A FRAMEWORK FOR PROCESSING K-BEST SITE QUERY

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 A FRAMEWORK FOR PROCESSING K-BEST SITE QUERY Yuan-K Huang * and Lien-Fa Lin Department f Infrmatin Cmmunicatin Ka-Yuan University; Kahsiung Cuntry, Taiwan R.O.C. ABSTRACT A nvel query in spatial databases is the K-Best Site Query (KBSQ fr shrt). Given a set f bjects O, a set f sites S, and a user-given value K, a KBSQ retrieves the K sites frm S such that the ttal distance frm each bject t its clsest site is minimized. The KBSQ is indeed an imprtant type f spatial queries with many real applicatins. In this paper, we investigate hw t efficiently prcess the KBSQ. We first prpse a straightfrward apprach with a cst analysis, and then develp the K Best Site Query (KBSQ) algrithm cmbined with the existing spatial indexes t imprve the perfrmance f prcessing KBSQ. Cmprehensive experiments are cnducted t demnstrate the efficiency f the prpsed methds. KEYWORDS spatial databases; K-Best Site Query; spatial indexes. INTRODUCTION With the fast advances f psitining techniques in mbile systems, spatial databases that aim at efficiently managing spatial bjects are becming mre pwerful and hence attract mre attentin than ever. Many applicatins, such as mbile cmmunicatin systems, traffic cntrl systems, and gegraphical infrmatin systems, can benefit frm efficient prcessing f spatial queries [- 7]. In this paper, we present a nvel and imprtant type f spatial query, namely the K-Best Site Query (KBSQ fr shrt). Given a set f bjects O, a set f sites S, and a user-given value K, a KBSQ retrieves the K sites s, s,, s K frm S such that d ( i, s j ) is minimized, where d( i, s j ) refers t the distance between bject i and its clsest site s j. We term the sites retrieved by executing the KBSQ the best sites (r bs fr shrt). The KBSQ prblem arises in many fields and applicatin dmains. As an example f real-wrld scenari, cnsider a set O f sldiers n the battlefields that is fighting the enemy. In rder t immediately supprt the injured sldiers, we need t chse K sites frm a set S f sites t build the emergicenters. Nte that there are many sldiers fighting n the battlefields and many sites culd be the emergicenters. T achieve the fastest respnse time, the sum f distances frm each battlefield t its clsest emergicenter shuld be minimized. Anther real-wrld example is that the McDnald's Crpratin may ask what are the ptimal lcatins in a city t pen new McDnald's stres. In this case, the KBSQ can be used t find ut the K best sites amng a set S f sites s that every custmer in set O can rapidly reach his/her clsest stre. DOI : 0./ijdms.0.0 7 i O

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 Let us use an example in Figure t illustrate the KBSQ prblem, where six bjects,,, and fur sites s, s,, s are depicted as circles and rectangles, respectively. Assume that tw best sites (i.e., bs) are t be fund in this example. There are six cmbinatins (s, s ), (s, s ),, (s, s ), and ne cmbinatin wuld be the result f KBSQ. As we can see, the sum f distances frm bjects,, t their clsest site s is equal t 9, and the sum f distances between bjects,, and site s is equal t. Because the cmbinatin (s, s ) leads t the minimum ttal distance (i.e., 9 + = ), the tw sites s and s are the bs. s s s s Figure. An example f the KBSQ T prcess the KBSQ, the clsest site fr each bject needs t be first determined and then the distance between bject and its clsest site is cmputed s as t find the best cmbinatin f K sites. When a database is large, it is crucial t avid reading the entire dataset in identifying the K best sites. Fr saving CPU and I/O csts, we develp an efficient methd cmbined with the existing spatial indexes t avid unnecessary reading f the entire dataset. A preliminary versin f this paper is [8], and the cntributins f this paper are summarized as fllws. We present a nvel query, namely the K Best Site Query, which is indeed an imprtant type f spatial queries with many real applicatins. We prpse a straightfrward apprach t prcess the KBSQ and als analyze the prcessing cst required fr this apprach. An efficient algrithm, namely the K Best Site Query (KBSQ) algrithm, perates by the supprt f R*-tree [9] and Vrni diagram [0] t imprve the perfrmance f KBSQ. A cmprehensive set f experiments is cnducted. The perfrmance results manifest the efficiency f ur prpsed appraches. The rest f this paper is rganized as fllws. In Sectin, we discuss sme related wrks n prcessing spatial queries similar t the KBSQ, and pint ut their differences. In Sectin, the straightfrward apprach and its cst analysis is presented. Sectin describes the KBSQ algrithm with the used indexes. Sectin shws extensive experiments n the perfrmance f ur appraches. Finally, Sectin 6 cncludes the paper with directins n future wrk. 8

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0. RELATED WORK In recent years, sme queries similar t the KBSQ are presented, including the Reverse Nearest Neighbr Query (RNNQ) [], the Grup Nearest Neighbr Query (GNNQ) [], and the Min- Dist Optimal-Lcatin Query (MDOLQ) []. Several methds have been designed t efficiently prcess these similar queries. Hwever, the query results btained by executing these queries are quite different frm that f the KBSQ. Als, the prpsed methds cannt be directly used t answer the KBSQ. In the fllwing, we investigate why the existing methds fr prcessing the similar queries cannt be applied t the KBSQ separately... Methds Fr RNNQ Given a set f bject O and a site s, a RNNQ can be used t retrieve a set S f bjects cntained in O whse clsest site is s. Each bject in S is termed a RNN f s. An intuitive way fr finding the query result f KBSQ is t utilize the RNNQ t find the RNNs fr each site. Then, the K sites having the maximum number f RNNs (meaning that they are clser t mst f the bjects) are chsen t be the K best sites. Taking Figure as an example, the RNNs f site s can be determined by executing the RNNQ and its RNNs are bjects and. Similarly, the RNNs f sites s, s, and s are determined as and,, and, respectively. As sites s and s have the maximum number f RNNs, they can be the bs fr the KBSQ. Hwever, sites s and s lead t the ttal distance (i.e., d(, s ) + d(, s ) + d(, s ) + d(, s ) + d(, s ) + d(, s )), which is greater than the ttal distance as sites s and s are chsen t be the bs. As a result, the intuitin f using the RNNQ result t be the KBSQ result is infeasible. s s 8 s s and 6 are s s RNN and are s s RNN is s s RNN is s s RNN.. Methds Fr GNNQ Figure. An example f the RNNQ A GNNQ retrieves a site s frm a set f sites S such that the ttal distance frm s t all bjects is the minimum amng all sites in S. Here, the result s f GNNQ is called a GNN. T find the K best sites, we can repeatedly evaluate the GNNQ K times s as t retrieve the first K GNNs. It means that the sum f distances between these K GNNs and all bjects is minimum, and thus they can be the K bs. Hwever, in sme cases the result btained by executing the GNNQ K times is still different frm the exact result f KBSQ. Let us cnsider an example shwn in Figure, where bs are required. As shwn in Figure (a), the first and secnd GNNs are sites s and s, respectively. As such, the bs are s and s, and the 9

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 ttal distance d(, s ) + d(, s ) + d(, s ) + d(, s ) + d(, s ) + d(, s ) =. Hwever, anther cmbinatin (s, s ) shwn in Figure (b) can further reduce the ttal distance t. Therefre, using the way f executing GNNQ K times t find the K best sites culd return incrrect result. 6 s s s s The first The secnd The third GNN is s GNN is s GNN is s The last GNN is s (a) incrrect result s s s s.. Methds Fr MDOLQ (b) crrect result Figure. An example f the GNNQ Given a set f bjects O and a set f sites S, a MDOLQ returns a lcatin which, if a new site s nt in S is built there, minimizes d ( i, s j ) where d( i, s j ) is the distance between bject i and i O its clsest site SU{s}. At first glance, the MDOLQ is mre similar t the KBSQ than the s j ther queries mentined abve. Hwever, using the MDOLQ t btain the K best sites may still lead t incrrect result. Cnsider an example f using MDOLQ t find the K best sites in Figure. As bs are t be fund, we can evaluate the MDOLQ tw times t btain the result. In the first iteratin (as shwn in Figure (a)), the site s becmes the first bs because it has the minimum ttal distance t all bjects. Then, the MDOLQ is executed again by taking int accunt the remaining sites s, s, and s. As the site s can reduce mre distance cmpared t the ther tw sites, it becmes the secnd bs (shwn in Figure (b)). Finally, bs are s and s and the ttal distance is cmputed as d(, s ) + d(, s ) + d(, s ) + d(, s ) + d(, s ) + d(, s ) = 0. Hwever, the cmputed distance is nt 0

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 minimum and can be further reduced. As we can see in Figure (c), if s and s are chsen t be the bs, the ttal distance can decrease t 6. s 6 s s s s s s s (a) Step (b) Step s s s s. STRAIGHTFORWARD APPROACH (c) crrect result Figure. An example f the MDOLQ In this sectin, we first prpse a straightfrward apprach t slve the KBSQ prblem, and then analyze the prcessing cst required fr this apprach. Assume that there are n bjects and m sites, and the K bs wuld be chsen frm the m sites. The straightfrward apprach cnsists f three steps. The first step is t cmpute the distance d( i, s j ) frm each bject i ( i n) t each site s j ( j m). As the K best sites are needed t be retrieved, there are ttally C m K pssible cmbinatins and each f the cmbinatins cmprises K sites. The secnd step is t cnsider all f the cmbinatins. Fr each cmbinatin, the distance frm each bject t its clsest site is determined s as t cmpute the ttal distance. In the last step, the cmbinatin f K sites having the minimum ttal distance is chsen t be the query result f KBSQ. Figure illustrates the three steps f the straightfrward apprach. As shwn in Figure (a), the distances between bjects and sites are cmputed and stred in a table, in which a tuple represents the distance frm an bject t all sites. Then, the C m K cmbinatins f K sites are cnsidered s that C m K tables are generated (shwn in Figure (b)). Fr each table, the minimum attribute value f each tuple (depicted as gray bx) refers t the distance between an bject and its clsest site. As such, the ttal distance fr each cmbinatin can be cmputed by summing up the minimum attribute value f each tuple. Finally, in Figure (c) the cmbinatin f K sites can be the K bs because its ttal distance is minimum amng all cmbinatins.

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 n m n 6 6 m n K K n 6 6 8 cmbinatin n K K+ 6 n 8 cmbinatin n K m 6 6 n (a) Step (b) Step cmbinatin C m K n K K n 6 6 8 cmbinatin (c) Step Figure. Straightfrward apprach Since the straightfrward apprach includes three steps, we cnsider the three steps individually t analyze the prcessing cst. Let m and n be the numbers f sites and bjects, respectively. Then, the time cmplexity f the first step is m*n because the distances between all bjects and sites have t be cmputed. In the secnd step, C m K cmbinatins are cnsidered and thus the cmplexity is C m K *n*k. Finally, the cmbinatin having the minimum ttal distance is determined amng all cmbinatins s that the cmplexity f the last step is C m K. The prcessing m m cst f the straightfrward apprach is represented as m * n + C k * n * K + C k.. KBSQ ALGORITHM The abve apprach is perfrmed withut any index supprt, which is a majr weakness in dealing with large datasets. In this sectin, we prpse the KBSQ algrithm cmbined with the existing indexes R*-tree and Vrni diagram t efficiently prcess the KBSQ. Recall that, t prcess the KBSQ, we need t find the clsest site s fr each bject (that is, finding the RNN f site s). As the Vrni diagram can be used t effectively determine the RNN f each site [], we divide the data space s that each site has its wn Vrni cell. Fr example, in Figure 6(b), the fur sites s, s, s, and s have their crrespnding Vrni cells V,

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 V, V, and V, respectively. Taking the cell V as an example, if bject lies in V, then must be the RNN f site s. Based n this characteristic, bject needs nt be cnsidered in finding the RNNs fr the ther sites. Then, we use the R*-tree, which is a height-balanced indexing structure, t index the bjects. In a R*-tree, bjects are recursively gruped in a bttm-up manner accrding t their lcatins. Fr instance, in Figure 6(a), eight bjects,,, 8 are gruped int fur leaf ndes E t E 7 (i.e., the minimum bunding rectangle MBR enclsing the bjects). Then, ndes E t E 7 are recursively gruped int ndes E and E, that becme the entries f the rt nde E. E 7 8 s V V E E 7 E E s s V s E E 6 V (a) R-tree (b) Vrni diagram Figure 6. Spatial indexes Cmbined with the R*-tree and Vrni diagram, we design the fllwing pruning criteria t greatly reduce the number f bjects cnsidered in query prcessing. Pruning bjects: given an bject and the K sites s, s,, s K, if lies in the Vrni cell V i f ne site s i cntained in {s, s,, s K }, then the distances between bject and the ther K- sites need nt be cmputed s as t reduce the prcessing cst. Pruning MBRs: given a MBR E enclsing a number f bjects and the K sites s, s,, s K, if E is fully cntained in the cell V i f ne site s i cntained in {s, s,, s K }, then the distances frm all bjects enclsed in E t the ther K- sites wuld nt be cmputed. T find the K bs fr the KBSQ, we need t cnsider C m K cmbinatins f K sites. Fr each cmbinatin f K sites s, s,, s K with their crrespnding Vrni cells V, V,, V K, the prcessing prcedure begins with the R*-tree rt nde and prceeds dwn the tree. When an internal nde E (i.e., MBR E) f the R*-tree is visited, the pruning criterin is utilized t determine which site is the clsest site f the bjects enclsed in E. If the MBR E is nt fully cntained in any f the K Vrni cells, then the child ndes f E need t be further visited. When a leaf nde f the R*-tree is checked, the pruning criterin is impsed n the entries (i.e., bjects) f this leaf nde. After the traversal f the R*-tree, the ttal distance fr the cmbinatin f K sites s, s,, s K can be cmputed. By taking int accunt the ttal cmbinatins, the cmbinatin f K sites whse ttal distance is minimum wuld be the query result f the KBSQ. Figure 7 cntinues the previus example in Figure 6 t illustrate the prcessing prcedure, where there are eight bjects t 8 and fur sites s t s in data space. Assume that the cmbinatin (s, s ) is cnsidered and the Vrni cells f sites s and s are shwn in Figure 7(a). As the MBR E is nt fully cntained in the Vrni cell V f site s, the MBRs E and E still need t be visited. When the MBR E is checked, based n the pruning criterin the distances frm bjects and t site s wuld nt be cmputed because their clsest site is s. Similarly, the clsest

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 site f the bjects 7 and 8 enclsed in MBR E 7 is determined as site s. As fr bjects t, their clsest sites can be fund based n the pruning criterin. Having determined the clsest site f each bject, the ttal distance fr cmbinatin (s, s ) is btained. Cnsider anther cmbinatin (s, s ) shwn in Figure 7(b). The clsest site s f fur bjects t enclsed in MBR E can be fund when E is visited. Als, we can cmpute the ttal distance fr the cmbinatin (s, s ) after finding the clsest sites fr bjects t 8. By cmparing the distances fr all cmbinatins, the bs are retrieved. 8 s E 8 7 s E 7 E 7 s E s 7 E s E s E E E E 6 s E E 6 s (a) cmbinatin ( s, s ) (b) cmbinatin ( s, s ). PERFORMANCE EVALUATION Figure 7. Prcessing KBSQ with indexes We cnduct fur experiments fr the straightfrward apprach and the prpsed KBSQ algrithm in this sectin. The first three experiments are evaluated t study the perfrmance f the prpsed methds by measuring the CPU time fr prcessing a KBSQ. The last experiment demnstrates the usefulness f the KBSQ algrithm by cmparing the precisin f query result against its cmpetitrs... Experimental Setting All experiments are perfrmed n a PC with Intel.8 GHz CPU and GB RAM. The algrithms are implemented in Java. One synthetic dataset is used in ur simulatin. The synthetic dataset cnsists f 000 bjects whse lcatins are unifrmly spread ver a regin f 00000 * 00000 meters. In the experimental space, we als generate 0 query datasets, each f which cntains sites whse lcatins are in the same range as thse f the bjects mentined abve. Fr each query dataset, we perfrm a KBSQ t find the K best sites, where the default value f K is set t. The perfrmance is measured by the average CPU time in perfrming wrklad f the 0 queries. Table summarizes the parameters under investigatin, alng with their default values and ranges. We cmpare the prpsed KBSQ algrithm with the straightfrward apprach t investigate the perfrmance f prcessing a KBSQ. Als, we cmpare the precisin f the KBSQ algrithm against its cmpetitrs, including the RNNQ, the GNNQ, and the MDOLQ methds. Table. System parameters. Parameter Default Range Number f bjects (O) 000 00, 000, 000, 0000 Number f sites (S) 0,, 0, K,, 0, 0

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0.. Efficiency Of KBSQ Algrithm In this subsectin, we cmpare the KBSQ algrithm with the straightfrward apprach in terms f the CPU time. Three experiments are cnducted t investigate the effects f three imprtant factrs n the perfrmance f prcessing KBSQ. These imprtant factrs are the number f bjects O, the number f sites S, and the value f K. Figure 8 illustrates the perfrmance f the KBSQ algrithm and the straightfrward apprach as a functin f the number f bjects (ranging frm 00 t 0000). Nte that hereafter all figures use a lgarithmic scale fr the y-axis. As we can see frm the experimental result, the KBSQ algrithm significantly utperfrms the straightfrward apprach in the CPU time, even fr a smaller number f bjects (e.g., 00). This is mainly because fr the straightfrward apprach the distances f all bjects have t be cmputed which incurs high cmputatin cst. Mrever, the perfrmance gap between the KBSQ algrithm and the straightfrward apprach increases with the increasing number f bjects. The reasn is that mst distance cmputatins f bjects can be avided by using the KBSQ algrithm with the supprt f R*-tree, but these distance cmputatins are necessary fr the straightfrward apprach. Figure 8. Effect f number f bjects. Figure 9 demnstrates the effect f varius numbers f sites (i.e., varying S frm 0 t) n the perfrmance f the KBSQ algrithm and the straightfrward apprach. When the number f sites increases, the CPU verhead fr bth algrithms grws. The reasn is that as the number f sites becmes greater, the number f cmbinatins t be cnsidered increases s that mre distance cmputatins between bjects and sites are required. The experimental result shws that the KBSQ algrithm utperfrms its cmpetitr significantly in all cases, which cnfirms again that applying the KBSQ algrithm with R*-tree and Vrni diagram can greatly imprve the perfrmance f prcessing a KBSQ.

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 Figure 9. Effect f number f sites Finally in this subsectin, we study hw the value f K affects the perfrmance f the KBSQ algrithm and the straightfrward apprach, by varying K frm t 0. Similar t the previus experimental results, the KBSQ algrithm achieves significantly better perfrmance than the straightfrward apprach (as shwn in Figure 0). The KBSQ algrithm utperfrms the straightfrward apprach by a factr f 70 t 0 in terms f the CPU cst. In additin, an interesting bservatin frm Figure 0 is that a smaller K (e.g., ) r a larger K (e.g., 0) results in a lwer CPU time fr the KBSQ algrithm and the straightfrward apprach. This is because fr a smaller (r larger) value f K, less number f cmbinatins needs t be cnsidered in prcessing a KBSQ s that the required CPU time can be reduced. Figure 0. Effect f K 6

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0.. Precisin Of KBSQ Algrithm The fllwing experiment demnstrates the precisin f the KBSQ algrithm and its cmpetitrs (including the RNNQ, the GNNQ, the MDOLQ methds) under varius values f K, where the precisin is represented as fllws: #( bsresult I bsreal ) precisin = 00% # bs In the abve equatin, bs result refers t the set f K best sites retrieved by executing the KBSQ algrithm, the RNNQ methd, the GNNQ methd, r the MDOLQ methd. As fr bs real, it is the set f the real K best sites. In Figure, we vary K frm t 0 t investigate the precisin f the KBSQ algrithm, the RNNQ methd, the GNNQ methd, and the MDOLQ methd. As we can see, as the real K best sites can be precisely determined by executing the KBSQ algrithm, the precisin f the KBSQ algrithm is always equal t 00% under different values f K. Hwever, if the RNNQ, the GNNQ, and the MDOLQ methds are adpted t answer a KBSQ, sme f the real K best sites are missed. As shwn in the experimental result, the precisin fr the MDOLQ methd can nly reach 60% t 8%. Even wrse, the precisin fr the RNNQ and the GNNQ methds is belw 60% fr a smaller value f K, which means that mst f the retrieved best sites are incrrect. real 6. CONCLUSIONS Figure. Precisin fr different K In this paper, we fcused n prcessing the K Best Site Query (KBSQ) which is a nvel and imprtant type f spatial queries. We highlighted the limitatins f the previus appraches fr the queries similar t the KBSQ, including the RNNQ, the GNNQ, and the MDOLQ. T slve the KBSQ prblem, we first prpsed a straightfrward apprach and then analyzed its prcessing cst. In rder t imprve the perfrmance f prcessing the KBSQ, we further prpsed a KBSQ algrithm cmbined with R*-tree and Vrni diagram t greatly reduce the CPU and I/O csts. Cmprehensive set f experiments demnstrated the efficiency and the precisin f the prpsed appraches. 7

Internatinal Jurnal f Database Management Systems ( IJDMS ) Vl., N., Octber 0 Our next step is t discuss the space requirement f the prpsed methds and design a nvel index structure fr answering the KBSQ. Then, we will fcus n prcessing the KBSQ fr mving bjects with fixed r uncertain velcity. Mre cmplicated issues will be intrduced because f the mvement f bjects. Finally, we wuld like t extend the prpsed apprach t prcess the KBSQ in rad netwrk. ACKNOWLEDGEMENTS This wrk was supprted by Natinal Science Cuncil f Taiwan (R.O.C.) under Grants NSC 0-9-M- -00 and NSC 0-9-M- -00. REFERENCES [] Benetis, R.; Jensen, C.S.; Karciauskas, G.; Saltenis, S. Nearest neighbr and reverse nearest neighbr queries fr mving bjects. VLDB Jurnal 006,, 9-9. [] Hakkymaz, V. A specificatin mdel fr tempral and spatial relatins f segments in multimedia presentatins. Jurnal f Digital Infrmatin Management 00, 8, 6-6. [] Huang, Y.-K.; Chen, C.-C.; Lee, C. Cntinuus k-nearest neighbr query fr mving bjects with uncertain velcity. GeInfrmatica 009,, -. [] Huang, Y.-K.; Lia, S.-J.; Lee C. Evaluating cntinuus k-nearest neighbr query n mving bjects with uncertainty. Infrmatin Systems 009,, -7. [] Mkbel, M.F.; Xing, X.; Aref, W.G. Sina: Scalable incremental prcessing f cntinuus queries in spati-tempral databases. In Prceedings f the ACM SIGMOD 00. [6] Pagel, B.-U.; Six, H.-W.; Winter, M. Windw query-ptimal clustering f spatial bjects. In Prceedings f the ACM SIGMOD 99. [7] Papadias, D.; Ta, Y.; Muratidis, K.; Hui, C.K. Aggregate nearest neighbr queries in spatial databases. ACM Trans. Database Syst. 00, 0, 9-76. [8] Huang, Y.-K.; Lin, L.-F. Evaluating k-best site query n spatial bjects. In Prceedings f the NDT 0. [9] Guttman, A. R-trees: A dynamic index structure fr spatial searching. In Prceedings f the ACM SIGMOD 98. [0] Samet, H. The design and analysis f spatial data structures. Addisn-Wesley, Reading 990. [] Krn, F.; Muthukrishnan, S. Influence sets based n reverse nearest neighbr queries. In Prceedings f the ACM SIGMOD 00. [] Papadias, D.; Shen, Q.; Ta, Y.; Muratidis, K. Grup nearest neighbr queries. In Prceedings f the ICDE 00. [] Zhang, D.; Du, Y.; Xia, T.; Ta, Y. Prgressive cmputatin f the min-dist ptimal-lcatin query. In Prceedings f the VLDB 006. [] Zhang, J.; Zhu, M.; Papadias, D.; Ta, Y.; Lee, D.L. Lcatin-based spatial queries. In Prceedings f the ACM SIGMOD 00. 8