Learning-Based Top-N Selection Query Evaluation over Relational Databases

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1 Learnng-Based Top-N Selecton Query Evaluaton over Relatonal Databases Lang Zhu *, Wey Meng ** * School of Mathematcs and Computer Scence, Hebe Unversty, Baodng, Hebe , Chna, zhu@mal.hbu.edu.cn ** Department of Computer Scence, State Unversty of New York at Bnghamton, Bnghamton, NY 13902, USA, meng@cs.bnghamton.edu Abstract. A top-n selecton query aganst a relaton s to fnd the N tuples that satsfy the query condton the best but not necessarly completely. In ths paper, we propose a new method for evaluatng top-n selecton queres aganst relatonal databases. Ths method employs a learnng-based strategy. Intally, t fnds and saves the optmal search spaces for a small number of random top-n queres. The learned knowledge s then used to evaluate new queres. Extensve experments are carred out to measure the performance of ths strategy and the results ndcate that t s hghly compettve wth exstng technques for both low-dmensonal and hgh-dmensonal data. Furthermore, the knowledge base can be updated based on new user queres to reflect new query patterns so that frequently submtted queres can be processed most effcently. 1 Introducton As ponted out n a number of recent papers [1, 4, 5, 6, 7, 9, 10, 15], t s of great mportance to fnd the N tuples n a database table that best satsfy a gven user query, for some nteger N. Ths s especally true for searchng commercal products on the Web. For example, for a Web ste that sells used-cars, the problem becomes fndng the N best matchng cars based on a gven car descrpton. A top-n selecton query aganst a relaton/table s to fnd the N tuples that satsfy the query condton the best but not necessarly completely. A smple soluton to ths problem s to retreve all tuples n the relaton, compute ther dstances wth the query condton usng a dstance functon and output the N tuples that have the smallest dstances. The man problem wth ths soluton s ts poor effcency, especally when the number of tuples of the relaton s large. Fndng effcent strateges to evaluate top-n queres has been the prmary focus of top-n query research. In ths paper, we propose a new method for evaluatng top-n selecton queres aganst a relaton. The man dfference between ths method and exstng ones s that t employs a learnng-based strategy. A knowledge base s bult ntally by fndng the optmal (.e., the smallest possble) search spaces for a small number of random top-n selecton queres and by savng some related nformaton for each optmal soluton. * Ths work s completed when the frst author was a vstor at SUNY at Bnghamton.

2 Ths knowledge base s then used to derve search spaces for new top-n selecton queres. The ntal knowledge base can be contnuously updated whle new top-n queres are evaluated. Clearly, f a query has been submtted before and ts optmal soluton s stored n the knowledge base, then ths query can be most effcently processed. As a result, ths method s most favorable for repeatng queres. It s known that database queres usually follow a Zpfan dstrbuton [2]. Therefore, beng able to support frequently submtted queres well s mportant for the overall performance of the database system. What s attractve about ths method, however, s that even n the absence of repeatng queres, our method compares favorably to the best exstng methods wth comparable storage sze for storng nformaton needed for top-n query evaluaton. In addton, ths method s not senstve to the dmensonalty of the data and t works well for both low-dmensonal and hgh-dmensonal data. The rest of the paper s organzed as follows. In Secton 2, we brefly revew some related works and compare our method wth the exstng ones. In Secton 3, we ntroduce some notatons. In Secton 4, we present our learnng-based top-n query evaluaton strategy. In Secton 5, we present the expermental results. Fnally n Secton 6, we conclude the paper. 2. Related Work A large number of research works on the effcent evaluaton of top-n selecton queres (or N nearest-neghbors queres) are reported recently [1, 4, 5, 6, 7, 9, 10, 15]. Here we only revew and compare wth those that are most related to ths paper. The authors of [10] use a probablstc approach to query optmzaton for returnng the top-n tuples for a gven selecton query. The rankng condton n [10] nvolves only a sngle attrbute. In ths paper, we deal wth mult-attrbute condtons. In [6], hstogram-based approaches are used to map a top-n selecton query nto a tradtonal range selecton query. In [1], the hstogram-based strateges are extended to a new technque called Dynamc, expressed as dq(α) = drq + α(dnrq - drq). A sgnfcant weakness of hstogram-based approaches s that ther performance deterorates quckly when the number of dmensons of the data exceeds 3 [1, 14]. Therefore, hstogram-based approaches are sutable for only low-dmensonal data n practce. In [7], a samplng-based method s proposed to translate a top-n query nto an approxmate range query. Unlke hstogram-based approaches, ths method s sutable for hgh-dmensonal data and s easy to mplement n practce. However, ths method only provdes an approxmate answer to a gven top-n query,.e., t does not guarantee the retreval of all of the top-n tuples. In addton, for large databases, ths method may be neffcent. For example, for a relaton wth 1 mllon tuples and usng a 5% sample set (as reported n [7]), 50,000 tuples need to be evaluated n order to fnd the approxmate range query for a top-n query. Our learnng-based method s fundamentally dfferent from all exstng technques. It can learn from ether randomly generated tranng queres or real user queres so t can adapt to changes of query patterns. Furthermore, t delvers good performance for both low-dmensonal and hgh-dmensonal data.

3 3 Problem Defnton Let R n be a metrc space wth dstance (or, metrc) functon d(.,.), where R s the real lne. Suppose that R R n s a relaton (or dataset) wth M tuples and n attrbutes (A 1,, A n ). A tuple t R s denoted by t = (t 1,, t n ). Consder a pont (query) Q = (q 1,, q n ) R n. A top-n selecton query, or top-n query for short, s to fnd a sorted set of N tuples n R that are closest to Q accordng to the gven dstance functon. The results of a top-n query are called top-n tuples. Suppose Q = (q 1,, q n ) s a top-n query and r > 0 s a real number. We use S(Q, r) n to denote the n-square = [ q, + ] 1 r q r centered at Q wth sde length 2r. Our goal s to fnd a search dstance r as small as possble such that S(Q, r) contans the top-n tuples of Q accordng to the gven dstance functon. We use S(Q, r, N) to denote ths smallest possble n-square and the correspondng r s called the optmal search dstance for the top-n query Q. Some example dstance functons are [1, 7, 15]: Summaton dstance (.e., L 1 -norm or Manhattan dstance): n d 1 (x, y) = = x y 1. n 2 Eucldean dstance (.e., L 2 -norm dstance): d 2 (x, y) = = ( x y ). Maxmum dstance (.e., L -norm dstance): d (x, y) = max{ x y }. 1 n Frequently, t s not sutable to apply a dstance functon to the values of dfferent attrbutes drectly due to the unt/scalng dfference of the values n dfferent attrbutes. For example, when buyng a used car, a 100-dollar dfference n prce s probably more sgnfcant than a 100-mle dfference n mleage. Ths can be resolved by multplyng approprate mportance/scalng factors to the raw dstances based on the values of dfferent attrbutes [9, 15]. Wthout loss of generalty, n ths paper, we assume that all attrbutes are of the same mportance for rankng tuples so that the dstance functons can be appled drectly. 4 Learnng-based Top-N Query Evaluaton In ths secton, we ntroduce our learnng-based method for top-n query evaluaton. Frst, keep track of frequently submtted queres and save the evaluaton strateges for these queres n a knowledge base. Next, for each newly submtted query, f ts evaluaton strategy has been saved or t s smlar to some queres whose evaluaton strateges have been saved, then derve an evaluaton strategy for the new query from the saved relevant strategy/strateges. In ths paper, we study ssues related to ths method. 4.1 Query Informaton to Save Intally, the knowledge base s empty. When a new query arrves at the system, an exstng method such as the one proposed n [1] or n [7] s employed to evaluate the query. Let Q = (q 1,, q n ) be the query and t = (t 1,, t n ), 1 N, be the top-n tu- 1

4 ples, respectvely, then the search dstance of the smallest n-square s r = max {d (Q, t )} = max{max{ q j tj }}. When the smallest n-square s obtaned, several peces of nformaton are collected and saved. For a top-n query Q, let 1 N 1 N 1 j n r be the search dstance of S(Q, r, N) the smallest n-square that contans the top-n tuples of Q, f denote the frequency of S(Q, r, N),.e., the number of tuples n S(Q, r, N) (obvously N f ), c denote the number of tmes that Q has been submtted, and d denote the most recent tme when Q was submtted. The nformaton that we wll keep for each saved query Q s represented as ζ(q) = (Q, N, r, f, c, d) and t s called the profle of the query. After the system has been used for sometme, a number of query profles are created and saved n the knowledge base. Let P = {ζ 1, ζ 2,, ζ m } denote the set of all query profles,.e., the knowledge base, mantaned by the system. Queres n P wll be called profle queres. Generally, profles should be kept only for queres that are frequently submtted recently as reflected by the values of c and d n each profle. In our mplementaton, the ntal knowledge base s not bult based on real user queres. Instead, t s based on randomly selected queres from a possble query space (see Secton 5 for more detals). 4.2 New Query Evaluaton When a newly submtted top-n query Q s receved by the system, we need to fnd an approprate search dstance r for t. In a system that does not store query profles, ths query wll be processed just lke any new query and the methods dscussed n [1, 7] may be used to fnd r. When query profles are stored, t becomes possble to obtan the r for some new user queres from these profles Determnng the search dstance r The detals of obtanng the search dstance r for a new query are descrbed below. Fgure 1. Q = Q. Fgure 2. Q S(Q, r, N ). Fgure 3. Q S(Q, r, N ) Frst we dentfy Q = ( q, q,..., q ) from P that s the closest to Q under the dstance functon d(.,.). The followng cases exst. 1 2 n 1. d(q, Q ) = 0,.e., Q = Q. In ths case, fnd all profles n P whose query pont s Q, but have dfferent result sze N, say, N 1 < N 2 < < N k. An example of a 2- dmenson case s depcted n Fgure 1, where squares of sold-lnes represent the search spaces of profle queres (.e., those n P) and the square of dotted-lnes represents the search space of the new query Q. We now consder three subcases.

5 a. There s N {N 1, N 2,, N k } such that N = N. That s, there s a top-n query n P that s dentcal to the new query n both the query pont and result sze. In ths case, let r := r, where r s from the profle ζ = (Q, N, r, f, c, d ). b. There s no N {N 1, N 2,, N k } such that N = N but there s N {N 1, N 2,, N k } such that N > N and t s the closest to N among N 1, N 2,, N k (Fgure 1). In ths case, let r := r to guarantee the retreval of all the top-n tuples for Q. c. N k < N. In ths case, we assume that the search space for Q has the same local dstrbuton densty as that for Q. Based on ths assumpton, we have N/(2r) n = N k /(2r k ) n. As a result, we let r := ( n N N k )r k. If not enough top-n tuples are retreved, a larger r wll be used (see Secton 4.2.2). 2. d(q, Q ) 0,.e., Q Q. We consder two subcases. a. Q s n the search space S(Q, r, N ) of Q. Fnd out all query ponts n P whose search spaces contan Q. Let these query ponts be (Q 1,, Q k ) (see Fgure 2). To estmate the search dstance r for Q, we frst use a weghted average of the local dstrbuton denstes of the search spaces for Q 1,, Q k to estmate the local densty of search space for Q. The weght w for the search space correspondng to Q s computed based on ts sze and the dstance between Q and Q. Weght w s an ncreasng functon of the sze of the search space and a decreasng functon of the dstance. In ths paper, w s computed by the followng formula: w = v(s(q, r, N )) * 1/(d(Q, Q )) α where α s a parameter and α = 3n/4 s a good value based on our experments. Let ρ = f /(2 r ) n be the local densty of the search space for Q. Then the local densty of the search space for Q s estmated by: ρ = ( k = 1 w ρ ) / ( k = 1 w ). Based on the above ρ, we estmate the search dstance r to be ( n 2 N / ρ )/2. Note that to ncrease the possblty that all of the top-n tuples for Q are retreved, we replaced N by 2N n the estmaton for r (.e., am to retreve 2N tuples). b. Q s not n the search space S(Q, r, N ) of Q (see Fgure 3). Let h := d(q, Q ) be the dstance between Q and Q. Construct an n-square S(Q, h) and let (Q 1,, Q k ) be all the query ponts n P whose search spaces ntersect wth S(Q, h). Obvously, k 1 as Q s n ths query set. Now the same technque used above n step 2.a s used to estmate the search dstance r for Q. The search dstance r obtaned above (2.a and 2.b) may sometmes be ether too small or too large. To remedy ths, the followng adjustments to r are mplemented. The followng two cases are consdered. (1) N = N. () If r < r or r < d(q, Q ), then r may be too small. We use the followng formula to adjust r: r =max(r_medan, r_mean, r)/2 + (r +d(q, Q ))/2 where r_medan = ( n 2 N / N ) r medan medan, rmedan s the search dstance of the search space whose densty s the medan among all search spaces n P, and N s the N value of the correspondng profle; r_mean = medan

6 ( n 2 N / ρ mean )/2 and ρ mean s the average of all the denstes of the search spaces n P. () If r > r + d(q, Q ), then r s too large as r = r + d (Q, Q ) can already guarantee the retreval of all the top-n tuples of Q. In ths case, we smply lower r to r + d(q, Q ). (2) N N. Ths s handled n a smlar manner as n case (1) except that a constant factor λ = n N / N s utlzed to take nto consderaton the dfference between N and N. () If r < λ r or r < d(q, Q ), then r := max(r_medan, r_mean, r)/2 + (λr + d(q, Q ))/2. () If r > λ r + d(q, Q ), then r := λ r + d(q, Q ) Query Mappng Strateges For a gven top-n query Q = (q 1,, q n ), to retreve all tuples n S(Q, r), one strategy s to map each top-n query to a smple selecton range query of the followng format [1]: SELECT * FROM R WHERE (q 1 r A 1 q 1 + r) AND AND (q n r A n q n + r) If the query returns N results, sort them n non-descendng dstance values and output the top N tuples. A potental problem that needs to be handled s that the estmated search dstance r s not large enough. In ths case, the value of r needs to be ncreased to guarantee that there are at least N tuples n S(Q, r). One soluton to ths problem s provded below. Choose N query ponts n P, whch are closest to the top-n query Q, and sort them n ascendng order of ther dstances wth Q wth respect to the used dstance functon d(.,.). Let the order be Q 1,, Q N and ther correspondng profles be ζ 1, ζ 2,, ζ N. There exsts a number h, 1 < h N, such that N N h N. Durng the computaton of the sum, f S(Q, r, N ) S(Q j, r j, N j ) φ, then N + N j n the above sum was replaced by max{n, N j } to ensure that the search spaces of the frst h queres n (Q 1,, Q N ) contan at least N unque tuples. Let r := max {d(q, Q ) + r }.Thus, by usng ths r as the search dstance for Q to generate the restart query, the restart query wll guarantee the retreval of all the top-n tuples for Q. If there s a hstogram over the relaton R, by usng dnrq n [1], the search dstance r for the restart query can be obtaned as follows. If the sum of all N s n P s less than N, then set r to be dnrq. Otherwse, fnd the number h as mentoned above and let r := mn{ max {d(q, Q ) + r }, dnrq}. 1 h 1 h 5 Expermental Results 5.1 Data Sets and Preparatons All of our experments are carred out usng Mcrosoft's SQL Server 7.0 on a PC. To facltate comparson, the same datasets and parameters used n [1] and [7] are used n

7 ths paper. These datasets nclude data of both low dmensonalty (2, 3, and 4 dmensons) and hgh dmensonalty (25, 50, and 104 dmensons). For low-dmensonal datasets, both synthetc and real datasets used n [1] are used. The real datasets nclude Census2D and Census3D (both wth 210,138 tuples), and Cover4D (581,010 tuples). The synthetc datasets are Gauss3D (500,000 tuples) and Array3D (507,701 tuples). In the names of all datasets, suffx KD ndcates that the dataset has K dmensons. For hgh-dmensonal datasets, real datasets derved from LSI are used n our experments. They have 20,000 tuples and the same 25, 50 and 104 attrbutes as used n [7] are used to create datasets of 25, 50 and 104 dmensons, respectvely. Each experment uses 100 test queres (called a workload). The workloads follow two dstnct query dstrbutons, whch are consdered representatves of user behavors [1]: Based: Each query s a random exstng pont n the dataset used. Unform: The queres are random ponts unformly dstrbuted n the dataset used. The unform workloads are only used for low-dmensonal datasets to compare the results wth those of [1]. For convenence, we report results based on a default settng. Ths default settng uses a 100-query Based workload, N = 100 (.e., retreve top 100 tuples for each query) and max s the dstance functon. When a dfferent settng s used, t wll be explctly mentoned. The most basc way of processng a top-n query s the sequental scan method [4, 5]. In ths paper, we compare the followng four top-n query evaluaton technques: Optmum technque [1]. As a baselne, we consder the deal technque that uses the smallest search space contanng the actual top-n tuples for a gven query. The smallest search space s obtaned usng the sequental scan technque n advance. Hstogram-based technques [1]. We only cte the results produced by the dynamc (Dyn) mappng strategy descrbed n [1] for comparson purpose. Dyn s the best among all hstogram-based technques studed n [1]. Samplng-based technque [7]. In ths paper, we only cte the expermental results produced by the parametc (Para) strategy descrbed n [7]. The Para strategy s the best of all samplng-based strateges dscussed n [7]. Learnng-based (LB) technque. Ths s our method descrbed n Sectons 4. For a gven dataset D, the knowledge-base (or profle set) P s constructed as follows. Frst, a set of random tuples from D s selected. The sze of the random tuple set s decded such that the sze of P does not exceed the sze of the hstogram or the sze of the samplng set when the correspondng method s beng compared. For each query pont chosen for P, the sequental scan technque s used to obtan ts profle durng the knowledge-base constructon phase. For easy comparson, we use the followng performance metrcs used n [1] n ths paper. Percentage of restarts: Ths s the percentage of the queres n the workload for whch the assocated selecton range query faled to retreve the N best tuples, hence leadng to the need of a restart query. When presentng expermental results, Method(x%) s used to ndcate that when evaluaton strategy Method s used, the percentage of restart queres s x%. For example, LB(3%) means that there are 3% restart queres when the learnng-based method s used. Percentage of tuples retreved: Ths s the average percentage of tuples retreved from the respectve datasets for all queres n the workload. When at least N tuples are retreved, lower percentage of tuples retreved s an ndcaton of better effcency. We

8 report SOQ (Successful Orgnal Query) percentages and IOQ (Insuffcent Orgnal Query) percentages. The former s the percentage of the tuples retreved by the ntal selecton range query and the latter s the percentage of the tuples retreved by a restart query when the ntal query faled to retreve enough tuples. 5.2 Performance Comparson Comparson wth Dyn The expermental results that compared Dyn and Para were reported for lowdmensonal datasets and for both Based and Unform workloads n [1] and the dfferences between the two methods are very small. Therefore, t s suffcent to compare LB wth one of these two methods for low-dmensonal datasets. (a) Based Workloads (b) Unform Workloads Fgure 4. Comparson of LB and Dyn (a) Eucl-dstance, 25, 50, 104 dmensons (b) Ls50D, Sum, Eucl and Max Fgure 5. Comparson of LB and Para Fgure 4 compares the performance of LB and that of Dyn for dfferent datasets and for both Based and Unform workloads. From Fgure 4(a), t can be seen that when Based workloads are used, for datasets Gauss3D and Census3D, LB outperforms Dyn sgnfcantly; for Array3D, Census2D and Cover4D, LB and Dyn have smlar performance. For Unform workloads (Fgure 4(b)), LB s sgnfcantly better than Dyn for 4 datasets and s slghtly better for 1 dataset. However, LB has much hgher restart percentages for Gauss3D and Cover4D (95% and 31%, respectvely).

9 5.2.2 Comparson wth Para for Hgh-Dmensonal Datasets Note that the Para method does not am to guarantee the retreval of all top-n tuples. In [7], results for top-20 (.e., N=20) queres when retrevng 90% (denoted Para/90) and 95% (denoted Para/95) were reported. In contrast, LB guarantees the retreval of all top-n tuples for each query. Fgure 5 compares the performances of LB and Para for top-20 queres [7]. When Eucldean dstance functon s used, LB slghtly underperforms Para/95 for 25- and 50-dmensonal data but sgnfcantly outperforms Para/95 for 104-dmensonal data. Fgure 5(b) shows the results usng dfferent dstance functons for 50-dmensonal data. For the most expensve sum functon, LB s sgnfcantly better than Para/95; for Eucldean dstance functon, Para/95 s slghtly better than LB, and for the max functon, Para/95 s sgnfcantly better than LB. 5.3 Addtonal Expermental Results for LB We carred out a number of addtonal experments to gan more nsghts regardng the behavor of the LB method. In ths secton, we report the results of these experments. Effect of Dfferent Result Sze N. Based on dataset Census2D, we construct 4 dfferent knowledge bases usng top-50, top-100, top-250 and top-1000 queres, respectvely. For each knowledge base, the number of queres used s the same (1,459 queres). These knowledge bases are then used to evaluate a test workload of 100 top-100 queres. Our results ndcate that usng the knowledge bases of top-50 and top-100 queres yelds almost the same performance. The best and the worst performances are obtaned when the knowledge base of top-250 queres and that of top-1000 queres are used, respectvely. Effect of Dfferent Values of N. In the second set of experments, we use some top-100 queres to buld a knowledge base for each of the 5 low-dmensonal datasets. 178, 218 and 250 queres are used to buld the knowledge base for 2-, 3- and 4- dmensonal datasets, respectvely. Each knowledge base s then used to evaluate a workload of top-50, top-100, top-250 and top-1000 queres. Overall, the results are qute good except when top-1000 queres are evaluated. But even for top-1000 queres, on the average, no more than 2% of the tuples are retreved n the worst-case scenaro. For ths experment, our results ndcate that the LB method performs overall much better than the Dyn method. For example, n only two cases, more than 1% of the tuples are retreved by LB, but n Fgure 22(b) n [1], there are 10 cases where more than 1% of the tuples are retreved. Effect of Dfferent Dstance Functons. The effectveness of the LB method may change dependng on the dstance functon used. Our expermental results usng varous low-dmensonal datasets ndcate that the LB method produces smlar performance for the three wdely used dstance functons (.e., max, Eucldean and sum). By comparng our results wth Fgure 21(b) n [1], we fnd that the LB method sgnfcantly outperforms the Dyn method for datasets Gauss3D, Census3D and Cover4D; for the other two datasets, namely Array3D and Census2D, the two methods have smlar performance.

10 6 Conclusons In ths paper, we proposed a learnng-based strategy to translate top-n selecton queres nto tradtonal selecton queres. Ths technque s robust n several mportant aspects. Frst, t can cope wth a varety of dstance functons. Second, t does not suffer the much feared dmensonalty curse as t remans effectve for hgh-dmensonal data. Thrd, t can automatcally adapt to user query patterns so that frequently submtted queres can be processed most effcently. We carred out extensve experments usng a varety of datasets of dfferent dmensons (from 2 to 104). These results demonstrated that the learnng-based method compares favorably wth the current best processng technques for top-n selecton queres. Acknowledgments: The authors would lke to express ther grattude to Ncolas Bruno [1] and Chung-Mn Chen [7] for provdng us some of the test datasets used n ths paper and for provdng us some expermental results of ther approaches that made t possble for us to compare wth ther results drectly n ths paper. References 1. N. Bruno, S. Chaudhur, and L. Gravano. Top-k Selecton Queres over Relatonal Databases: Mappng Strateges and Performance Evaluaton, ACM Transactons on Database Systems, 27 (2), 2002, N. Bruno, S. Chaudhur and L. Gravano. STHoles: A Multdmensonal Workload-Aware Hstogram. ACM SIGMOD Conference, N. Bruno, L. Gravano, and A.Maran. Evaluatng Top-k Queres over Web-Accessble Databases. 18th IEEE Internatonal Conference on Data Engneerng, M. Carey, and D. Kossmann. On sayng Enough Already! n SQL. ACM SIGMOD Conference, M. Carey, and D. Kossmann. Reducng the brakng dstance of an SQL query engne. Internatonal Conference on Very Large Data Bases S. Chaudhur, and L. Gravano. Evaluatng top-k selecton queres. Internatonal Conference on Very Large Data Bases, C. Chen, and Y. Lng, A samplng-based estmator for top-k selecton query. 18th Internatonal Conference on Data Engneerng, C. Chen and N. Roussopoulos. Adaptve selectvty estmaton usng query feedback. ACM SIGMOD Conference, 1994, Y. Chen, and W. Meng. Top-N Query: Query Language, Dstance Functon and Processng Strateges. Internatonal Conference on Web-Age Informaton Management, D. Donjerkovc, and R. Ramakrshnan. Probablstc optmzaton of top N queres. Internatonal Conference on Very Large Data Bases, R. Fagn. Combnng fuzzy nformaton from multple systems. ACM Symposum on Prncples of Database Systems, 1996, R. Fagn. Fuzzy queres n multmeda database systems. ACM Symposum on Prncples of Database Systems, R. Fagn, A. Lotem, and M. Naor. Optmal aggregaton algorthms for mddleware. ACM Symposum on Prncples of Database Systems, J. Lee, D. Km, and C. Chung. Mult-dmensonal Selectvty Estmaton Usng Compressed Hstogram Informaton. ACM SIGMOD Conference, 1999, C. Yu, G. Phlp, and W. Meng. Dstrbuted Top-N Query Processng wth Possbly Uncooperatve Local Systems. VLDB, Berln, Germany, September 2003,

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