New On-Line Algorithms for the Page Replication Problem. Susanne Albers y Hisashi Koga z. Abstract

Transcription

1 New On-Line Algoithm fo the Page Replication Poblem Suanne Albe y Hiahi Koga z Abtact We peent impoved competitive on-line algoithm fo the page eplication poblem and concentate on impotant netwok topologie fo which algoithm with a contant competitive atio can be given. We develop an optimal andomized on-line eplication algoithm fo tee and unifom netwok; it competitive atio i appoximately Thi pefomance hold againt obliviou adveaie. We alo give a andomized memoyle eplication algoithm fo tee and unifom netwok that i 2-competitive againt adaptive on-line adveaie. Futhemoe we conide on-line eplication algoithm fo ing and peent geneal technique that tanfom c-competitive algoithm fo tee into 2c-competitive algoithm fo ing. A a eult we obtain a andomized on-line algoithm fo ing that i 3.16-competitive. We alo deive two 4-competitive on-line algoithm fo ing which ae eithe deteminitic o andomized and memoyle. Again, the andomized eult hold againt obliviou adveaie. Apat fom thee technique, we nally give a andomized memoyle eplication algoithm fo ing that i 4-competitive againt adaptive on-line adveaie. 1 Intoduction Thi pape deal with poblem that aie in the memoy management of lage multipoceo ytem. Such multipoceing envionment typically conit of a netwok of poceo, each of which ha it local memoy. A global haed memoy i modeled by ditibuting the phyical page among the local memoie. Accee to the global memoy ae accomplihed by acceing the local memoie. Suppoe a poceo p want to ead a memoy adde fom page B. If B i toed in p' local memoy, then thi ead opeation can be accomplihed locally. Othewie, p detemine a poceo q holding the page and end a equet to q. The deied infomation i then tanmitted fom q to p, and the communication cot incued theeby i popotional to the ditance fom q to p. If p ha to acce page B fequently, it may be wothwhile to move o Thi pape combine and extend eult of two confeence pape. Fit pape: H. Koga. Randomized on-line algoithm fo the page eplication poblem. In Poc. 4th Intenational Annual Sympoium on Algoithm and Computation (ISAAC93). Second pape: S. Albe and H. Koga. New on-line algoithm fo the page eplication poblem. In Poc. 4th Scandinavian Wokhop on Algoithm Theoy (SWAT94). y Max-Planck-Intitut fu Infomatik, Im Stadtwald, Saabucken, Gemany. Wok wa uppoted in pat by the ESPRIT Baic Reeach Action of the EU unde contact No (poject ALCOM II). albe@mpi-b.mpg.de z Fujitu Laboatoie Limited., Kawaaki, Japan. Thi wok wa done while the econd autho wa a tudent in the Depatment of Infomation Science, Univeity of Tokyo. koga@ab.fujitu.co.jp 1

2 copy B fom q to p becaue ubequent accee will become cheape. Howeve, tanmitting an entie page incu a high communication cot popotional to the page ize time the ditance fom q to p. If a page i witable, it i eaonable to toe only one copy of the page in the entie ytem. Thi avoid the poblem of keeping multiple copie of the page conitent. The migation poblem i to decide in which local memoy the ingle copy of the witable page hould be toed o that a equence of memoy accee can be poceed at low cot. On the othe hand, if a page i ead-only, it i poible to keep eveal copie of the page in the ytem, i.e., a page may be copied fom one local memoy to anothe. In the eplication poblem we have to detemine which local memoie hould contain copie of the ead-only page. Finding ecient migation and eplication tategie i an impotant poblem that ha been tudied fom a pactical and theoetical point of view [2, 4, 6, 7, 8, 11, 13]. In thi pape we tudy on-line algoithm fo the page eplication poblem. We analyze the pefomance of on-line algoithm uing competitive analyi [12], the wot cae atio of the cot incued by an on-line algoithm and the cot incued by an optimal o-line algoithm. Awebuch et al. [2] peented a deteminitic on-line eplication tategy fo geneal gaph that achieve an optimal competitive atio of (log n), whee n i the numbe of poceo. Howeve, fo many impotant topologie, thi bound i not vey expeive. Black and Sleato [6], who initiated the theoetical tudy of the eplication poblem, popoed a 2-competitive deteminitic on-line algoithm fo tee and unifom netwok. A unifom netwok i a complete gaph in which all edge have the ame length. Black and Sleato alo poved that no deteminitic on-line eplication algoithm can be bette than 2-competitive. Batal et al. [4] peented a andomized 2(2 + p 3)-competitive eplication algoithm againt adaptive on-line adveaie fo the cae that the netwok topology fom a ing. We note that 2(2 + p 3) 7:5. Uing the 2(2 + p 3)-competitive algoithm by Batal et al., one can contuct a deteminitic eplication algoithm fo the ing that achieve a competitive atio of 2 2 (2 + p 3) 2 55:7, ee [5]. Howeve, that algoithm i vey complicated and not ueful in pactical application. In thi pape we develop a numbe of new deteminitic and andomized on-line eplication algoithm. We concentate on netwok topologie that ae impotant in pactice and fo which on-line algoithm with a contant competitive atio can be developed. In Section 4.1 we peent a andomized on-line eplication algoithm fo tee and unifom netwok, called GEOMETRIC, which i ( +1 )-competitive againt obliviou adveaie. Hee = 1 and i the page ize facto. Fo lage value of, which occu in pactice, GEOMETRIC' competitivene i appoximately e 1:58. We alo how that GEOMETRIC i optimal. Specically we pove e 1 that no andomized on-line eplication algoithm can be bette than ( )-competitive againt 1 obliviou adveaie. Inteetingly, ou algoithm GEOMETRIC ue only one andom numbe duing an initialization phae and un completely deteminitically theeafte. Lund et al. [9] have independently developed the ame eult fo tee and unifom netwok uing a dieent appoach. Moeove, we give a andomized memoyle on-line eplication algoithm fo tee and unifom netwok that i 2-competitive againt adaptive on-line adveaie. Thi i the bet competitivene that can be achieved againt adaptive on-line adveaie. In Section 5 we conide eplication algoithm fo ing. We peent a deteminitic technique that tanfom c-competitive algoithm fo tee into 2c-competitive algoithm fo ing. 2

3 Combining thi technique with the algoithm GEOMETRIC, we obtain a andomized algoithm fo ing that achieve a competitive atio of ( 2 ) 3:16. We alo deive two 4-competitive 1 algoithm fo ing which ae eithe deteminitic o andomized and memoyle. The andomized pefomance hold againt obliviou adveaie. Ou 4-competitive deteminitic algoithm i vey imple and geatly impove the competitive atio of 55.7 mentioned above. We alo peent a andomized veion of ou deteminitic technique fo contucting ing algoithm; thi vaiant achieve the ame pefomance. Finally, uing an appoach dieent fom the above technique, we develop a andomized memoyle eplication algoithm fo ing that i 4-competitive againt adaptive on-line adveaie. 2 Poblem tatement and competitive analyi Fomally, the page eplication poblem can be decibed a follow. We ae given an undiected gaph G. Each node in G coepond to a poceo and the edge epeent the inteconnection netwok. Aociated with each edge i a length that i equal to the ditance between the connected poceo. We aume that the edge length atify the tiangle inequality. In the page eplication poblem we geneally concentate on one paticula page. We ay that a node v ha the page if the page i contained in v' local memoy. A equet at a node v occu if v want to ead an adde fom the page. The equet can be atied at zeo cot if v ha the page. Othewie the equet i eved by acceing a node w holding the page and the incued cot equal the ditance fom v to w. Afte the equet i atied, the page may be eplicated fom node w to any othe node v 0 that doe not hold the page (node v 0 may coincide with node v). The cot incued by thi eplication i time the ditance fom w to v 0. Hee denote the page ize facto. In pactical application, i a lage value, uually eveal hunded o thouand. (The page may only be eplicated afte a equet becaue it i impoible to delay the evice of the memoy acce while the entie page i copied.) We tudy the page eplication poblem unde the aumption that a node having the page neve dop it. A page eplication algoithm i uually peented with an entie equence of equet that mut be eved with low total cot. A page eplication algoithm i on-line if it eve evey equet without knowledge of any futue equet. We analyze the pefomance of on-line page eplication algoithm uing competitive analyi [12]. In a competitive analyi, the cot incued by an on-line algoithm i compaed to the cot incued by an optimal o-line algoithm. An optimal o-line algoithm know the entie equet equence in advance and can eve it with minimum cot. Let C A () and C OP T () be the cot of the on-line algoithm A and the optimal o-line algoithm OPT on a equet equence. Uually an on-line algoithm i called c-competitive if thee exit a contant a uch that C A () c C OP T () + a hold fo evey equet equence. Note, howeve, that if the contant a depend on and the numbe of poceo in the netwok, then an on-line eplication algoithm can be 0-competitive by eplicating the page initially to all poceo and aigning the total cot of initial eplication to a. On the othe hand, if a doe not depend on, an additive contant cannot educe the competitivene of an on-line eplication algoithm becaue can be lage elative to the cot of eving a numbe of accee. Theefoe, we ue a tonge denition. 3

4 We call an on-line eplication algoithm c-competitive if C A () c C OP T () fo all equet equence. If A i a andomized algoithm, then C A () mut be eplaced by the expected cot incued by A, whee the expectation i taken ove the andom choice made by A. In thi pape we evaluate andomized on-line algoithm only againt obliviou and adaptive on-line adveaie, ee [5] fo detail. An obliviou adveay ha to geneate a equet equence in advance and i not allowed to ee the andom choice made by the on-line algoithm. An adaptive on-line adveay may ee the andom choice made by the on-line algoithm, i.e., when geneating a new equet the adveay can ee all the on-line algoithm' andom choice on pat equet. Howeve, an adaptive on-line adveay alo ha to eve the equet equence on-line. 3 Baic denition and technique Befoe decibing ou new algoithm in the following ection, we intoduce ome baic denition fo tee. Thee will be ueful thoughout the pape, ince, even when conideing unifom netwok o ing, we will often educe the algoithm and thei analye to the cae that the undelying topology fom a tee. The oot of the given tee i geneally denoted by. We aume that initially, only ha the page. Conide an undiected edge e = fv; wg in the tee. The node in fv; wg that i fathe away fom the oot i called the child node of e. The length of e i denoted by l(e). Given two node u and v in the tee, let l(u; v) denote the length of the (unique) path fom u to v. In the following we will alway aume that if an algoithm (on-line o o-line) eplicate the page fom a node w to a node v, then the page i alo eplicated to all node on the path fom w to v. Thi doe not incu exta cot. Thu, the node with the page alway fom a connected component of the given tee. Note that if a node v doe not have the page, then the cloet node w with the page lie on the path fom v to the oot, and all path fom v to a node with the page pa though w. Theefoe, we may aume without lo of geneality that a eplication algoithm alway eve equet at a node not holding the page by acceing the cloet node with the page. Thi cannot inceae the total cot incued in eving the whole equet equence. We peent a technique that we will fequently ue to analyze on-line eplication algoithm fo tee. Let T be a tee and be a equet equence fo T. We uually analyze an on-line eplication algoithm A by patitioning the cot that ae incued by A and by OPT into pat that ae incued by each edge of the tee. Suppoe an algoithm eve a equet at a node v. Then an edge e incu a cot equal to the length of e if e belong to the path fom v to the cloet node with the page. If e doe not belong to that path, then e incu a cot of zeo. An edge alo incu the cot of a eplication aco it. Let C A (; e) denote the cot that i incued by edge e when A eve. Analogouly, let C OP T (; e) be the cot that i incued by e when OPT eve. (If A i a andomized algoithm, then C A (; e) i the expected cot incued by e.) The pefomance of an on-line algoithm A i geneally evaluated by compaing C A (; e) to 4

5 C OP T (; e) fo all edge e of the tee. In ode to analyze C A (; e), we intoduce ome notation. Let = (1); (2); : : :; (m) be a equet equence of length m and let (t), 1 t m, be the equet at time t. Suppoe (t) i a equet at node v. We et a (e; t) = 1 if e belong to the path fom v to the oot. Othewie we et a (e; t) = 0: If a (e; t) = 1, we ay that (t) caue an acce at edge e. Let a (e) = mx t=1 a (e; t); i.e., a (e) i the numbe of equet that caue an acce at edge e. The following imple lemma i cucial in ou analye. Lemma 1 Let A be an on-line eplication algoithm that, given an abitay tee T and a equet equence fo T, atie C A (; e) c minfa (e); g l(e) (1) fo all edge e. Then the algoithm A i c-competitive. (If A i a andomized algoithm, then C A (; e) i the expected cot incued by e and the competitive atio of c hold againt any obliviou adveay.) Poof: We pove that fo any edge e, C OP T (; e) = minfa (e); g l(e). By Eq. (1), thi implie C A (; e) c C OP T (; e) fo all edge e, and hence A i c-competitive. If a (e) <, then OPT doe not eplicate the page aco e and e incu a cot of a (e)l(e). Hence C OP T (; e) = a (e) l(e) = minfa (e); gl(e): On the othe hand, if a (e), then OPT eplicate the page aco e immediately, befoe eving any equet, and e incu a cot of l(e). Thu C OP T (; e) = l(e) = minfa (e); gl(e): 2 4 Algoithm fo tee and unifom netwok Fit, in Section 4.1, we decibe and analyze two andomized on-line algoithm fo tee. The t of thee algoithm achieve an optimal competitive atio againt any obliviou adveay. We alo give an algoithm that i competitive againt any adaptive on-line adveay. In Section 4.2 we demontate that both ou algoithm can be eaily applied to unifom netwok, while maintaining thei competitive pefomance. Thoughout thi ection let = Tee Algoithm GEOMETRIC (fo tee): The algoithm t chooe a andom numbe fom the et f1; 2; : : :; g. Specically, the numbe i i choen with pobability p i = i 1, whee = 1. While poceing the equet equence, the algoithm maintain a count on each 1 5

6 edge of the tee. Initially, all count ae et to 0. If thee i a equet at a node v that doe not have the page, then all count along the path fom v to the cloet node with the page ae incemented by 1. When a count eache the value of the andomly choen numbe, the page i eplicated to the child node of the coeponding edge. Befoe we analyze the pefomance of GEOMETRIC, we mention a few obevation and emak. The algoithm i called GEOMETRIC becaue p i+1 =p i = i contant fo all i = 1; 2; : : :; 1. It i eay to veify that P p i = 1. Suppoe that GEOMETRIC pocee a equet equence. We can eaily pove by induction on the numbe of equet poceed o fa that the count on a path fom the oot to a node v ae alway monotonically non-inceaing. Futhemoe, afte each equet, a node (except fo the oot ) ha the page if and only if it i the child node of an edge whoe count i equal to the value of the andomly choen numbe. Theoem 1 Fo any tee, the algoithm GEOMETRIC i ( )-competitive againt any obliviou 1 adveay. Note that goe to e 1:58 a tend to innity. Futhemoe, GEOMETRIC ue 1 e 1 only one andom numbe duing an initialization phae and un completely deteminitically theeafte. Poof: Conide an abitay tee T and a equet equence fo T. Let e be an edge of the tee and let E[C G (; e)] denote the expected cot incued by edge e when GEOMETRIC eve. We will how that E[C G (; e)] ( 1 ) minfa (e); g l(e) (2) fo any edge e of T. Lemma 1 implie the theoem. Let k = a (e) and (t 1 ); (t 2 ); : : :; (t k ) be the equet in that caue an acce at the edge e. Note that the algoithm GEOMETRIC inceae the count of e exactly at the equet (t 1 ); (t 2 ); : : :; (t k ), povided that the page ha not been eplicated aco e o fa. P Fit, aume that k >. Since p i = 1, GEOMETRIC ha eplicated the page aco e befoe the equet (t +1 ). Thu the edge e incu the ame cot a if we had k =. Fo thi eaon it uce to conide the cae that k atie 1 k and how E[C G (; e)] c k l(e), whee c = 1. Thi pove (2). So uppoe we have 1 k. The algoithm GEOMETRIC t chooe a andom numbe i fom the et f1; 2; : : :; g. If i atie i k, the edge e incu a cot of + i. Othewie e incu a cot of k. Thu E[C G (; e)] = l(e)( = l(e)( ( + i)p i + i 1 + = l(e)( (k 1) 1 X i=k+1 kp i ) i i 1 + X i=k+1 k i 1 ) + kk+1 (k + 1) k k( k ) ): ( 1) 2 1 6

7 We have 1 = 1. Thu E[C G (; e)] = l(e) 1 ((k 1) + k k ( k 1) + k( k )) = l(e) 1 (k ) = 1 k l(e): 2 We now pove that GEOMETRIC' competitive atio i optimal fo all value of. Theoem 2 Let A be a andomized on-line eplication algoithm. Then A cannot be bette than )-competitive againt any obliviou adveay, even on a gaph coniting of two node. ( 1 Poof: Let and t be two node connected by an edge of length 1. We aume that initially, only node ha the page. We will contuct a equet equence coniting of equet at node t uch that the expected cot incued by A i at leat time the optimal o-line cot. 1 Fo i = 1; 2; : : :, let q i be the pobability that A eplicate the page fom to t afte exactly i equet, given a equet equence that conit only of equet at node t. In the following we compae the algoithm A to the algoithm GEOMETRIC. Let E[C A ()] and E[C G ()] denote the expected cot incued by A and GEOMETRIC on a equet equence. Futhemoe, fo i = 1; 2; : : :;, let p i = i 1. We conide two cae. Cae 1: Thee exit an l, whee 1 l, uch that P l q i P l p i. Let k be the mallet numbe atifying the above inequality, i.e., P k q i P k p i and P j q i < P j p i fo all j with 1 j < k. Let be the equet equence that conit of k equet at node t. We how that the inequality E[C A ()] E[C G ()] 0 hold. Thi implie E[C A ()] E[C G ()] = k and A cannot be bette than P ( )-competitive becaue the 1 1 k optimal o-line cot on equal k. Since, with pobability 1 q i, A ha not eplicated the page to t afte the evice of the equet equence, we have Similaly, we alo have E[C A ()] = E[C G ()] = Hence E[C A ()] E[C G ()] = Since P k q i P k p i and k 0, we obtain E[C A ()] E[C G ()] ( + i)q i + k(1 ( + i)p i + k(1 i(q i p i ) + ( k) i(q i p i ) = ( q i ): p i ): (q i p i ). j=i q j j=i p j ): 7

8 P i 1 j=1 q j P i 1 P P i 1 Fo i = 2; 3; : : :; k we have j=1 q i 1 j < P P P P P P j=1 p k k k k j and hence j=i q j j=i p j > j=i q j j=i p j + j=1 p k k j = j=1 q j j=1 p j : We conclude E[C A ()] E[C G ()] ( j=i q j j=i p j ) ( j=1 q j Cae 2: Fo all k = 1; 2; : : :;, the inequality P k q i < P k p i i atied. j=1 p j ) 0: Let be the equet equence that conit of 2 equet P at node t. Let A 0 be the on-line algoithm with qi 0 = q i, fo i = 1; 2; : : :; 1, and q 0 1 = 1 q i. Then E[C A ()] = = X2 1 X ( + i)q i + 2(1 X2 q i ) 1 X ( + i)q 0 i + 2q 0 = E[C A 0()]: Since P q 0 i = P p i = 1 and P j q0 i < P j ( + i)q i + 2(1 p i fo all j with 1 j <, Cae 1 immediately implie E[C A ()] E[C A 0()] E[C G ()] = ; and A cannot be bette than ( 1 )- 1 competitive becaue the optimal o-line cot equal. 2 Next we peent anothe on-line eplication algoithm fo tee. Thi algoithm ha the advantage of being memoyle, i.e., it doe not need any memoy (fo intance fo count) in ode to detemine when a eplication hould take place. Alo, it competitive pefomance hold againt adaptive on-line adveaie. Algoithm COINFLIP (fo tee): If thee i a equet at a node with the page, then the algoithm pefom no action. If thee i a equet at a node v without the page, the algoithm eve the equet by acceing the cloet node u with the page. Then with pobability 1, the algoithm eplicate the page fom u to v. 1 X q i ) Theoem 3 The algoithm COINFLIP i 2-competitive againt any adaptive on-line adveay. Poof: We ue a potential function to analyze COINFLIP. Fo any equet equence geneated by an adaptive on-line adveay ADV, we compae imultaneou un of COINFLIP and ADV on by meging the action of both algoithm into a ingle equence of event. Thi equence contain two type of event: (Type I) ADV eplicate the page. (Type II) A equet i eved by COINFLIP and ADV; thi event may be accompanied by COINFLIP eplicating the page to the equeting node. Fo any event, let C CF and C ADV denote the cot incued by COINFLIP and ADV duing the event, and let denote the change in potential. We will how that fo any event, Summing up thi inequality fo all event, we obtain E[C CF ] + E[] 2C ADV : (3) E[C CF ()] + E[ end ] E[ tat ] 2C ADV (); (4) 8

9 whee tat and end denote the initial and nal potential. Since we will chooe the potential function uch that i alway non-negative and uch that the initial potential i 0, (4) implie that COINFLIP i 2-competitive. We dene the potential function. Let E be the et of edge e in the tee T uch that ADV ha eplicated the page to the child node of e but COINFLIP ha not eplicated the page to the child node. Let X = 2 l(e): e2e In the following we pove (3) fo all event. Let ch(e) denote the et of the child node of all edge contained in E. Type I: ADV eplicate the page. Suppoe that the page i eplicated fom node u to node v (and to all node along the path fom u to v). Then C ADV = l(u; v) and C CF = 0. Thu we mut how 2l(u; v). Thee ae two cae to conide depending of whethe v 2 ch(e) afte the eplication. Cae 1: v =2 ch(e) afte the eplication. Then = 0. Cae 2: v 2 ch(e) afte the eplication. If u wa in ch(e) befoe the eplication, then = 2l(u; v). Othewie 2l(u; v). Type II: A equet i eved by COINFLIP and ADV. Let v be the node at which the equet occu. We have to conide two cae. Cae 1: In the tee maintained by COINFLIP, node v aleady ha the page. Then C CF = 0 and C ADV 0. Alo = 0 becaue COINFLIP doe not eplicate the page. Inequality (3) i atied. Cae 2: In the tee maintained by COINFLIP, node v doe not have the page. Let u CF be the node cloet to v in the tee to which COINFLIP ha eplicated the page. Recall that u CF lie on the path fom v to the oot. COINFLIP incu a cot of l(u CF ; v) in eving the equet. Then with pobability 1, COINFLIP alo eplicate the page fom u CF to v. Theefoe E[C CF ] = l(u CF ; v) + 1 l(u CF ; v) = 2l(u CF ; v). Fo the evaluation of C ADV, we have to conide thee cae, depending on ADV' conguation of node with the page. Fit uppoe that ADV ha not eplicated the page beyond u CF when the equet occu. Then C ADV l(u CF ; v) = 1 E[C 2 CF ]. Note that E[] = 0 becaue doe not change egadle of whethe COINFLIP eplicate the page o not. Inequality (3) hold. Next imagine that ADV ha eplicated the page beyond u CF but not beyond v. Let u ADV be the node cloet to v to which ADV ha eplicated the page. Then C ADV = l(u ADV ; v). We have to how that the expected change in potential i E[] = 2l(u CF ; u ADV ). Thi implie E[C CF ] + E[] = 2l(u CF ; v) 2l(u CF ; u ADV ) = 2l(u ADV ; v) = 2C ADV. We have = 0 if COINFLIP doe not eplicate the page; othewie = 2l(u CF ; u ADV ). Hence E[] = 1 ( 2l(u CF ; u ADV )) = 2l(u CF ; u ADV ). Finally, uppoe that ADV ha eplicated the page beyond v. In thi cae C ADV = 0. If COINFLIP doe not eplicate the page, then = 0. Othewie = 2l(u CF ; v). Theefoe, E[] = 1 ( 2l(u CF ; v)) = 2l(u CF ; v) = E[C CF ]. Inequality (3) hold. 2 The COINFLIP algoithm achieve the bet poible pefomance. No andomized on-line algoithm A can be bette than 2-competitive againt any adaptive on-line adveay, even on 9

10 a gaph coniting of two node. Thi can be een a follow. Conide two node and t connected by an edge of length 1 and aume that ha the page initially. An adaptive on-line adveay iue equet at t until A eplicate the page to t. With pobability 1, the adveay 2 initially eplicate the page to t, and with pobability 1 it eve all the equet by acceing 2. Suppoe that A eplicate the page afte k equet. Then A' cot i k +, wheea the expected cot of the adveay i 1 (k + ). If A neve eplicate the page to t, then, by making 2 the equet equence uciently long, we can achieve a lowe bound of 2, fo any > Unifom netwok Any eplication algoithm fo tee can be eaily applied to unifom netwok. Conide an abitay unifom netwok and let be the node that ha the page initially. Since all edge in the gaph have the ame length, we may aume without lo of geneality that a eplication algoithm (on-line o o-line) eve equet and eplicate the page only along edge f; vg. Hence the netwok can be educed to a tee by neglecting the edge fv; wg with v 6=, w 6=. Run on thi tee, any on-line algoithm fo tee can maintain it competitive pefomance. The eult given in Section 4.1 imply the following coollaie. Coollay 1 The algoithm GEOMETRIC fo unifom netwok i ( )-competitive againt 1 any obliviou adveay. Thi i the bet competitive atio that a andomized on-line eplication algoithm can achieve againt thi type of adveay. Coollay 2 The algoithm COINFLIP fo unifom netwok i 2-competitive againt any adaptive on-line adveay. Thi i the bet competitive atio that a andomized on-line eplication algoithm can achieve againt thi type of adveay. 5 Algoithm fo the ing In thi ection we aume that the given net of poceo fom a ing. Fit we will peent technique that tanfom c-competitive algoithm fo tee into 2c-competitive algoithm fo ing. Uing thee technique, we obtain a deteminitic ing algoithm and andomized ing algoithm that ae competitive againt any obliviou adveay. Then we will develop a andomized eplication algoithm fo ing that i competitive againt any adaptive on-line adveay. We aume that initially, only one node of the ing, ay, ha the page. Let n be the numbe of node in the ing and let v 1 ; v 2 ; : : :; v n be the node if we can the ing in clockwie diection tating fom, i.e., v 1 =. Fo i = 1; 2; : : :; n, let e i = fv i ; v i+1 g be the undiected edge fom v i to v i+1. Natually, v n+1 equal v 1. Again, fo any edge e i, l(e i ) i the length of e i. Let x and y be any two point on the ing; x and y need not neceaily be poceo node. We denote by (x; y) the ac of the ing that i obtained if we tat in x and go to y in clockwie diection. Let l(x; y) be the length of the ac (x; y). 10

11 5.1 Geneal technique Fit we peent a deteminitic tategy fo contucting ing algoithm. Algoithm RING: Let P, P 6=, be the point on the ing atifying l(; P ) = l(p; ), i.e., P i the point \oppoite" to. The algoithm t cut the ing at P. It egad the eulting tuctue a a tee T with oot = v 1. The ac (; P ) epeent one banch of the tee and the ac (P; ) epeent anothe banch of the tee (ee Figue 1). We aume that the point P become pat of the ac (; P ). Thi i ignicant if P coincide with one of the poceo node v i. The algoithm RING then ue an on-line eplication algoithm A fo tee in ode to eve a equet equence. That i, RING aume that i a equet equence fo T and eve the equet equence uing the tee algoithm A. = v 1 ṣ P - Fig. 1: A cut of the ing = @ P Theoem 4 Let A be an on-line eplication algoithm that i c-competitive fo an abitay tee. If the algoithm RING ue A a tee algoithm, then the eulting algoithm i 2c-competitive. (If A i a andomized on-line algoithm, then the competitive atio of 2c hold againt any obliviou adveay.) Befoe we pove thi theoem, we mention ome impotant implication. Theoem 1 immediately implie the following eult. Coollay 3 If RING ue the algoithm GEOMETRIC a the tee algoithm, then the eulting algoithm i c-competitive againt any obliviou adveay, whee c = 2 1 : We obeve that c goe to 2e 3:16 a tend to innity. Alo note that if RING ue the e 1 GEOMETRIC algoithm, then only one andom numbe i ued duing an initialization phae. Next we conide the deteminitic eplication algoithm fo tee popoed by Black and Sleato [6]. The algoithm achieve an optimal competitive atio of 2. Algoithm DETERMINISTIC COUNT: The algoithm wok in the ame way a the algoithm GEOMETRIC. Howeve DETERMINISTIC COUNT doe not chooe a andom numbe in ode to detemine when a eplication hould occu. Rathe it eplicate the page to the child node of an edge when the coeponding count eache. Coollay 4 If the algoithm RING ue DETERMINISTIC COUNT a the tee algoithm, then the eulting algoithm i 4-competitive. 11

12 We emak that the combination of RING and DETERMINISTIC COUNT i a complete deteminitic on-line algoithm. Theoem 4 and Theoem 3 imply the following eult. Coollay 5 If RING ue the algoithm COINFLIP a tee algoithm, then the eulting algoithm i 4-competitive againt any obliviou adveay. Note that the combination of RING and COINFLIP i memoyle. Next we peent a andomized vaiant of the algoithm RING and a tatement analogou to Theoem 4. Algoithm RING(RANDOM): The algoithm wok in the ame a the algoithm RING. Howeve, intead of cutting the ing at the point oppoite to, the algoithm RING(RANDOM) chooe a point P unifomly at andom on the ing and cut the ing at that point P. Theoem 5 Let A be an on-line eplication algoithm that i c-competitive fo an abitay tee. If the algoithm RING(RANDOM) ue A a tee algoithm, then the eulting algoithm i 2c-competitive againt any obliviou adveay. Theoem 5 implie that if the cutting point P i choen andomly, the ame competitive pefomance i obtained a if the cutting point i choen deteminitically to be the point oppoite to. Theefoe, tatement analogou to Coollaie 3-5 hold. Note, howeve, that a combination of RING(RANDOM) and DETERMINISTIC COUNT i not a puely deteminitic algoithm. It emain to pove the above theoem. In the following we peent a detailed poof of Theoem 4. Since Theoem 5 i an inteeting tatement but doe not yield tonge eult than Theoem 4, we omit a poof of Theoem 5. The poof of Theoem 5 i imila to that of Theoem 4. Poof of Theoem 4: Let be a equet equence fo the ing. We tat with ome obevation on how the optimal o-line algoithm OPT eve. Conide the tate of the ing afte OPT ha eved. Let u a and u b be the node fathet fom to which OPT ha eplicated the page in clockwie and counte-clockwie diection, epectively. Figue 2(a) illutate thi ituation. We may aume without lo of geneality that OPT eplicate the page fom to u a and fom to u b at the beginning of the equet equence, befoe any equet ae eved. Thi doe not incu a highe cot a if the eplication i done while equet ae poceed. ṣ ṣ u b u b u a u a Fig. 2. (a) Q (b) Any equet at a node that belong to (; u a ) o (u b ; ) can then be eved at zeo cot. Let Q be the point on (u a ; u b ) which atie l(u a ; Q) = l(q; u b ), ee Figue 2(b). Any equet at a 12

13 node v that belong to (u a ; Q) i eved by acceing u a and the incued cot equal l(u a ; v). Any equet at a node v that belong to (Q; u b ) i eved by acceing u b, and the incued cot equal l(v; u b ). Let C R () be the cot incued by RING in eving. Futhemoe, let T be the tee that i obtained if the ing i cut at point P. ṣ ṣ u b ua Fig. 3. u a P (a) P (b) u b Cae 1: Suppoe that P belong eithe to (; u a ) o to (u b ; ), ee Figue 3. Let TOPT be the o-line algoithm that eve optimally, i.e. with minimal cot, on the tee T. By aumption, ince the tee algoithm A i c-competitive, C R () c C T OP T (). Alo, C T OP T () 2 l(; P ): Since C OP T () l(; P ), we obtain and the theoem i poved. C R () c C T OP T () 2c l(; P ) 2c C OP T () Cae 2: Now uppoe that P belong neithe to (; u a ) no to (u b ; ). We only conide the cae that l(; u a ) l(u b ; ). The cae l(u b ; ) l(; u a ) i ymmetic. Now, let TOPT be the algoithm that t eplicate the page fom to u a and fom to u b in clockwie and counte-clockwie diection, epectively, and then eve the equet equence a follow. Any equet at a node v that belong to (u a ; P ) i eved by acceing u a, and any equet at a node v that belong to (P; u b ), v 6= P, i eved by acceing u b. Since both RING and TOPT ue T a undelying tee and ince tee algoithm A i c-competitive, we have In the following we will how that C R () c C T OP T (): C T OP T () 2 C OP T (): (5) Thi implie the theoem. ṣ u b u a Q P Fig

14 We compae the cot incued by TOPT and OPT. Note that only equet at node on (P; Q) ae eved in dieent way by TOPT and OPT. Fo each equet on (P; Q), TOPT incu a cot that i by at mot 2l(P; Q) geate than the cot incued by OPT. Thee occu at mot equet on (P; Q), ince othewie OPT would have eplicated the page fom u a beyond P. Thu C T OP T () C OP T () + 2l(P; Q): We have l(p; Q) = 1 2 (l(; u a) l(u b ; )) 1 2 l(; u a). Thu, and (5) i poved. 2 C T OP T () C OP T () + l(; u a ) 2 C OP T () 5.2 A andomized algoithm againt adaptive adveaie We peent an on-line eplication algoithm that i 4-competitive againt any adaptive on-line adveay. Thi algoithm ha the additional advantage of being memoyle. The eplication tategy ued by the algoithm i motivated by the HARMONIC k-eve algoithm [10]. Algoithm HARM-RING: If thee i a equet at a node with the page, then the algoithm pefom no action. If thee i a equet at a node v without the page, then let w a and w b be the node fathet fom to which HARM-RING ha eplicated the page in clockwie and counteclockwie diection, epectively. With pobability l(wa;v) l(w a;w b the algoithm eve the equet by ) acceing w b, and with pobability l(v;w b) the algoithm eve the equet by acceing l(w w a;w b ) a. Then with pobability 1, HARM-RING eplicate the page to v fom the node that wa actually acceed duing the evice of the equet. Theoem 6 The algoithm HARM-RING i 4-competitive againt any adaptive on-line adveay. Poof: Fo any equet equence geneated by an adaptive on-line adveay, we compae imultaneou un of HARM-RING and ADV on. A in the poof of Theoem 3, the action of HARM-RING and ADV can be claied into two type of event. (Type I) ADV eplicate the page. (Type II) A equet i eved by HARM-RING and ADV; thi event may be accompanied by HARM-RING eplicating the page to the equeting node. A befoe, we will give a nonnegative potential function, that i initially 0, o that E[C HR ] + E[] 4 C ADV (6) fo all event. Thi implie the theoem. Hee C HR and C ADV denote the cot incued by HARM-RING and ADV duing the event; i the change in potential. We dene the potential function. At any given time, let u a and u b be the node fathet fom to which ADV ha eplicated the page in clockwie and counte-clockwie diection, epectively. Recall that w a and w b ae the node fathet fom to which HARM-RING ha eplicated the page. Let = 4 maxf0; l(; u a ) l(; w a )g + 4 maxf0; l(u b ; ) l(w b ; )g: 14

15 Intuitively, i the length of the ange of the ing at which ADV ha the page but HARM-RING ha not. We will how (6) fo all event. (Type I) ADV eplicate the page. Suppoe that the page i eplicated fom node u to node v. Aume without lo of geneality that the page i eplicated in clockwie diection, i.e., the ac (; u a ) i extended. The cae that the ac (u b ; ) i extended i analogou. We have C ADV = l(u; v) and C HR = 0. We mut how E[] 4l(u; v). If neithe node u no v ha the page in the ing maintained by HARM-RING, then = 4l(u; v). Othewie 4l(u; v). Inequality (6) hold. (Type II) A equet i eved by HARM-RING and ADV. Let v be the node equeting the page. If v ha the page in the ing maintained by HARM-RING, then C HR = 0, C ADV 0 and = 0 becaue HARM-RING doe not eplicate the page. Inequality (6) i atied. In the emainde we aume that v doe not have the page in the ing maintained by HARM-RING, i.e., v lie on the ac (w a ; w b ) and w a 6= v 6= w b. Then E[C HR ] = l(v; w b) l(w a ; w b ) (l(w a; v) + 1 l(w a; v)) + l(w a; v) l(w a ; w b ) (l(v; w b) + 1 l(v; w b)) = 4 l(w a; v) l(v; w b ) : l(w a ; w b ) We may aume without lo of geneality that in the ing maintained by ADV, v lie on the ac (; u a ) o ADV eve the equet by acceing u a. The cae that v lie on the ac (u b ; ) o that ADV eve the equet by acceing u b i ymmetic. Fo the evaluation of E[] and C ADV we invetigate thee cae egading the elative poition of u a ; w a and v, a illutated in Figue 5. We intoduce an odeing on the node of the ing uch that fo two node x; y: x < y if l(; x) < l(; y): ṣ ṣ ṣ u a wa wa wa u a u a v v v (1) (2) (3) Fig. 5: Thee cae egading the location of u a, w a, and v Cae 1: Suppoe that w a < v u a. We have C ADV = 0. The change in potential i 4l(w a ; v) if HARM-RING eplicate the page fom w a to v. If HARM-RING eplicate the page fom w b to v, then the change in potential i non-poitive. Thu E[] 1 l(v; w b ) l(w a ; w b ) ( 4l(w a; v)) = 4 l(w a; v) l(v; w b ) = E[C HR ] l(w a ; w b ) 15

16 and (6) hold. Cae 2: Suppoe that w a u a < v. In thi cae C ADV = l(u a ; v). If HARM-RING eplicate the page fom w a to v, then the potential change i 4l(w a ; u a ). Again, if HARM-RING eplicated the page fom w b to v, then the change in potential i non-poitive. Theefoe We obtain E[] 1 l(v; w b ) l(w a ; w b ) ( 4l(w a; u a )) = 4 l(w a; u a ) l(v; w b ) : l(w a ; w b ) E[C HR ] + E[] 4 l(u a; v) l(v; w b ) l(w a ; w b ) 4l(u a ; v) = 4C ADV : The econd inequality hold becaue l(v;w b) l(w a;w b 1. Inequality (6) i atied. ) Cae 3: Suppoe that u a < w a < v. Hee C ADV = l(u a ; v) and 0. We have E[C HR ] = 4 l(w a; v) l(v; w b ) l(w a ; w b ) 4l(w a ; v) 4l(u a ; v) = 4C ADV : A befoe, the t inequality follow fom the fact that l(v;w b) l(w a;w b 1. Again, Eq. (6) hold. 2 ) 6 Concluion and open poblem We have invetigated the page eplication poblem fo impotant netwok topologie uch a tee, unifom netwok and ing. Fo thee topologie we have developed deteminitic and andomized on-line algoithm that achieve a contant competitive atio. Ou andomized algoithm fo tee and unifom netwok achieve the bet poible competitive atio. While the competitivene achieved by deteminitic and andomized algoithm i ettled fo tee and unifom netwok, a numbe of open poblem emain with epect to the ing topology. One inteeting poblem i to tighten the gap fo deteminitic algoithm. We have peented a 4- competitive deteminitic eplication algoithm. Black and Sleato [6] mention (without poof) that no deteminitic on-line algoithm fo ing can be bette than 5 -competitive. Moeove, 2 no lowe bound ae known on the competitivene achieved by andomized on-line algoithm on ing. An inteeting poblem i to develop lowe o impoved uppe bound fo ing. Thi pape (and almot all othe elated peviou wok) tudie the page eplication unde the aumption that the local memoie of the poceo have innite memoy capacity. That i, wheneve an algoithm want to eplicate a given page into the local memoy of a poceo, thee i oom fo it; no othe page need to be dopped. An impotant poblem i to tudy the page eplication unde the aumption that the local memoie have bounded capacity. Batal et al. [4] howed that in thi model, no on-line eplication algoithm in any topology can be bette than (m)-competitive, whee m i the total numbe of page that can be toed in the netwok. They alo gave an O(m)-competitive algoithm fo unifom netwok. One appoach to ovecome the (m) bound might be to conide pecial memoy type. Albe and Koga [1] tudied the page migation poblem fo diect-mapped memoie, i.e., the poceo ue a hah function in ode to locate page in thei local memoie. Awebuch et al. [3] invetigated ditibuted paging poblem when an on-line algoithm ha lightly moe memoy capacity than the o-line algoithm. 16

17 Topology Ring Abitay Poblem Detemine the competitive atio c achieved by deteminitic and andomized on-line algoithm. Bet bound known fo { det. alg.: 2:5 c 4 { and. alg. againt obliviou adv.: c 3:16 { and. alg. againt adaptive on-line adv.: c 4 Find memoy model fo which on-line algoithm with a competitive atio of o(m) can be developed. Table 1: Summay of open poblem Refeence [1] S. Albe and H. Koga. Page migation with limited local memoy capacity. In Poc. 4th Intenational Wokhop on Algoithm and Data Stuctue (WADS95), Spinge LNCS, Vol. 955, page 147{158, [2] B. Awebuch, Y. Batal and A. Fiat. Competitive ditibuted le allocation. In Poc. 25th Annual ACM Sympoium on Theoy of Computing, page 164{173, [3] B. Awebuch, Y. Batal and A. Fiat. Ditibuted paging fo geneal netwok. In Poc. 7th Annual ACM-SIAM Sympoium on Dicete Algoithm, page 574{538, [4] Y. Batal, A. Fiat and Y. Rabani. Competitive algoithm fo ditibuted data management. In Poc. 24th Annual ACM Sympoium on Theoy of Computing, page 39{50, [5] S. Ben-David, A. Boodin, R.M. Kap, G. Tado and A. Wigdeon. On the powe of andomization in on-line algoithm. Algoithmica, 11(1):2{14, [6] D.L. Black and D.D. Sleato. Competitive algoithm fo eplication and migation poblem. Technical Repot Canegie Mellon Univeity, CMU-CS , [7] M. Chobak, L.L. Lamoe, N. Reingold and J. Wetbook. Page migation algoithm uing wok function. In Poc. 4th Intenational Annual Sympoium on Algoithm and Computation, Spinge LNCS Vol. 762, page 406{415, [8] D. Downey and D. Fote. Compaative model of the le aignment poblem. Computing Suvey, 14(2):287{313, [9] C. Lund, N. Reingold, J. Wetbook and D. Yan. On-line ditibuted data management. In Poc. of the 2nd Annual Euopean Sympoium on Algoithm, Spinge LNCS Vol. 855, page 202{214, [10] P. Raghavan and M. Sni. Memoy veu andomization in on-line algoithm. In Poc. 16th Intenational Colloquium on Automata, Language and Pogamming, Spinge LNCS Vol. 372, page 687{703,

18 [11] C. Scheuich and M. Duboi. Dynamic page migation in multipoceo with ditibuted global memoy. IEEE Tanaction on Compute, 38(8):1154{1163, [12] D.D. Sleato and R.E. Tajan. Amotized eciency of lit update and paging ule. Communication of the ACM, 28:202{208, [13] J. Wetbook. Randomized algoithm fo the multipoceo page migation. SIAM Jounal on Computing, 23:951{965,