PARALLEL database systems are essential to important

Transcription

1 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 13, NO. 12, DECEMBER Load Baanced and Optia Disk Aocation Strategy for Partia Match Queries on Mutidiensiona Fies Saja K. Das, Meber, IEEE, and Cristina M. Pinotti, Meber, IEEE Coputer Society Abstract A utidiensiona fie is one whose data are characterized by severa attributes, each specified in a given doain. A partiaatch query on a utidiensiona fie extracts a data whose attributes atch the vaues of one or ore attributes specified in the query. The disk aocation probe of a utidiensiona fie F on a database syste with utipe disks accessibe in parae is the probe of distributing F aong the disks such that the data quaifying for each partiaatch query are distributed as eveny as possibe aong the disks of the syste. We propose an optia soution to this probe for utidiensiona fies with pairwise prie doains based on a arge and fexibe cass of axiu distance separabe codes, naey, the redundant residue codes. We aso introduce a new faiy of residue codes, caed the redundant nonpairwise prie residue codes, to dea with fies whose attribute doains are nonpairwise prie. Index Ters Nonunifor utidiensiona fie, partiaatch query, stricty optia disk aocation, redundant pairwise-prie residue code, redundant nonpairwise prie residue code. æ 1 INTRODUCTION PARALLEL database systes are essentia to iportant rea-ife appications that require to anage extreey arge voues of data, often of the order of terabytes, which cannot be stored in a singe disk. The appications incude spatia databases, airine reservation systes, cartography, Word Wide Web, etc. Queries on such arge databases invove the retrieva of a big portion of data, often under critica tie constraints. Therefore, to retrieve quaifying data in parae is andatory. Before proceeding further, et us define a few terinoogy and notations. In this paper, we consider database systes of utidiensiona fies, each ade up of buckets, each, in turn, ade up of records. Arecord is an n-tupe <r 1 ;...;r n > of attributes such that the vaue of the jth attribute r j beongs to the interva I j ¼½L j ;R j Š, for 1 j n. In order to define a bucket, et us unifory partition the vaues of the jth attribute of a record into j groups, each of size R j L j þ 1 ; j and et j be tered the doain of r j. The bucket < x 1 ;...;x n > of respective doains 1 ;...; n is the set of the records <r 1 ;...;r n > such that the vaue of r j, for 1 j n, beongs to the x j th group of I j. More foray,. S.K. Das is with the Departent of Coputer Science and Engineering, University of Texas at Arington, Arington, TX E-ai: das@cse.uta.edu.. C.M. Pinotti is with the Departent of Coputer Science and Teecounications, University of Trento, Povo 38050, Itay. E-ai: pinotti@science.unitn.it. Manuscript received 21 Aug. 2001; revised 8 Apr. 2002; accepted 22 Apr For inforation on obtaining reprints of this artice, pease send e-ai to: tpds@coputer.org, and reference IEEECS Log Nuber <x 1 ;...;x n >¼ 8 >< >: <r 1;...;r n> L j þ R j L j þ1 j L jþ R j L j þ1 j x j r j L j þ R j L j þ1 j x jr jr j ðx j þ1þ 1 if 0x j j 2; if x j¼ j 1; Thus, the bucket <x 1 ;...;x n > stores at ost n R j L j þ 1 j¼1 j 9 >= >; : records. Finay, a utidiensiona fie F ð 1 ;...; n Þ of doains 1 ;...; n is the set of a n i¼1 i buckets <x 1 ;...;x n > where x i 2½0;...; i 1Š and 1 i n. Exape 1. Consider two attributes that range, respectivey, in I 1 ¼½ Š and I 2 ¼½1...50Š. Fixing the doains 1 ¼ 20 and 2 ¼ 10, partition I 1 and I 2, respectivey, into 20 and 10 groups of equa size. Specificay, the ith group of I 1 contains the vaues ½5i þ 1; 5i þ 2; 5i þ 3; 5i þ 4; 5ði þ 1ÞŠ, for 0 i 19, of the first attribute. Thus, the bucket < 2; 3 > contains the records <r 1 ;r 2 >, where 11 r 1 15 and 16 r The utidiensiona fie F ð20; 10Þ consists of a the 200 buckets <x 1 ;x 2 >, where 0 x 1 19 and 0 x 2 9, and each bucket stores 25 records. A unifor p-ary utidiensiona fie F ðp;...;pþ is defined as one in which a the doains assue the identica vaue p. Otherwise it is caed a nonunifor utidiensiona fie. If the n doains 1 ;...; n are pairwise prie, F ð 1 ;...; n Þ is said to be a nonunifor, pairwise prie utidiensiona fie. The ost coon operation on a database syste is the inforation retrieva. Foray, Definition 1. Given a utidiensiona fie F ð 1 ;...; n Þ,a atch query q ¼< q 1 ;...;q n > on F retrieves the bucket 8j /02/$17.00 ß 2002 IEEE

2 1212 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 13, NO. 12, DECEMBER 2002 <q 1 ;...;q n >.Apartiaatch query (PMQ) is one whose ith attribute q i is either specified or unspecified, and retrieves fro F the set of buckets that atch the query q on the specified attributes. Such buckets are said to quaify for q. The query response set, QRðqÞ, of a PMQ with the unspecified attributes q i1 ;q i2 ;...;q ik consists of ¼ k j¼1 i j buckets that quaify for q. As an exape, given the fie F ð20; 10Þ, the PMQ q ¼ < 5; > extracts Nð< 5; >Þ ¼10 buckets defined as f< 5;i>j0 i 9g. In rea database systes, however, the nuber of attributes is uch arger than two. Typicay, there are thousands of buckets, and a consideraby arge nuber of the ay quaify for each PMQ. In such a context, database systes need to store their fies on utipe disks. A parae database syste is a database syste that stores utidiensiona fies on utipe disks that can be read or written siutaneousy. This additiona capabiity ay iprove significanty the tie required for data transission if a suitabe schee is adopted for data distribution aong the disks. In other words, if the data that quaify for a query are aost eveny distributed aong the disks avaiabe in the database syste, the response tie is reduced by a factor equa to the nuber of disks. In fact, distributing F ð 1 ;...; n Þ into a parae database syste of D disks, denoted as d 0 ;...;d D 1, the tie to retrieve the query response set QRðqÞ of a partiaatch query q is proportiona to the nuber of buckets that quaify for q in each disk. Precisey, the response tie of a query is proportiona to RT ðqþ ¼ax 0iD 1 fn i ðqþg, where N i ðqþ is the nuber of buckets that quaify for q stored on disk d i. Thus, the response tie is iniu when RT ðqþ ¼ ; D i.e., when the query response set QRðqÞ is baanced aong a the avaiabe disks in the syste. Therefore, an optia soution to the disk aocation probe is equivaent to finding a oad baanced distribution of the buckets aong the utipe disks of the syste. We adopt the foowing criteria for optiaity. Definition 2. [1], [7]. Given a syste with D disks that can be accessed siutaneousy, a disk aocation strategy is tered optia for a specific partiaatch query q if it eveny distributes aong the D disks, thus achieving a response tie of D for q. Moreover, a disk aocation strategy is caed stricty optia if it is optia for a partiaatch queries. Fro now on, whenever the doains of the attributes are cear fro the context, F ð 1 ;...; n Þ wi be referred to as sipy F. 2 PREVIOUS WORK ON THE DISK ALLOCATION PROBLEM The probe of iniizing the response tie of PMQs on unifor utidiensiona fies has received a ot of attention in the past. For exape, one of the first approaches to the probe proposed of distributing F aong D disks with the hep of a (pseudo) rando nuber generator, with 1=D as the probabiity of assigning a bucket to a particuar disk. This soution had no constraints on the nuber of disks as we as on the cardinaity of the doains of the attributes. However, it did not guarantee optiaity for any cass of partiaatch queries. Subsequenty, Fang et a. [6] interpreted each bucket of Fð 1 ;...; n Þ as a point of the n-diensiona space 1... n, and suggested to partition the buckets into two geoetrica siiar groups; that is, in two groups with aost the sae spanning tree and aost the sae set of short spanning paths. They proved that, as a consequence of the geoetrica siiarity in each group, aost the sae nuber of buckets quaifies for each PMQ. This soution, however, can at ost have the query response tie. Moreover, the partitioning of F in ore than two groups sees quite intricate and, therefore, such a ethod is not suitabe for generaizations. A different approach, due to Du and Soboewski [5], assued a D-disk syste and assigned the bucket <x 1 ;...;x n > to the disk d j, where j P n i¼1 x i od D. This ethod has been shown to be aways stricty optia for PMQs with exacty one unspecified attribute. Moreover, for PMQs with two unspecified attributes, the ethod is stricty optia when the disk syste as D ¼ 2 or D ¼ 3, or when the attribute doains satisfy i 0 or 1 od D, for 1 i n. Later, Ki and Praanik [8] soved the disk aocation probe optiay for both PMQs with exacty two unspecified attributes and for PMQs with the doain of one unspecified attribute arger than D. Their soution assigns the bucket <x 1 ;...;x n > to the disk d j, such that j ¼ðþ ½ Š n i¼1 x iþ od D, where ½þŠ denote the bitwise Excusive-OR operation. In [7], Faoutsos and Metaxas foowed a copetey different approach. Given a binary utidiensiona fie F with n-attributes and D ¼ 2 k disks, their soution assigns to each disk a set of buckets which for a inear ðk 1Þ-error correcting binary code C. The code definition ipies that any two buckets stored in the sae disk differ by at east k attributes. This approach is aways optia for PMQs with at ost ðk 1Þ unspecified attributes and stricty optia when C is a axiu distance separabe (MDS) code [10]. Unfortunatey, for any pair of vaues n; k, there are no MDS codes; so, rarey, a stricty optia soution can be found. Abde-Ghaffar and E Abbadi [1] extended the resuts of [7] to unifor p-ary utidiensiona fies, with p>2, and foray estabished the equivaence between stricty optia aocation strategies and axiu distance separabe codes. Based on such an equivaence and using the fact that the Reed-Sooon codes for a arge faiy of p-ary MDS codes appicabe to p-ary utidiensiona fies with at ost p 1 attributes, the designed strategy is stricty optia for fies with at ost p p 1 buckets and uses D ¼ p t disks, where

3 DAS AND PINOTTI: LOAD BALANCED AND OPTIMAL DISK ALLOCATION STRATEGY FOR PARTIAL MATCH QUERIES t p 1. This soution does not work for nonunifor fies as we as for fies whose records have ore than p 1 attributes. 3 OUR CONTRIBUTIONS This paper presents the first systeatic soution to the disk aocation probe for nonunifor pairwise-prie utidiensiona fies. It can be considered as an extension of the resuts in [7], [1]. Our ethod takes fu advantage of a (possiby a very arge nuber) disks and, hence, the response tie is drasticay reduced. The soution is based on a very arge and fexibe cass of seiinear MDS codes, caed the redundant residue (pairwise prie) codes [2]. The existence of such a faiy of codes guarantees stricty optia soution and aeviates the iitations of the previous code-based disk aocation strategies which ainy suffer fro the ack of enough MDS codes for fies of different sizes. Our soution handes nonunifor utidiensiona fies that ode rea data uch better than the unifor utidiensiona fies, as argued beow with an exape. Exape 2. Let us consider a utidiensiona fie F which has three integer attributes whose intervas are very skewed. Let the interva of the first attribute be I 1 ¼½1::1000Š, that of the second be I 2 ¼½1::21Š, and that of the third be I 3 ¼½1::9Š. Thus, there are records atogether. Assuing that each disk page contains no ore than 100 records, et us deterine a suitabe doain for each attribute such that a bucket has aost the sae size as a disk page and each bucket is fu. First, we consider the scenario when a the three attributes have the sae doain vaue 1 ¼ 2 ¼ 3 ¼ p. Then, the bucket <x 1 ;x 2 ;x 3 >, with 0 x i p 1 for 1 i 3, contains the records <r 1 ;r 2 ;r 3 > such that L i þ R i L i þ 1 x i r i L i þ R i L i þ 1 ðx i þ 1Þ 1: p p Specificay,. For p ¼ 4, we obtain very arge buckets that do not fit in one disk page. Indeed, there are records stored in each bucket.. For p ¼ 10, each bucket consists of records, which is sti too arge in size.. For p ¼ 20, each bucket has at ost records. Athough a bucket perfecty fits in a page, the fie F is apped into 20 3 buckets, ost of which are epty (consider, for exape, a the buckets with x 3 1). On the other hand, if different doains are aowed a uch better oad factor can be obtained:. For 1 ¼ 20, 2 ¼ 11 and 3 ¼ 9, the bucket <x 1 ;x 2 ;x 3 >, for 0 x 1 19, 0 x 2 10 and 0 x 3 8, consists of the records <r 1 ;r 2 ;r 3 > such that þ x 1 r 1 ðx 1 þ 1Þ 1000 ; þ x 2 r 2 in ðx 2 þ 1Þ 21 ; 21 ; and þ x 3 r 3 ðx 3 þ 1Þ: Hence, each bucket <x 1 ;x 2 ;x 3 > consists at ost of 100 records and the fie F has ¼ 1; 980 buckets, a of which are fu. This exape shows that partitioning a attributes of the fie with the sae doain, one can get any epty buckets. On the contrary, if different doains are used for different attributes, a good oad factor of the buckets can be obtained. Back to the proposed disk aocation strategy: Note that the faiy of redundant residue codes appies ony to fies whose attributes have pairwise-prie doains. It is worth to point out that this is not a serious iitation since the attribute doains are seected when the disk aocation strategy is designed. For the sake of copeteness, however, we introduce a new cass of Redundant Nonpairwise Prie Residue Codes in order to hande attributes with nonpairwise prie doains. This ay ead to a sowdown with respect to the optia soution by a predeterined factor. The reainder of the paper is organized as foows: Section 4 reviews basic concepts of the coding theory that are reevant to our work. Based on the redundant pairwise prie residue codes, Section 5 proposes a stricty optia disk aocation strategy for nonunifor utidiensiona fies whose attributes have pairwise-prie doains. In Section 5.1, we study how to guarantee strict optiaity when either the nuber of attributes of the fie or the nuber of disks in the syste changes. Finay, a new faiy of redundant residue codes, caed the redundant nonpairwise prie residue codes, is introduced in Section 6 to sove the disk aocation strategy for nonunifor utidiensiona fies whose attributes have nonpairwise prie doains. Finay, concusions are offered in Section 7. 4 CODING THEORY FRAMEWORK Given a set of n positive integer radices, denoted as 1 ; 2 ;...; n, et S ¼ n be the space of a n-tupes of size M ¼ n i¼1 i.acode C is a subset of S. Each n-tupe of S which is aso in C is caed a codeword. Since S is defined on n radices, C has ength n. Let the size of C be its cardinaity. If a radices are equa to p 2, C is caed a p-ary code. A code C is inear if it is cosed under the addition and subtraction operations, as defined beow. Given two codewords x ¼< x 1 ;x 2 ;...;x n > and y ¼< y 1 ;y 2 ;...;y n > of C, both the n-tupes x þ y and x y as defined beow are aso codewords of C. x þ y ¼< ðx 1 þ y 1 Þ od 1 ; ðx 2 þ y 2 Þ od 2 ;...; ðx n þ y n Þ od n >; and x y ¼< ðx 1 y 1 Þ od 1 ; ðx 2 y 2 Þ od 2 ;...; ðx n y n Þ od n >:

4 1214 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 13, NO. 12, DECEMBER 2002 Let the inforation set I Sk ¼f i1 ; i2 ;...; ik g of a code C be a set of k indices such that for any k-tupe a i1 ;a i2 ;...;a ik, where a ij 2½0; ij 1Š, there is a unique codeword <x 1 ;x 2 ;...;x n > 2 C such that x ij ¼ a ij for 1 j k. Hence, if a code C has an inforation set I Sk, then the size of C is arger than or equa to ¼ k j¼1 i j. Such a code C is said to be a systeatic code. For a inear code C, the Haing weight of a codeword x ¼< x 1 ;x 2 ;...;x n > is the nuber of nonzero coponents x i, and the Haing distance between two codewords is the nuber of coponents in which they differ. The iniu distance d of C is the iniu Haing distance between a pairs of distinct codewords in C. Since the a-zero codeword aways beong to a inear code, the iniu distance d>0 and the iniu Haing weight of the codewords of C coincide. The concept of iniu distance is essentia for defining the error contro capabiities of a code. In fact, representing the error as an n-tupe e ¼< e 1 ;e 2 ;...;e n > and denoting a codeword x subject to error e as x þ e, a code of iniu distance d is abe 1) to detect at east a errors e with Haing weight at ost d 1 and 2) to correct a errors e with Haing weight bðd 1Þ=2c. It foows directy that, aong a the codes of a given ength n and size, the codes with the highest error contro are those with the argest iniu distance d. Fro now on, C ¼½n; ; dš wi denote a inear code of ength n, size and iniu distance d. Furtherore, when ax k i¼1 j i <ax kþ1 i¼1 j i, the inequaity d n k þ 1 hods, which is known as the Singeton bound [10]. The cass of codes that satisfy the equaity d ¼ n k þ 1 pays an iportant roe in our soution approach as evident fro the foowing definition. Definition 3. [10] A code C ¼½n; ; dš, where is the product of any k radices, with iniu distance d ¼ n k þ 1 is caed a axiu distance separabe (MDS) code which is aso a systeatic code. Any k radices of an MDS code C for an inforation set. Aong the MDS codes, of particuar interest are the Redundant Pairwise-Prie Residue Codes, which are defined on a Redundant Residue Nuber Syste (RRNS). An RRNS has n pairwise-prie positive radices, 1 ; 2 ;...; k ; kþ1 ;...; n ; caed odui. Let the first k odui be tered as the nonredundant odui, whie the reaining n k odui be the redundant odui. Let M IS ¼ k i¼1 i and M R ¼ n i¼kþ1 i be the product of, respectivey, the nonredundant and the redundant odui. Aso, et M ¼ n i¼1 i be the product of aodui. It is we known that every integer X 2½0;MÞ can be uniquey represented by a residue vector x ¼< x 1 ;x 2 ;...;x n > such that x i X od i, for 1 i n. Ceary, 0 x i < i. Siiary, given a residue vector x, the corresponding integer X can be uniquey deterined by appying the Chinese Reainder Theore [2]. The residue representations of the integers in the range ½0;MÞ can be partitioned into codes as foows. Definition 4. For fixed k 2½1;...;n 1Š and given a set of n pairwise-prie redundant odui, 1 ; 2 ;...; k ; kþ1 ;...; n ; et M IS ¼ k i¼1 i, M R ¼ n i¼kþ1 i, and M ¼ n i¼1 i. For fixed 2½0;M M IS Þ, the redundant residue RRðn; k; Þ-code (or briefy, RR-ðn; kþ-code) consists of a the residue vectors representing integers in the range ½; þ M IS Þ. In particuar, for ¼ 0, RR-ðn; k; 0Þ-code consists of a the residue vectors representing integers in ½0;M IS Þ. Every codeword of an RR-ðn; k; Þ-code has an inforation part consisting of the first k residue digits and a parity part, consisting of the reaining n k residue digits. Moreover, as proven in [9], C is seiinear, i.e., it is inear under certain conditions. A the ain properties of the inear codes aso hod for the seiinear codes. In particuar, Lea 1. [9] An RR-ðn; kþ-code C has a iniu distance d if and ony if ax d i¼1 j i >MR ax d 1 ; for 1 ji n: i¼1 j i Therefore, Lea 2. [2], [9] For fixed k 2½1;...;n 1Š, the RR-ðn; kþ-code C defined by the odui 1 ;...; k ; kþ1 ;...; n such that 1 < 2 <...< k < kþ1 <...< n has iniu distance d ¼ n k þ 1 and, hence, it is an MDS code. Moreover, varying the nuber k of nonredundant odui fro 1 to n 1, we obtain n 1 different MDS codes. Fact 1. Given n odui 1 ;...; k ; kþ1 ;...; n and a fixed integer k 2½1;...;n 1Š, et M IS ¼ k i¼1 i and M R ¼ n i¼kþ1 i. Then, the entire space S ¼ k kþ1... n can be partitioned into M R subspaces, denoted as C 0 ;C 1 ;...C MR 1 such that C j ¼ RR-ðn; k; jm IS Þ. 5 DISK ALLOCATION STRATEGY BASED ON REDUNDANT RESIDUE CODES We are now ready to present the Decuster procedure, which soves the disk aocation probe for utidiensiona fies whose attributes have pairwise prie doains. For fixed k 2½1;...;n 1Š, the procedure Decuster in Fig. 1 distributes a fie Fð 1 ;...; k ; kþ1 ;...; n Þ aong a set of D ¼ n i¼kþ1 i disks denoted as d 0 ;d 1 ;...;d D 1, with the hep of the RR-ðn; k; 0Þ-code ¼ C 0 ¼½n; k i¼1 i; dš, whose codewords are the residue representations of the integers in the range ½0; k i¼1 iþ. In this paper, C 0 wi be caed the seed code. Let us prove soe properties of the above procedure. Lea 3. The Decuster procedure guarantees that every partia atch query (PMQ) q on the fie F with at ost d 1 unspecified attributes has response tie RT ðqþ ¼1. Proof. No ore than one bucket in each disk quaifies for q because the seed code has iniu distance d and, thus, two buckets stored in the sae disk ust differ in at east d attributes. tu

5 DAS AND PINOTTI: LOAD BALANCED AND OPTIMAL DISK ALLOCATION STRATEGY FOR PARTIAL MATCH QUERIES Fig. 1. The Decuster procedure. Theore 1. Let the pairwise-prie doains of the attributes of the fie F ð 1 ;...; k ; kþ1 ;...; n Þ be arranged in an increasing order such that 1 <...< k < kþ1 <...< n : For fixed k 2½1::n 1Š, given the seed code C 0 ¼½n; k i¼1 i;n k þ 1Š; the associated Decuster procedure eads to a stricty optia disk aocation strategy for every PMQ q on F. Hence, using D ¼ n i¼kþ1 i disks, the response tie for every PMQ q whose response set has size is given by RTðqÞ ¼ D. Proof. Fixed k: For any partiaatch query with at ost n k unspecified attributes, the resut foows fro Lea 3 and the fact that D. For PMQs with ore than n k unspecified attributes, et q be an arbitrary query whose attributes q i1 ;...;q iu are specified, and the reaining n u attributes satisfy q ij ¼ a ij for u þ 1 j n. Let ¼ n j¼uþ1 i j be the product of the doains of the unspecified attributes of q. By the Chinese Reainder Theore [2], there is a unique integer X in the range ½0; Þ such that X a iuþj od iuþj, for 1 j n u. In the range ½0;MÞ, where M ¼ n i¼1 i, there are M strings <x 1 ;...;x n > such that x iuþj ¼ a iuþj for 1 j n u. Those n-tupes correspond to the integers X þ k where 0 k< M, and verify ðx þ k ÞXod iuþj for 1 j n u. Reca that, by Fact 1, the code C j associated with the disk d j, for 0 j D 1, consists of the residue representations of the integers in j k i¼1 i; ðj þ 1Þ k i¼1 i. There are at ost k i¼1 i buckets quaifying for q aong the codewords of C j. Hence, the response tie is RTðqÞ ¼ k i¼1 i. Since ¼ M and D ¼ n i¼kþ1 i, for every PMQ q, the cai foows: RTðqÞ ¼ k i¼1 i ¼ M=D ¼ M=M s ¼ : D D ut 5.1 Capturing Dynaic Scenario Our soution to the disk aocation probe can efficienty anage situations in which either 1) the nuber of disks avaiabe in the syste changes, or 2) the nuber of attributes in a fie changes, without redistributing the entire utidiensiona fie fro scratch aong the disks. Nonetheess, in either cases, soe buckets ust be oved fro one disk to another. Assue a pairwise-prie utidiensiona fie with F ð 1 ;...; k ; kþ1 ;...; n Þ 1 <...< k < kþ1 <...< n ; that has been distributed by the Decuster procedure aong D ¼ n i¼kþ1 i ¼ M R disks using the MDS seed code C 0 ¼½n; M IS ¼ k i¼1 i;n k þ 1Š. First, suppose the nuber of avaiabe disks increases fro D to k D. To reduce the response tie, fewer buckets wi be assigned to each disk. The fie F wi be decustered using the new MDS seed code C þ ¼½n; M Is k ;n k þ 2Š, which assigns M Is k buckets to each disks in the new configuration. This eans that each code stored at the beginning into a singe disk d j, for 0 j M R 1, is now spread out aong the k disks, f k d j ; k d j þ 1;...; ð k þ 1Þd j 1g; Moreover, the new soution is stricty optia because the new response tie is RT þ ðqþ ¼ D k. Siiary, if the nuber of avaiabe disks reduces fro M R to MR kþ1, a new MDS seed code C ¼½n; M I kþ1 ;n kš wi be used to decuster F. The buckets distributed in kþ1 consecutive disks in the initia disk syste are now coapsed into a singe disk of the new disk syste. The new disk aocation reains stricty optia. Note that the response tie increases because fewer disks are avaiabe and ony the PMQ with no ore than n k 1 unspecified attributes can now exhibit constant response tie. Finay, observe that when the nuber ðdþ of disks avaiabe is not a utipe of the attribute doains, the fastest disk aocation strategy achievabe with our strategy wi invove D ¼ n i¼n uþ1 i disks where D is the product of the u argest doains of F such that n i¼n u i >D n i¼n uþ1 i. For every PMQ q, the response tie RTðqÞ using ony D disks is sower than the response tie RT ðqþ achieved using a the D disks. However, the sowdown is upper bounded by n u because RTðqÞ <RT ðqþ n u. Suppose now that D reains unchanged, whie the nuber of attributes of the fie F changes. Athough this

6 1216 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 13, NO. 12, DECEMBER 2002 situation ay have ess ipact in practice, this coud be the case when ore refined searches, using a new attribute of F, are perfored or a previousy used attribute of the fie becoes obsoete. If F is searched according to a new attribute n þ 1, whose doain is nþ1, each bucket <a 1 ;...;a n > is partitioned into the nþ1 buckets. These are <a 1 ;...;a n ; 0 >; < a 1 ;...;a n ; 1 >;...;<a 1 ;...;a n ; nþ1 1 >; one for each vaue of the ðn þ 1Þth attribute. The nuber of buckets increases fro M ¼ n i¼1 i to M 0 ¼ M nþ1. Since the nuber of disks is unchanged, ore buckets ust be assigned to each disk. Assuing the MDS C ¼½n; M IS ;n k þ 1Š, as the seed code according to which F was distributed, the new seed code wi be C ¼ ½n þ 1;M IS nþ1 ;tš where t n k þ 1. Depending on the vaue of nþ1, C ay or ay not be an MDS code and, therefore, optiaity is no onger guaranteed. About the reocation process, the bucket <a 1 ;...;a n > initiay stored into the disk d j is now partitioned into nþ1 buckets, as entioned above. Precisey, the buckets <a 1 ;...;a n ;x nþ1 >, which is the residue representation of X þ km for fixed k 2½0; nþ1 1Š, wi be assigned to the disk dj k, according to Fact 1. XþkM M Is nþ1 A siiar reasoning appies when the nuber of attributes of F decreases. 6 REDUNDANT NONPAIRWISE PRIME RESIDUE CODES The soution proposed in the ast section works for pairwise-prie fies. This is not a strong constraint because the doains of the attributes are seected whie the database is designed depending on the origina interva of the attributes and on the size of the disk page (see Exape 2). Nevertheess, for the sake of copeteness, et us extend our soution to the case of nonpairwise prie doains. Consider a set of n nonpairwise prie odui 1 ; 2 ;...; k ; kþ1 ;...; n and aso consider a the n-tupes <x 1 ;...;x n > for x i 2½0; i Þ, of the n-diensiona space S ¼ 1... n. Let M ¼ n i¼1 i and ¼ :c::ð 1 ; 2 ;...; n Þ be, respectivey, the product and the east coon utipe (.c..) of the n odui. In contrast to the pairwise-prie case, there are n-tupes <x 1 ;...;x n > 2 S which do not represent any integer X. That is, there is no integer X such that x i X od i, for 1 i n. In fact, a genera for of the Chinese Reainder Theore [11] ipies that, if S consists of trings, ony the integers in ½0;Þ have a residue representation <x 1 ;...;x n > 2 S. Precisey, if g ij ¼ g:c:dð i ; j Þ is the greatest coon divisor (g.c.d.) of the two odui i and j, ony those n-tupes which satisfy the congruence x i x j od g ij ; for 1 i; j n are vaid integer representations in the n-diensiona space S. Note that those vaid residue representations for a subset of S, which is a code by definition. However, the iniu distance of such a code is unknown (if the code itsef is not generated). To overcoe this iitation, soe expicit redundancy can be added as foows. Definition 5. Given n nonpairwise prie odui 1 ;...; k ; kþ1 ;...; n ; for fixed k 2½1;...;n 1Š, et the first k odui be caed nonredundant and the reaining n k odui be caed redundant. Moreover, et ¼ :c:ð 1 ; 2 ;...; k Þ and ¼ :c:ð 1 ; 2 ;...; n Þ. The nonpairwise prie redundant residue code or NPRR-ðn; kþ-code C consists of the set of vaid integer representations, i.e., the set of representations of a integers in ½0; Þ. It is easy to see that the NPRR-ðn; kþ-code C is a seiinear code. To derive the iniu distance of the NPRR-ðn; kþ-code C, et. } denote the.c.. of the n k redundant odui;. denote the.c.. of the k nonredundant odui;. denote the.c.. of a n odui;. partitioning the n odui into two arbitrary sets s d 1 and s d 1 of sizes d 1 and n d þ 1 respectivey, et sd 1 and sd 1 be the.c. of the odui in s d 1 and s d 1, respectivey. In other words, sd 1 ¼ :c::ð ij j ij 2 s d 1 Þ and sd 1 ¼ :c::ð ij j ij 2 s d 1 Þ;. partitioning the n odui in two arbitrary sets s d and s d of sizes d and n d, respectivey, et sd and sd be the.c. of the odui in s d and s d, respectivey. That is, sd ¼ :c::ð ij j ij 2 s d Þ and sd ¼ :c::ð ij j ij 2 s d Þ;. S d be the set of a possibe s d subsets aong f 1 ;...; n g;. S d 1 be the set of a possibe s d 1 subsets aong f 1 ;...; n g. Theore 2. Given the radices 1 ;...; n for fixed k 2 ½1::n 1Š, the NPRR-ðn; kþ-code C has the iniu distance d if and ony if the foowing reation hods:! sd } ax sd2s d > gcdð sd ; sd Þ gcdð;}þ! sd 1 ax sd 1 2S d 1 : gcdð sd 1 ; sd 1 Þ Proof. The seiinear NPRR-ðn; kþ-code C has iniu distance d if no codeword has Haing weight d 1 (except for the a-zero codeword) and if there is at east one codeword of Haing weight d. A codeword with d 1 nonzero digits represents an integer X ¼ X 0 sd 1, for soe s d 1. The saest vaue of X is obtained for X 0 ¼ 1 and in sd 12S d 1 sd 1.

7 DAS AND PINOTTI: LOAD BALANCED AND OPTIMAL DISK ALLOCATION STRATEGY FOR PARTIAL MATCH QUERIES Fig. 2. The Partition procedure. Consequenty, the corresponding word is not a codeword if and ony if in sd 1 2S d 1 sd 1 ; or equivaenty where ax sd 12S d 1 ¼ ; sd 1 gcdð sd 1 ; sd 1 Þ sd 1 sd 1 gcdð sd 1 ; sd 1 Þ : Now, if a codeword of Haing weight d exists, the above condition ust be denied for any subset of d digits. Hence, ax sd2s d < : sd gcdð sd ; sd Þ Fro the above two conditions and recaing that can be expressed as ¼ } gcdð;}þ, the cai foows. tu For nonpairwise prie redundant residue codes, Fact 1 can be rewritten as foows: Fact 2. For fixed k 2½1;...;n 1Š and given a set of n nonpairwise prie radices 1 ;...; n, et 1 ;...; k and kþ1 ;...; n be, respectivey, the nonredundant and the redundant odui. Considering the NPRR-ðn; kþ-code C ¼½n; ;dš, a the integers in ½0;Þ, can be partitioned into codes, each having the sae size and the sae distance as C. More precisey, code C j for 0 j 1, consists of the set of residue representations of the integers X 2 ½j; ðj þ 1Þ 1Š. Note that C 0 ¼ C. Proof. The proof foows directy by observing that the Haing distance of the two codewords x 1 and x 2 beonging to C 0 is the sae as the Haing distance between the two codewords x 1 þ j and x 2 þ j beonging to C j, where 0 j 1. tu Finay, et the utipe-su set, C½xŠ ¼fxþwjw2 Cg, be obtained by the oduar su, digit-by-digit, of the n-tupe x over a the n-tupes of the code C. Now, for a given NPRR-ðn; kþ-code C 0, the entire n-diensiona space S ¼ 1... n can be partitioned by the agorith in Fig. 2. To prove the correctness of the above procedure, et us prove the foowing: Lea 4. The set of subsets fc j ½TðiÞŠ j 0 j = 1 and 0 i M= 1g fors a partition of the space S ¼ 1... n. Moreover, each subset C j ½TðiÞŠ is a code ½n; ;dš. Proof. First note that C j ½Tð0ÞŠ ¼ C j, for 0 j = 1. Next, et us prove, by contradiction, that C j ½TðiÞŠ \ C r ½TðsÞŠ ¼ ; for any choice of i; j; r; s. Suppose that the two n-tupes Xðj; iþ 2C j ½TðiÞŠ and Y ðr; sþ 2C r ½TðsÞŠ, obtained by the residue vectors X; Y 2 C 0, are equa. W..o.g. assue j r. Note that Xðj; iþ is the su of the residue representation of ðx þ jþ and of the n-tupe T ðiþ, whie Y ðr; sþ is the su of the residue representation of ðy þ rþ and of the n-tupe T ðsþ. Now, Xðj; iþ is equa to Y ðr; sþ if and ony if X Y þ ðj rþ ¼ TðsÞ TðiÞ beongs to C j r. This cannot be true since it is against the criteria used for the seection of the n-tupes in T. Hence, the subsets in fc j ½TðiÞŠ j 0 j = 1 and 0 i M= 1g for a partition of the space S ¼ 1... n. It sti reains to be proven that, for any pair j; i, the set of subsets C j ½TðiÞŠ has the sae iniu distance as C 0. Consider two arbitrary n-tupes Xðj; iþ and Y ðj; iþ, both beonging to C j ½TðiÞŠ, and derived fro X and Y in C 0. Repeating the above reasoning, it is easy to see that the n-tupes Xðj; iþ Yðj; iþ and X Y are the sae. Hence, a the subsets in fc j ½TðiÞŠ j 0 j = 1 and 0 i M= 1g inherit the sae iniu distance d as the set C 0. tu Finay, the decuster procedure for nonpairwise pries is described in Fig. 3.

8 1218 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 13, NO. 12, DECEMBER 2002 Fig. 3. The Nonpairwise Prie Decuster procedure. Theore 3. For a fixed k 2½1::n 1Š, et F ð 1 ;...; k ; kþ1 ;...; n Þ be a fie whose attributes are not pairwise prie and et C 0 ¼ ½n; ;dš be the seed code. Distributing F aong D ¼ M disks using the Nonpairwise Prie Decuster procedure, the response tie for a PMQ q with s specified attributes q i1 ;...;q is is given by RTðqÞ ¼, where ¼ :c:ð i1 ;...; is Þ: Finay, the ratio between the response tie of this procedure, say RTðqÞ, and the best achievabe response tie, say RT ðqþ, is given by RTðqÞ RT ¼ ðqþ 2 6 s j¼1 i j 3 : 7 Proof. Consider the n-tupe q 0 beonging to S ¼ n and obtained by substituting a the unspecified attributes of the PMQ q with 0. By the previous discussion, there ust be i and j such that q 0 2 C j ½TðiÞŠ, and q 0 is stored into disk d i þj and there exists x 2 C 0 such that q 0 ¼ x þ j þ TðiÞ. Then, d i þj stores q 0 aong with the buckets q 0 þ u ¼ x þ j þ TðiÞþu, for a vaues of u such that 0 ðxþu Þ. Note that u has aways a vaid residue presentation since u <. Therefore, at ost buckets quaify for the query q on the sae disk. It reains to show how far is the response tie for nonpairwise prie decuster procedure fro the optia response tie. Using D ¼ M disks, the optia response tie for q woud be RT ðqþ ¼ & M & ¼ ¼ D s j¼1 : i j s j¼1 i j M Therefore, our soution is s j¼1 i j ties sower than the stricty optia soution. tu Exape 3. Consider the redundant residue nuber syste (RNS) associated with the nonpairwise prie radices, 1 ¼ 3, 2 ¼ 6, 3 ¼ 5. For fixed k ¼ 2, et 1 ; 2 be the nonredundant odui and et 3 be the redundant oduus. According to our definitions, M ¼ 90, ¼ 30, and ¼ 6. In other words, the space S consists of nuber of 3-tupes, but the RNSs associated with the odui 1 ; 2 ; 3 and 1 ; 2, can represent the integers in ½0; 29Š and ½0; 5Š, respectivey. Let the code C ¼½3; 6; 2Š consist of the residue representations of the integers in ½0; 5Š in the RNS of radices 1 ¼ 3, 2 ¼ 6, 3 ¼ 5. That is, C ¼f< 0; 0; 0 >; < 1; 1; 1 >; < 2; 2; 2 >; < 0; 3; 3 >; < 1; 4; 4 >; < 2; 5; 0 >g: Fro Fact 2, the vaid residue representations are partitioned into 90 6 ¼ 15 codes. Precisey, code C j½0; 0; 0Š corresponds to the residue representations of the integers in the range ½6j; 6ðj þ 1Þ 1Š, where 0 j 4. In particuar, C 1 ¼f< 0; 0; 1 >; < 1; 1; 2 >; < 2; 2; 3 >; < 0; 3; 4 >; < 1; 4; 0 >; < 2; 5; 1 >g; C 2 ¼f< 0; 0; 2 >; < 1; 1; 3 >; < 2; 2; 4 >; < 0; 3; 0 >; < 1; 4; 1 >; < 2; 5; 2 >g; C 3 ¼f< 0; 0; 3 >; < 1; 1; 4 >; < 2; 2; 0 >; < 0; 3; 1 >; < 1; 4; 2 >; < 2; 5; 3 >g; and C 4 ¼f< 0; 0; 4 >; < 1; 1; 0 >; < 2; 2; 1 >; < 0; 3; 2 >; < 1; 4; 3 >; < 2; 5; 4 >g: Now, et T ¼fTð0Þ ¼< 0; 0; 0 >; Tð1Þ ¼< 1; 0; 3 >; Tð2Þ ¼ < 2; 3; 0 >g be the set of 3-tupes such that T ð1þ, T ð2þ, Tð1Þ Tð2Þ ¼< 2; 3; 3 >, and Tð2Þ Tð1Þ ¼< 1; 3; 2 > does not beong to Z ¼ C 0 [ C 1 [ C 2 [ C 3 [ C 4. Hence, the codes C j ½1; 0; 3Š ¼f< 1; 0; 3 > þ C j ½0; 0; 0Šg and C j ½2; 3; 0Š ¼f< 2; 3; 0 > þ C j ½0; 0; 0Šg; for 0 j 4, copete the partition of S. Finay, the nonpairwise prie decuster procedure assigns the code C j ½TðiÞŠ to the disk d 5iþj. Atogether there are 15 codes and hence D ¼ 15 disks are required. Now, consider the PMQ q ¼< ; ; 2 >. Since ¼ 5, there are at ost ¼ 2 quaifying buckets for each disk. Hence, RTðqÞ ¼2 ¼ RT ðqþ in this case.

9 DAS AND PINOTTI: LOAD BALANCED AND OPTIMAL DISK ALLOCATION STRATEGY FOR PARTIAL MATCH QUERIES CONCLUSION In this paper, we studied the disk aocation probe for distributing nonunifor utidiensiona fies on to parae database systes that expoit the abiity of accessing utipe disks siutaneousy. Based on a arge and fexibe cass of axiu distance separabe codes, caed the redundant residue codes, a stricty optia aocation ethod is derived for every query q when the attribute doains of the utidiensiona fie are pairwise prie. We aso introduced a new faiy of residue codes, caed the redundant nonpairwise prie residue codes, that can be appied with utidiensiona fies that have attribute doains nonpairwise prie. ACKNOWLEDGMENTS This work is partiay supported by the Texas Advanced Research Progra TARP and a MIUR REAL- WINE research grant in Itay. A preiinary version of this paper appeared in [4]. C.M. Pinotti is gratefu to Professor Barsi for introducing her to the area of the redundant residue nuber systes. The authors woud aso ike to thank the reviewers for hepfu suggestions and the associate editor for tiey handing the paper. REFERENCES [1] K.A.S. Abde-Ghaffar and A. E Abbadi, Optia Disk Aocation for Partia Match Queries, ACM Trans. Database Systes, vo. 18, no. 1, pp , Mar [2] F. Barsi and P. Maestrini, Error Correcting Properties of Redundant Residue Nuber Systes, IEEE Trans. Coputers, vo. 22, pp , [3] F. Barsi and P. Maestrini, Error Codes in Residue Nuber Systes with Non-Pairwise-Prie Modui, Inforation and Contro, vo. 46, no. 1, pp , Juy [4] S.K. Das and M.C. Pinotti, An Optia Disk Aocation Strategy for Partia Match Query of Non-Unifor Cartesian Product Fies, Proc. IEEE Int Parae Processing Syp., pp , Apr [5] H.C. Du and J.S. Soboewski, Disk Aocation for Cartesian Product Fies, ACM Trans. Database Systes, vo. 7, no. 1, pp , Mar [6] M.F. Fang, R.C.T. Lee, and C.C. Chang, The Idea of De- Custering and its Appications, Proc. 12th Int Conf. Very Large Databases, pp , Aug [7] C. Faoutsos and D. Metaxas, Disk Aocation Methods Using Error Correcting Codes, IEEE Trans. Coputers, vo. 40, no. 8, pp , Aug [8] M.H. Ki and S. Praanik, Optia Fie Distribution for Partia Match Retrieva, Proc. ACM-SIGMOD Int Conf. Manageent of Data, pp , [9] H. Khrisna, K. Lin, and J. Sun, A Coding Theory Approach to Error Contro in Redundant Residue Nuber Systes Part I: Theory and Singe Correction, IEEE Trans. Circuits and Systes, II, vo. 39, no. 1, pp. 8-17, Jan [10] F.J. MacWiias and N.J.A. Soane, The Theory of Error-Correcting Codes, Parts I and II. New York: North-Hoand, [11] O. Ore, The Genera Chinese Reainder Theore, A. Math. Monthy, pp , Saja K. Das received the PhD degree in Coputer Science in 1988 fro the University of Centra Forida, Orando. Currenty, he is a fu professor of coputer science and engineering and aso the Founding Director of the Center for Research in Wireess Mobiity and Networking (CReWMaN) at the University of Texas at Arington (UTA). Prior to 1999, he was a professor of coputer science at the University of North Texas (UNT), Denton, where he founded the Center for Research in Wireess Coputing (CReW) in 1997, and aso served as the director of the Center for Research in Parae and Distributed Coputing (CRPDC) fro He is a recipient of the UNT Student Association s Honor Professor Award in 1991 and 1997 for best teaching and schoary research; UNT s Deveoping Schoars Award in 1996 for outstanding research; and UTA s Outstanding Senior Facuty Research Award in Coputer Science in He has visited nuerous universities, research organizations, and industry research abs for coaborative research and invited seinar taks. He was a visiting scientist at the Counci of Nationa Research in Pisa, Itay, and Sovak Acadey of Sciences in Bratisava, and was aso a Visiting Professor at the Indian Statistica Institute, Cacutta. His current research interests incude resource and obiity anageent in wireess networks, obie coputing, QoS provisioning and wireess utiedia, obie Internet, network architectures and protocos, distributed/parae processing, perforance odeing, and siuation. He has pubished ore than 185 research papers in these areas, directed severa projects funded by industry and governent, and fied four US patents in wireess obe networks. He received the best paper awards for significant research contributions in the ACM Fifth Internationa Conference on Mobie Coputing and Networking (MobiCo 99), Third ACM Internationa Workshop on Modeing, Anaysis, and Siuation of Wireess and Mobie Systes (MSWiM 2000), and ACM/IEEE Internationa Workshop on Parae and Distributed Siuation (PADS 97). Dr. Das serves on the editoria boards of the Journa of Parae and Distributed Coputing, Parae Processing Letters, Journa of Parae Agoriths and Appications, and the Coputer Networks journa. Each year he serves on nuerous IEEE and ACM conferences as a technica progra coittee eber, progra chair, or genera chair. He is a eber of the IEEE TCPP Executive Coittee and Advisory Boards of severa cutting-edge copanies. He is a eber of the IEEE. Cristina M. Pinotti received the PhD degree (cu aude) in coputer science fro the University of Pisa, Itay, in Since 2000, she has been an associate professor in the Departent of Coputer Science and Teecounications of the University of Trento, Itay. Prior to 2000, she was a researcher with the Consigio Nazionae dee Ricerche at the Istituto di Eaborazione de Inforazione, Pisa. She was a visiting researcher at the Departent of Coputer Science, University of North Texas, Denton, and at the Departent of Coputer Science, Od Doinion University, Norfok, Virginia. Her research interests incude parae data structures and utiprocessor interconnection networks, the design and anaysis of parae agoriths, I/O externaeories, obie and wireess networks, VLSI ayouts, coputer arithetic, and residue nuber systes. Each year she serves on the progra coittee of severa IEEE and ACM workshops and conferences. She is a eber of the IEEE Coputer Society.. For ore inforation on this or any coputing topic, pease visit our Digita Library at