CERIAS Tech Report Replicated Parallel I/O without Additional Scheduling Costs by Mikhail J. Atallah Center for Education and Research

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1 CERIAS Tech Report Repicated Parae I/O without Additiona Scheduing Costs by Mikhai J. Ataah Center for Education and Research Information Assurance and Security Purdue University, West Lafayette, IN

2 Repicated Parae I/O without Additiona Scheduing Costs Mikhai Ataah and Keith Frikken Purdue University Abstract. A common technique for improving performance in a database is to decuster the database among mutipe disks so that data retrieva can be paraeized. In this paper we focus on answering range queries in a mutidimensiona database (such as a GIS), where each of its dimensions is divided uniformy to obtain ties which are paced on different disks; there has been a significant amount of research for this probem (a subset of which is [1,2,3,4,5,6,7,8,9,11,12,13,14,15]). A decustering scheme woud be optima if any range query coud be answered by doing no more than # of ties inside the range/# of disks retrievas from any one disk. However, it was shown in [1] that this is not achievabe in many cases even for two dimensions, and therefore much of the research in this area has focused on deveoping schemes that performed cose to optima. Recenty, the idea of using repication (i.e. pacing records on more than one disk) to increase performance has been introduced. [7, 12,13,15]. If repication is used, a retrieva schedue (i.e. which disk to retrieve each tie from) must be computed whenever a query is being processed. In this paper we introduce a cass of repicated schemes where the retrieva schedue can be computed in time O(# of ties inside the query s range), which is asymptoticay equivaent to query retrieva for the non-repicated case. Furthermore, this cass of schemes has a strong performance advantage over non-repicated schemes, and severa schemes are introduced that are either optima or are optima pus a constant additive factor. Aso presented in this paper is a stricty optima scheme for any number of coors that requires the owest known eve of repication of any such scheme. 1 Introduction A typica botteneck in many systems is I/O; to reduce the effect of this botteneck data can be decustered onto mutipe disks to faciitate parae retrieva of the data. In a muti-dimensiona database, such as a GIS or a spatio-tempora database, the dimensions can be tied uniformy to form a grid, and when answering a range query in such a system, ony the ties that contain part of the Portions of this work were supported by Grants EIA and ISS from the Nationa Science Foundation, Contract N from the Office of Nava Research, by sponsors of the Center for Education and Research in Information Assurance and Security, by Purdue Discovery Park s e-enterprise Center, and by the GAANN feowship. V. Mařík et a. (Eds.): DEXA 2003, LNCS 2736, pp , c Springer-Verag Berin Heideberg 2003

3 224 M. Ataah and K. Frikken query need to be retrieved. In such an environment, a decustering scheme attempts to pace the ties on disks in such a way that the average range query is answered as efficienty as possibe. If the database is treated ike a grid and the disks as coors, then this can be stated as a grid cooring probem. For the rest of the paper we use record and tie synonymousy and ikewise use decustering and cooring interchangeaby. Given a database decustered on k disks and a range query Q contained on m ties, Q is answered optimay if no more than m k ties are retrieved from any one disk. A decustering is caed stricty optima if a range queries can be answered optimay, however it was shown in [1] that this is not achievabe except in a few imited circumstances. Thus there has been a significant amount of work to deveop decustering schemes that have cose to optima performance, a samping of which are in [2,3,4,5,6,9,11,14]. To improve performance further the idea of using repication (i.e. pacing each tie on mutipe disks) has been introduced [7,12,13,15]. When repication is used each tie in a query can be retrieved from mutipe paces which aows greater fexibiity when answering the query. In order to use repication an agorithm for computing an optima retrieva schedue is required (i.e. which disk do you retrieve each tie from). Agorithms for computing this schedue are given in [7, 12,13,15]. The most genera of these runs in time O(rm 2 + mk) where r is the most number of disks that a tie is stored on, m is the number of ties to be retrieved, and k is the number of disks. (For more information on work using repication see Section 2). One probem with repication is that it adds a nonnegigibe overhead to query response. In this paper we define a cass of cooring techniques, which we ca the grouping schemes, for which a schedue of retrievas can be computed in time O(# of ties to be retrieved) (from here on we refer to this as O(# of ties)), which is asymptoticay equivaent to the time required to compute a schedue for a non-repicated scheme. This technique essentiay transforms existing cooring schemes into repicated schemes by pacing disks into groups and pacing ties on a disks in a group; when the group size is two this is equivaent to RAID eve 1. Previousy, the ony genera strict optima soution for any number of disks was the Compete Cooring [7,15], which paces each tie on every disk. We introduce severa new schemes that are either have stricty optima performance for a queries or wi answer any query in no more time than a stricty optima scheme pus an additive constant; these new schemes have the owest known eve of repication for such performance bounds. Furthermore, these grouping schemes are shown to have stronger experimenta performance than schemes without repication. The outine of this paper is as foows: Section 2 discusses previous work in this area, in Section 3 the grouping schemes are introduced and schemes that achieve stricty optima performance or are a constant additive factor above an optima soution, Section 4 contains experimenta data showing the performance of the grouping schemes, and Section 5 concudes the paper.

4 2 Reated Work Repicated Parae I/O without Additiona Scheduing Costs 225 Given an n-dimensiona database with each dimension divided uniformy to form ties, if ties are paced on different disks, then the retrieva of records during query processing can be paraeized. The I/O time in such a system is time that it takes to retrieve the maximum number of ties stored on the same disk. The probem of pacing the records so that the response times for range queries is minimized has been we studied; this section presents a survey of this work. Given a database decustered onto k disks and a range query Q contained on m ties, an optima tie decustering woud require no more than m k retrievas from any one disk. It was shown in [1] that this bound is unachievabe for a range queries in a grid except in a few imited circumstances. Since there are many cases where no scheme can achieve this optima bound, severa schemes have been deveoped to achieve performance that is cose to optima. To quantify cose to optima, define the additive error of a decustering scheme to be the maximum over a range queries Q of the vaue (rettime(q) m k ), where rettime(q) is defined as the retrieva time for query Q (i.e. it is the maximum number of ties in Q retrieved from a singe disk). These schemes incude Disk Moduo DM [6], Fiedwise excusive (FX) or [9], the cycic schemes (incuding RPHM, GFIB, and EXH) [11], GRS [4], a technique deveoped by Ataah and Prabhakar [2] which we wi ca RFX, and severa techniques based on discrepancy theory [5,14] (for an introduction to discrepancy theory see [10]). Note that these are just a subset of the decustering techniques that have been deveoped for this probem. Suppose we are given k coors. The DM approach [6] assigns tie (x, y) to (x + y) modk. The FX approach [9] assigns tie (x, y) to(x y) modk. Cycic aocation schemes [11] choose a skip vaue s such that gcd(k, s) = 1 and assigns tie (x, y) to(x + sy) modk. The choice of the skip vaue, s, is what defines the scheme. In RPHM (Reativey Prime Haf Moduo), s is defined to be the integer nearest to k 2 that is reativey prime to k. The GFIB (Generaized FIBonacci) scheme defines s to be an approximate of the previous Fibonacci number (by using the cosed formua) that is reativey prime to k. The EXH (Exhaustive) scheme takes a vaues of s where gcd(s, k) = 1 and finds the one that optimizes a certain criterion, for exampe minimizing the additive error is a possibe criterion. Another cass of schemes are the permutation schemes [4], in these schemes a permutation φ of the numbers in {0,..., k 1} is chosen and then tie (x, y) is assigned coor (x φ 1 ((y) modk)). Exampes of permutation schemes are DM, the cycic schemes, and GRS. In the GRS scheme [4] the permutation is computed 2i as foows: i) i {0,..., k 1} compute the fractiona part of 1+, and ca 5 it k i and then ii) sort the vaues k i and use this to define the permutation. A scheme based on the Corput set is defined in [14] that is simiar to GRS except that the k i vaues are a0 2 + a1 4 + a a k 1 where a 2 k k 1...a 1 a 0 is the binary representation of i. In [2], the RFX scheme was presented that was ater found in [3] to be equivaent to (x y R )modk, where y R is the ( og k )-bit reversa of y. For brevity, the detais of higher dimensiona schemes are not provided.

5 226 M. Ataah and K. Frikken It was shown in [14] that the additive error for k coors in two dimensions is Ω(og k), and that in d( 3) dimensions it is Ω(og d 1 2 k). In two dimensions, schemes have been deveoped (RFX, GRS, and schemes based in discrepancy theory [2,14]) that have a provabe upper bound of O(og k) on additive error. For higher dimensions d( 3), two schemes are given in [5] with additive error O(og (d 1) k), which are the schemes with the owest proven asymptotic bound on additive error. A recent trend has been to use repication [7,12,13,15] to increase performance further. Severa query scheduing agorithms have been given in previous work, but the ony genera agorithm that works for any type of repication is in [7] and runs in time O(rm 2 +mk) where r is the most number of disks that a tie is stored on, m is the number of ties to be retrieved, and k is the number of disks. In [12,13] it was proven that if ties are stored on two random disks then the probabiity of requiring more than ( (# of ties/# of disks) +1) retrievas from a singe disk for a random query approaches 0 as the number of disks gets arge. In [15] repication was used to achieve optima soutions for up to 15 disks. A stricty optima scheme, caed Compete Cooring (CC), for any number of disks by storing a ties on a disks was introduced in [7,15]. The SRCDM scheme was introduced in [7], and has an additive error no arger than 1, but requires the number of disks be a perfect square (n 2 ) and requires that each tie is paced on n disks. 3 Grouping Repication Scheme In this section the grouping schemes are introduced. Section 3.1 defines some notations that wi be needed before defining this cass of schemes. In Section 3.2, the grouping schemes are defined aong with an agorithm that computes the retrieva schedue in time O(#of ties). Section 3.3 contains severa schemes that have an additive error that is 0 or is O(1). Finay, in Section 3.4 we provide a stricty optima cooring scheme that works for any number of coors. 3.1 Notations and Terminoogy Before we can formay define the grouping schemes we need to define some notation and terminoogy. A non-repicated cooring function C for d dimensions and m disks is a function C : ℵ d {0,..., m 1}, essentiay C maps a tie to a disk. A repicated cooring function C with eve of repication r for d dimensions and m disks is a function C : ℵ d r {0,..., m 1} i, essentiay C maps a i=1 tie to the set of disks (with size no more than r) that contain the tie. Since the repicated cooring function is a generaization of the non-repicated cooring function, we assume a cooring functions are repicated for the rest of the paper. A convenient shorthand notation for cooring schemes is (C, m, r, d) which states the cooring function C decusters a d dimensiona grid onto m disks with a eve of repication r. Two cooring schemes (C, m, r, d) and (D, m, r, d) are said to be equivaent if and ony if there is a bijection f : {0,..., m 1} {0,..., m 1} such

6 Repicated Parae I/O without Additiona Scheduing Costs 227 that i C(x 1,..., x d )ifff(i) D(x 1,..., x d ). Essentiay schemes are equivaent if there is a rearrangement of the coors that wi make them identica, and it is obvious that equivaent schemes have identica retrieva time for any query. 3.2 Definition of Grouping Schemes A scheme is considered to be formed with groups if the coors are partitioned into sets and ties are assigned to partitions where assigning a partition to a tie is equivaent to pacing it on a disks in that partition. The motivation for this cass of schemes is to be abe to distribute the additive error of the cooring that assigns ties to partitions among the different members of the partition. Hence, the additive error of any one member of the partition wi be smaer, and thus reducing the additive error of the scheme. Formay, a cooring scheme (C, m, r, d) is considered formed by groups if the coors can be partitioned into sets S 1,S 2,..., S k with at east one set where S i > 1 such that if C(x 1,x 2,..., x d )=S and the foowing hods: if (S i S), then S i S. Such a scheme is caed simpe if the ast constraint is changed to: if (S i S), then S i = S. A scheme formed by groups is said to have equa partitions if each partition S i is identica in size, or equivaenty S i = m k for a i. It is possibe to transform any cooring scheme (C, m, r 1,d) into a scheme formed by groups with equa partitions (of size r 2 )(C,mr 2,r 1 r 2,d), where C is defined as: C (x 1,..., x d )= {im + s 0 i<r 2 }, we denote this s C(x 1,...,x d ) transformation process by GROUP ((C, m, r 1,d),r 2 )((C, m, r 1,d) is referred to as the base scheme in what foows). A scheme defined with GROUP is simpe if r 1 = 1. Now any scheme defined with GROUP is a scheme formed by groups with equa partitions, but any scheme formed by groups with equa partitions is equivaent to a scheme that can be defined with GROUP (proof omitted). We ca the set of schemes defined by GROUP the grouping schemes. There have been scheduing agorithms defined for any repicated agorithm that wi work for any repicated scheme, but for simpe grouping schemes (represented by GROUP ((C, m, 1,d),r)) there is a scheduing agorithm that runs in time O(m) and executes with one pass over the ties (see RetrieveTies beow). The agorithm uses a function SetSchedue(tie,disk) which sets the schedue to retrieve tie from disk. begin RetrieveTies(Query, (C,mr,r,d) =GROU P ((D, m, 1,d),r)) A[] := array initiaized to 0 of size m. fora t =(t 1,..., t d ) in Query do c := C(t) SetSchedue(t,c + A[c]) A[c] :=((A[c]+m) mod(mr)) endfor end RetrieveTies

7 228 M. Ataah and K. Frikken Thus simpe grouping schemes can be used without having to incur the additiona scheduing costs of other repicated schemes. In addition to this, the additive error of a grouping scheme with eve of repication is bounded by the o where o is the additive error of the base scheme (see Theorem 3-3). Before this can be proven we need Lemmas 3-1 and 3-2. Lemma 3-1: If the cooring scheme (C, m, r, d) has an additive error of o, then GROUP ((C, m, r, d),) wi have a response time no arger than x k +o for x records. Proof: For the x records the origina cooring scheme wi have at most ( x k + o) instances of any one coor which means there wi be at most ( x k + o) instances of any group. These vaues can be distributed equay among the coors in that group to obtain a maximum response time of x k +o. QED Lemma 3-2: x k +o x k + o Proof: Let x = a(k)+bk + c, where 0 b<and 0 c<k. There are two cases to consider: (b = 0 and c =0)or(b 0orc 0). Case 1: (b = 0 and c = 0): x k +o = a(k) k +o = a+o = a + o = a(k) k + o = x k + o Case 2: (b 0orc 0): x k +o = ak+bk+c k +o = a+b+ c k +o a + b+1+o a + +o = a +1+ o = a(k)+bk+c k + o = x k + o. In either case x k +o x k + o. QED Theorem 3-3: If the cooring scheme (C, m, r,d) has an additive error of o, then GROUP ((C, m, r,d),r) wi have an additive error no arger than o r. Proof: Foows directy from Lemma 3-1 and Lemma 3-2. QED This ast theorem impies that the additive error for a cooring scheme can be reduced by using this grouping method. Since the additive error can be reduced the expected vaue above optima wi aso be reduced. To summarize this section, a cass of repicated schemes can be defined with the GROU P transformation, which we ca the grouping schemes. A subset of these schemes are simpe and for this subset there are two significant benefits compared to non-repicated schemes incuding: i) queries can be processed in time proportiona to the number of records which is asymptoticay optima, and ii) there is a performance increase. 3.3 Achieving Optima and Constant Additive Error In this section schemes with 0 and O(1) additive error are introduced. Coroary 3-4: If a scheme (C, m, r 1,d) is stricty optima so is GROUP ((C, m, r 1,d),r 2 ). Proof: Since (C, m, r 1,d) is stricty optima the additive error wi be 0, and thus by Theorem 3-3, the additive error of GROUP ((C, m, r 1,d),r 2 ) wi aso be 0, and thus is stricty optima. QED The previous coroary impies that any scheme with eve of repication r that is optima for c disks can be transformed using GROUP into a scheme that is optima for ck disks with eve of repication kr. It is possibe to coor a

8 Repicated Parae I/O without Additiona Scheduing Costs 229 two dimensiona grid optimay with 1, 2, or 3 (or 5 in 2-D) coors using RPHM in two dimensions and DM in higher dimensions. Hence, it is possibe to coor a grid with r, 2r, or3r, (or 5r in 2-D) with eve of repication r in a stricty optima fashion (by Coroary 3-4). Furthermore, these schemes are simpe and so RetrieveTies can be used to retrieve the ties in time proportiona to the number of records. The CC scheme described in [7,15] is the scheme defined above that uses a base scheme with ony 1 coor. In addition to these optima schemes there are grouping schemes that achieve an additive error that is O(1). Coroary 3-5: If a scheme (C, m, r 1,d) has an additive error of O(f(m)) for some function f then is GROUP ((C, m, r 1,d),x), where x>f(m) has an additive error that is O(1). Proof: Since the scheme (C, m, r 1,d) has an additive error of O(f(m)), then the additive error is bounded by af(m) for some constant a. Now by theorem 3-3, GROUP ((C, m, r 1,d),x) wi have an additive error no arger than af(m) x a and thus is O(1). QED The foowing is a tabe of grouping schemes with base schemes with m coors that have an additive error which is O(1) but can be schedued with RetrieveTies (it is assumed m is a power of 2 for the RFX scheme): Leve of Additive Base Scheme Dimensions Repication Error LHDM [8] d (m 1) d RFX, GRS, and other schemes [2,3,14] 2 og m O(1) RFX [2,3] 2 2 og m 3 1 Schemes in [5] d og d 1 (m) O(1) 3.4 Generaizing Optima Additive Error In the previous section schemes were introduced that were stricty optima, but these schemes are appied in the situation where the number of coors was a mutipe of the number of coors in a base scheme that is optima (i.e. 1, 2, or 3 (or 5 in 2-D)). The CC cooring is an instance of the previous scheme and is stricty optima for any number of coors, but it requires that the eve of repication be the number of coors, which may be unreasonabe for many appications. In this section a stricty optima scheme for any number of coors is given with a eve of repication cose to haf the number of coors. The scheme presented here is a generaization of GROU P ((C,2, 1, d), k) where C is the DM cooring scheme that is stricty optima for any number of coors. In the case, where the number of coors is even, we are triviay done using schemes discussed in the previous section. Suppose the number of coors is odd (i.e. m =2k + 1), to create a scheme for m coors pace 2k of the coors using the grouped DM scheme with eve of repication k and then pace the entire database on the ast disk. Note that the eve of repication for such a scheme for m disks is m 2 +(m mod 2) which is about m 2. It can be proven that this scheme is stricty optima, but we omit this proof due to space constraints.

9 230 M. Ataah and K. Frikken A possibe criticism of this scheme is that if you can pace the entire database on a singe disk, then why not use the CC mechanism for simpicity. However, in this case it is ony required that a singe disk be arge enough to hod the entire database. This may not be reasonabe for arge databases, but is reasonabe in some situations. This generaized approach can be extended to grouping schemes with base schemes with 3 (and 5 in 2-D) disks in a simiar fashion (we omit the detais due to space constraints). It is not true however that if you have an optima scheme for k coors that if you put a ties on another disk that the soution wi be optima for k + 1 disks. The scheme defined in this section constitutes a genera stricty optima schemes with the owest known eve of repication. With some modification to our scheduing agorithm the schedue can be computed in time O(# of records). We give a verba description of the agorithm here for when there are 2k + 1 coors and the scheme described above is used. Essentiay there are two groups of k coors and 1 extra coor. We know that an optima schedue is achievabe so we determine what optima is, and ca it o. Assign up to the first ko ties of each group to disks in that group, such that no more than o ties are assigned to any one disk, and if there are any ties remaining after this has been done to both groups assign these eftovers to the ast disk which is not in either group. 4 Experiments For this section, experiments were performed to compare the performance between grouped schemes with eve of repication as 2 and non-grouped schemes. The comparison criterion that is used is the expected deviation from optima for a queries. To compute the expected deviation from optima for a cooring scheme (C, m, r, d) we compute the expected vaue from optima of a wraparound queries in an d dimensiona grid with side engths equa to m. There is a finite number of queries in such a grid so this vaue can be computed exacty for smaer m vaues, but is estimated for arger vaues. This estimation is done by taking a random samping of queries and computing the expected deviation from optima of these queries. It is worth noting that when an exact vaue is computed that the maximum additive error found wi be the maximum additive error in any grid (see [8], which can be generaized to grouping schemes, but this generaization is omitted). To perform the comparison between the repicated and non-repicated schemes we use a hybrid cooring. Given a set of coorings this hybrid cooring uses the cooring that minimizes the expected deviation from optima for a specific number of disks, i.e. the hybrid cooring uses the best cooring in the set for a specific situation. The comparison is figure 1 is between the hybrid cooring of a set of non-repicated schemes and the hybrid cooring for these schemes transformed with GROU P using eve of repication of 2. The set of non-repicated cooring schemes used are DM [6], FX (for powers of 2) [9], RFX (for powers of 2) [2], RPHM [11], GFIB [11], GRS [4], and a scheme

10 Repicated Parae I/O without Additiona Scheduing Costs 231 Fig. 1. Expected Deviation from Optima for 2-D Schemes based on the Corput set [14]. For the grouping schemes we use the grouped version of these schemes with eve of repication 2. When the number of disks is no more than 40, exact vaues were computed, but estimates were used for up to 140 disks. These estimates were made by ooking at 5000 queries (chosen randomy with uniform distribution) in the grid using the mean deviation as the estimate. The resuts are dispayed in Figure 1. Figure 1 is interesting for severa reasons. First, it shows that the estimate is accurate for predicting the expected deviation for vaues up to 40. Aso, it shows that the grouping schemes perform far better than the non-repicated schemes, since the expected deviation from optima is 2-3 times arger for non-repicated coorings than for grouping schemes. Thus if a repication eve of 2 is used, then range query performance wi be improved substantiay. 5 Concusions When decustering data, there are three inhibiting factors that may prevent the usage of repication: i) there is not enough disk space on each disk to contain enough information, ii) the sow down that occurs with query scheduing for repication is too overwheming, and iii) the benefit from repication is not significant. We have introduced a cass of schemes, caed the grouping schemes, which eiminate conditions (ii) and (iii). Condition (ii) is eiminated because the grouping schemes can be schedued in time O(number of ties), and it was shown

11 232 M. Ataah and K. Frikken in Section 4, that these techniques perform extremey we even if the eve of repication is 2 which eiminates condition (iii). Thus an important concusion about the usage of repication can be stated: If there is enough room on the disks to faciitate repication, then repication shoud be used. Furthermore, a stricty optima scheme for any number of coors with the owest known eve of repication for such a soution was presented aong with severa schemes with additive error that is O(1) were given that have fewer restrictions on the number of disks and have a ower eve of repication than previous schemes that achieve an O(1) bound on additive error (the authors know of ony one such previous scheme, which is SRCDM). References 1. K. Abde-Ghaffar and A. E. Abbadi. Optima aocation of two-dimensiona data. In Internationa Conference on Database Theory, pages , M. J. Ataah and S. Prabhakar. (amost) optima parae bock access to range queries. In Proceedings of the nineteenth ACM SIGMOD-SIGACT-SIGART symposium on Principes of database systems, pages ACM Press, R. Bhatia, R. Sinha, and C.-M. Chen. Hierarchica decustering schemes for range queries. In In 7th Int Conf. on Extending Database Technoogy, R. Bhatia, R. K. Sinha, and C.-M. Chen. Decustering using goden ratio sequences. In ICDE, pages , C.-M. Chen and C. T. Cheng. From discrepancy to decustering: near-optima mutidimensiona decustering strategies for range queries. In Proceedings of the twentyfirst ACM SIGMOD-SIGACT-SIGART symposium on Principes of database systems, pages ACM Press, H. Du and J. Soboewski. Disk aocation for cartesian product fies on mutipe disk systems. ACM Transactions on Database System, pages , K. Frikken, M. Ataah, S. Prabhakar, and R. Safavi-Naini. Optima parae i/o for range queries through repication. In Proceedings of 13th Int. Conf. on Database and Expert Systems Appication (LNCS 2453), pages B. Himatsingka, J. Srivastava, J.-Z. Li, and D. Rotem. Latin hypercubes: A cass of mutidimensiona decustering techniques, M. H. Kim and S. Pramanik. Optima fie distribution for partia match retrieva. In Proceedings of the 1988 ACM SIGMOD internationa conference on Management of data, pages ACM Press, J. Matousek. Geometric discrepancy, an iustrated guide. Springer-Verag, S. Prabhakar, K. Abde-Ghaffar, D. Agrawa, and A. E. Abbadi. Cycic aocation of two-dimensiona data. Technica Report TRCS97-08, 1, P. Sanders. Reconciing simpicity and reaism in parae disk modes. In Proceedings of the twefth annua ACM-SIAM symposium on Discrete agorithms, pages ACM Press, P. Sanders, S. Egner, and J. Korst. Fast concurrent access to parae disks. In Proceedings of the eeventh annua ACM-SIAM symposium on Discrete agorithms, pages ACM Press, R. K. Sinha, R. Bhatia, and C.-M. Chen. Asymptoticay optima decustering schemes for range queries. In Internationa Conference on Database Theory, A. Tosun and H. Ferhatosmanogu. Optima parae i/o using repication. Technica Report OSU-CISRC-11/01-TR26, 2001.