemma 7 w e get a omp etitiv e ratio of c · log b · f ( k ) , what (applying the ab o v e orollary) giv es us the follo wing result: Corollary 12. Supp ose the metri sp a e is a µ -HST having only de gr e es of at most b . Then ther e exists a c h 2 - omp etive r andomize d online algorithm, wher e h is the depth of the tr e e and c 2 = c 2 ( b ) is some onstant. 4 F urther questions In the eld of online optimization the onept of buying extra resoures is also in v esti- gated [9℄. The quan tit y min ℓ { opt s ( ℓ, ) + ℓ ∆ } an b e seen as the optimal ost of a mo del where one has to buy the serv ers, for a ost of ∆ ea h. This problem on uniform spaes w as studied in [9 ℄. In this ase D s ( ) is the n um b er of serv ers b ough t in an optimal so- lution. No w onsidering to sequene i , the b eha viour of the asso iated sequene D s ( i ) is unlear at the momen t. It is an in teresting question whether the ab o v e sequene is monotonially inreasing, or do es it hold that | D s ( ) i − D s ( ) i +1 | ≤ 1 for ea h i . The presen ted pro ofs w ould substan tially simplify in b oth ases. Another in teresting question is that whether the log k fator in the omp etitiv e ratio p er lev el of the HST is una v oidable, or an o v erall omp etitiv e ratio of Θ(log k ) holds for an y HST. A kno wledgemen t. The author wish to thank Béla Csaba for his guidane and sug- gestions, furthermore P éter Ha jnal, Csanád Imreh and András Pluhár for their v aluable remarks. Referenes [1℄ Y. Bartal, Probabilisti appro ximation of metri spae and its algorithmi applia- tion. 37th A nnual Symp osium on F oundations of Computer Sien es , pp. 183193, Otob er 1996 10 [2℄ Y. Bartal, On appro ximating arbitrary metris b y tree metris. Pr o e e dings of the 30th A CM Symp osium on The ory of Computing , pp. 161168, 1998 [3℄ Y. Bartal, B. Bollobás, m. Mendel, A Ramsey-t yp e theorem for metri spaes and its appliation for metrial task system and related problems. Pr o e e dings of the 42nd A nnual Symp osium on F oundations of Computer Sien e , Las V egas, Nev ada, 2001 [4℄ Y. Bartal, M. Mendel, Randomized k -serv er algorithms for gro wth-rate b ounded graphs. Journal of A lgorithms , 55(2) :192202, 2005 [5℄ S. Ben-Da vid, A. Boro din, R. Karp, G. T ardos, A. Wigderson, On the p o w er of randomization in on-line algorithms. A lgorithmi a , 11 :214, 1994 [6℄ A. Blum, H. Karlo, Y. Rabani, M. Saks, A deomp osition theorem and b ounds for randomized serv er problems. Pr o e e dings of the 33r d Symp osium on F oundations of Computer Sien e , pp. 197207, Otob er 1992 [7℄ B. Csaba, S. Lo dha, A randomized on-line algorithm for the k-serv er problem on a line. R andom Strutur es and A lgorithms , 29(1) :82104, 2006 [8℄ B. Csaba, A. Pluhár, A randomized algorithm for on-line w eigh ted bipartite mat h- ing problem. Will app ear in Journal of She duling [9℄ J. Csirik, Cs. Imreh, J. Noga, S. S. Seiden, G. W o eginger, Buying a onstan t om- p etitiv e ratio for paging, Pr o e e dings of ESA 2001, LNCS 2161 , 98108, 2001 [10℄ A. Fiat, R. M. Karp, M. Lub y , L. MGeo h, D. Sleator, N. E. Y oung, Comp etitiv e paging algorithms. Journal of A lgorithms , 12(4) :685699, 1994 [11℄ J. F ak haro enphol, S. Rao, K. T alw ar, A tigh t b ound on appro ximating arbitrary metris b y tree metris, J. Comput. System Si. 69(3) , 485497, 2004 [12℄ E. K outsoupias, C. P apadimitriou, On the k-serv er onjeture. Journal of the A CM , 42(5) :971983, 1995 [13℄ L. MGeo h, D. Sleator, A strongly omp etitiv e randomized paging algorithm. A l- gorithmi a , 6(6) :816825, 1991 [14℄ M. Manasse, L. MGeo h, D. Sleator, Comp etitiv e algorithms for serv er problems. Journal of A lgorithms , 11(2) :208230, 1990 [15℄ S. S. Seiden, A general deomp osition theorem for the k-serv er problem. Information and Computation , 174(2) :193202, 2002 11
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment