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1 Powered by TCPDF ( Swinburne Research Bank htt://researchbank.swinburne.edu.au Author: Isla, Md. Saiful; Liu, Chengfei; Rahayu, Wenny; Anwar, Tariue Title: Q lus Tree: An efficient Quad Tree based data indexing for arallelizing dynaic and reverse Skylines Editor: S. Mukhoadhyay and C. X. Zhai Conference nae: The 5th ACM International Conference on Inforation and Knowledge Manageent (CIKM') Conference location: United States Conference dates: - October Place ublished: United States Publisher: ACM Year: Pages: -3 URL: htt://hdl.handle.net/5.3/337 Coyright: Coyright ACM. This is the author's version of the work. It is osted here for your ersonal use. Not for redistribution. The definitive Version of Record was ublished in the Proceedings of the 5th ACM International Conference on Inforation and Knowledge Manageent (CIKM'), htts://doi.org/.5/ This is the author s version of the work, osted here with the erission of the ublisher for your ersonal use. No further distribution is eritted. You ay also be able to access the ublished version fro your library. The definitive version is available at: htts://doi.org/.5/ Swinburne University of Technology CRICOS Provider D swinburne.edu.au

2 : An Efficient Quad Tree based Data Indexing for Parallelizing Dynaic and Reverse Skylines Md. Saiful Isla #, Chengfei Liu, Wenny Rahayu #3 and Tariue Anwar # La Trobe University, Melbourne, Australia Swinburne University of Technology, Melbourne, Australia {.isla5, 3 w.rahayu}@latrobe.edu.au, { cliu, tanwar}@swin.edu.au ABSTRACT Skyline ueries lay an iortant role in ulti-criteria decision aking alications of any areas. Given a dataset of objects, a skyline uery retrieves data objects that are not doinated by any other data object in the dataset. Unlike standard skyline ueries where the different asects of data objects are coared directly, dynaic and reverse skyline ueries adhere to the around-by seantics, which is realized by coaring the relative distances of the data objects w.r.t. a given uery. Though, there are a nuber of works on arallelizing the standard skyline ueries, only a few works are devoted to the arallel coutation of dynaic and reverse skyline ueries. This aer resents an efficient uad-tree based data indexing schee, called, for arallelizing the coutations of the dynaic and reverse skyline ueries. We coare the erforance of with an existing uad-tree based indexing schee. We also resent several otiization heuristics to irove the erforance of both of the indexing schees further. Exerientation with both real and synthetic datasets verifies the efficiency of the roosed indexing schee and otiization heuristics. Keywords Quad Tree, Aggressive Partitioning, Dynaic Skyline, Reverse Skyline, Load Balancing and Parallel Coutation.. INTRODUCTION The skyline uery has been first roosed by Börzsönyi et al. []. Since then, this uery has received lots of attention aong the counity and is studied extensively in doinance based data retrieval ([], [], [], [3], [], [], [], [7] for survey). Given a dataset of objects P, the standard skyline uery retrieves all data objects P that are not doinated by any other data objects P. A data object doinates another data object iff it is as good as in every asects of, but better than in at least one asect. Given P and a uery object, a dynaic skyline uery [] retrieves all data objects P that are not dynaically Perission to ake digital or hard coies of all or art of this work for ersonal or classroo use is granted without fee rovided that coies are not ade or distributed for rofit or coercial advantage and that coies bear this notice and the full citation on the first age. Coyrights for coonents of this work owned by others than ACM ust be honored. Abstracting with credit is eritted. To coy otherwise, or reublish, to ost on servers or to redistribute to lists, reuires rior secific erission and/or a fee. Reuest erissions fro erissions@ac.org. c ACM. ISBN DOI:.5/35 ID Di Di (a) Products, P ID Di Di u u u 3 u u 5 u u 7 u u u 3 (b) Users, U Figure : A dataset of roducts and users doinated by another data object P w.r.t. the uery. Unlike standard skyline ueries where the asects of is directly coared with the corresonding asects of without considering any uery object, the dynaic skyline uery adheres to the around-by seantics where the absolute differences of the asects of and the uery are coared with the corresonding absolute differences of the asects of and the uery object in deciding the doinance between and. Consider the dataset of roducts P given in Fig. (a), the standard skyline retrieves and fro P (without considering any uery) as no other objects in P doinate the. On the other hand, given a uery =<, >, the dynaic skyline of retrieves, 5 and fro P as no other objects in P can doinate the w.r.t.. Both the standard [] and dynaic [] skyline ueries retrieve data objects fro P considering the user s oint of view, i.e., objects incoarably referable to a user. Dellis et al. [3] roose a new tye of skyline uery called, the reverse skyline uery, which retrieves data objects fro the database considering the anufacturer s oint of view. Given a dataset of roducts P and a uery, the onochroatic reverse skyline uery retrieves all roducts P that includes in their dynaic skylines. Consider the dataset of roducts P given in Fig. (a) and a uery =<, >, the onochroatic reverse skyline of retrieves,, 5 and as these objects include in their dynaic skylines. Given datasets of roducts P, users U and a uery, a bichroatic reverse skyline [] uery retrieves all users u U who find the uery in their dynaic skylines. Consider the dataset of roducts P and users U given in Fig. and a uery =<, >, the bichroatic reverse skyline of retrieves u, u 5 and u 7 as they include in their dynaic skylines. Like skyline ueries [][], the reverse skyline ueries also receive lots of attention in the counity, secifically in influence-based rocessing of arket research ueries for easuring the attractiveness of a roduct aong the users ([], [], [], [], [], [5] for survey).

3 c7 u 7 c Grid Index: 3x3, = c 5 c c c5 c 7 7 c c c c u 5 u 3 u u 5 u u u u u (a) Standard skyline (b) Dynaic skyline (c) Monochroatic Grid Index: 3x3, RSL = c7 u 7 (d) Bichroatic RSL Figure : The (a) standard skyline (b) dynaic skyline (c) onochroatic reverse skyline and (d) bichroatic reverse skyline of the uery = (, ) Due to the abundance of data in today s data intensive systes including doinance based data retrieval systes, there is a growing interest in arallelizing the skyline ueries. Though, there are a nuber of works on arallelizing the standard skyline ([], [], [7] [], [5] for survey), arallelizing dynaic and reverse skylines receives little attention aong the counity. The only work on arallelizing the dynaic and (onochroatic) reverse skylines based on uad-tree structure exists in [] (robabilistic version [3]). The existing uad-tree based data indexing schee, we call it here, has erforance bottleneck when there is a skewed data distribution. In addition of it, the otial value of slit threshold and ideal saling ethod are hard to know for uad-tree based data indexing. This aer resents an advanced uad tree based data indexing schee, called, which alleviates uch of the aforeentioned robles. We also resent several otiization heuristics such as aggressive artitioning and load balancing to exedite the erforance of the uad-tree based data indexing schees for arallelizing the dynaic and reverse skyline ueries. Our ain contributions are as follows:. We resent an efficient uad tree based data indexing schee, called, for arallelizing the coutations of dynaic skyline, onochroatic reverse skyline and bichroatic reverse skyline ueries.. We resent several otiization heuristics to exedite the erforance of the uad-tree based indexing schees for arallelizing all of the skyline ueries. 3. We coare the efficiency of with an existing uad-tree based indexing schee by conducting extensive exerients with both real and synthetic datasets. The rest of the aer is organized as follows. Section resents the reliinaries and the couting environent. Section 3 discusses the related work. Section resents the uad tree based data indexing schee for arallelizing the dynaic and reverse skyline ueries. Section 5 resents the otiization heuristics roosed in this aer. Section resents the exeriental evaluation of all indexing schees for rocessing the dynaic and reverse skyline ueries in arallel. Finally, Section 7 concludes the aer.. BACKGROUND This section rovides the reliinaries, a background on the skyline ueries and the couting enviorent.. Preliinaries 5 5 We assue that the dataset D consists of two different grou of objects and these are: roducts P and users U. We consider each roduct P, uery and user u U as a d-diensional data object. Without any loss of generality, we assue that each data object stores only nueric values in its diensions. The i th diensional values of a roduct, uery and user u are denoted by i, i and u i, resectively. In general, we use o to denote any kind of data object in D. (Dynaic) Doinance. A data object o doinates another data object o, denoted by o o, iff: (a) i [...d], o i o i and (b) j [...d], o j < o j. On the other hand, a data object o dynaically doinates another data object o w.r.t. a third data object o 3, denoted by o o3 o, iff: (a) i [...d], o i o i 3 o i o i 3 and (b) j [...d], o j o j 3 < o j o j 3. Consider the datasets of roducts P given in Fig. (a) and the uery =<, >. The roduct doinates the roduct, i.e.,, as (a) (= ) < 3 (= ) and also, (b) (= ) < 3 (= ). Now, the roduct dynaically doinates roduct w.r.t., i.e.,, as (a) (= 3) < (= ) and (b) (= 3) < (= ). Orthants and Midoints. Given an object o and a uery, the orthant O of o w.r.t., denoted by O (o), is couted as: O(o) i = iff o i i, otherwise O(o) i =. A d-diensional uery has d orthants in total, e.g., the orthants of =<, > are shown in Fig (c)-(d). The idoint of a roduct w.r.t. a uery is couted as: i = ( i + i )/. For exale, the idoints of,, 5,, 7, and are shown in Fig (c)-(d).. Skyline Queries (Dynaic) Skyline. Given a dataset of roducts P, the standard skyline, denoted by SKL, retrieves all roducts P that are not doinated by other roducts P. Given a dataset of roducts P and a uery, the dynaic skyline of, denoted by DSL(), retrieves all P that are not dynaically doinated by any P w.r.t., i.e., P :. Consider the dataset of P given in Fig. (a) and the uery =<, >. The SKL of P consists of and, shown in Fig. (a), as no other roducts in P doinate the. The DSL() consists of, 5 and, shown in Fig. (b), as no other roducts in P dynaically doinate the w.r.t.. The DSL() can be couted inefficiently by any SKL algorith [] having all P transfored into a new sace where is treated as the origin and the relative distances to are used as the aing functions as shown in Fig. (b). The aing function f is defined as f i ( i ) = i i. The transfored

4 Master Worker Worker... Worker Figure 3: The silified couting environent for arallelizing dynaic and reverse skylines is denoted by in this aer. Reverse Skyline. Given a dataset of roducts P and a uery, the onochroatic reverse skyline of, denoted by MRSL(), retrieves all P such that is in the DSL( ). Matheatically, the MRSL() retrieves all P such that P and the following holds (a) i [...d], i i i i and (b) i [...d], i i < i i. On the other hand, given a dataset of roducts P, users U and a uery, the bichroatic reverse skyline of, denoted by BRSL(), retrieves all u U such that is not dynaically doinated by any P w.r.t. u. Matheatically, the BRSL() retrieves all u U such that the following holds (a) i [...d], i u i i u i and (b) i [...d], i u i < i u i, P. Consider the dataset of P and U given in Fig. and the uery =<, >. The MRSL() consists of,, 5 and as shown in Fig. (c) as is in their DSLs. On the other hand, the BRSL() consists of u, u 5 and u 7, shown in Fig.(d), as no other P can doinate w.r.t. the. We use RSL() to denote any kind of reverse skylines of. Mid Skyline. The id skyline of P w.r.t., denoted by MSL(), consists of all idoints that are not doinated by any other idoint w.r.t. in the sae orthant. For exale, the MSL of =<, > consists of,, 5,, 7 and as they are not doinated by other idoints w.r.t. in the sae orthant (see in Fig. (d)). Lea. A user u U aears in BRSL() iff MSL() such that (a) O (u) = O () and (b) u..3 Couting Environent We assue an oversilified couting environent, as shown in Fig. 3, where a aster rocessor is resonsible for coordinating and anaging the indeendent tasks carried out by the worker rocessors. The worker rocessor receives the necessary inut data fro the aster and the task tye, finish the task accordingly and finally, ay send the rocessed result back to the aster. The aster rocessor ay index and re-rocess the inut data before sending the to the workers. We assue that the counication and the synchronization between the aster and the worker are integral arts of this environent. We also assue that the couting ower of all workers are the sae. This odel can be siulated through Java Multithreading, MPIs and the state of the art MaReduce technology. 3. RELATED WORK Parallelizing the Standard Skyline. There are a nuber of works on arallelizing the standard skyline. Vlachou et al.[] exloit the hyersherical coordinates of the data oints to roose an angle-based sace artitioning for arallelizing the standard skyline uery rocessing. Zhang et al.[] aly object-based sace artitioning techniue for rocessing skyline ueries in arallel. Kohler et al.[7] resent a hyerlane data rojection techniue, which is indeendent of the data distribution, for couting skyline in arallel. Mullesgaard et al.[] resents a grid-based Table : The ratios of runed areas in different data indexing schees of the dataset given in Fig. Data Indexing Schee(s) DSL MRSL BRSL.75%.%.75% Otiized (AVG) 5.% 5.7% 3.%.%.3% 55.3% Otiized (AVG).% 3.%.% data artitioning schee for couting the standard skyline in MaReduce. Pertesis and Doulkeridis [5] roose a novel techniue for rocessing the skyline uery in SatialHadoo. Recently, Zhang et al.[3] roose a two-hase MaReduce aroach for arallelizing the standard skyline uery rocessing by alying filtering techniues and anglebased artitioning. None of these aroaches are devoted to the arallel coutation of dynaic and reverse skylines. Parallelizing the Dynaic and Reverse Skylines. Though uch of work are devoted to arallelizing the standard skyline, very few works exist on arallelizing the dynaic and reverse skylines [], [3]. Park et al. [] resents an aroach for arallelizing the dynaic and reverse skylines based on uad-tree index (robabilistic version [3]). However, the uad-tree index is uch deendent on the saling ethod and the slit threshold. The selection of otiu slit threshold and the ideal saling ethod are hard in reality. Also, the basic uad-tree based data indexing schee [] has erforance bottleneck if the underlying data distribution is skewed. Much of these robles of uad-tree based data indexing and the [] can be overcoed as evident fro Table (take these results granted until we reach Section and Section 5), i.e., the ratios of runed areas of the basic are uch less coared to the indexing schee roosed in this aer. Our work. We resent an efficient uad-tree based data indexing schee, called to alleviate uch of the robles of the uad-tree based indexing schees []. We also resent several otiization heuristics for the uad-tree based indexing schees to exedite the erforance of couting dynaic and reverse skyline ueries in arallel.. THE Q+TREE The uad-tree indexing schee for couting dynaic and reverse skylines is an indexing schee in which the given data sace D are recursively divided into d artitions in each orthant of a uery, until the nuber of objects in it eets a certain threshold, ρ. The uery is the root and reresents the whole data sace D. Each internal node reresenting a artition has exactly d children. The range of values covered by a node n is denoted by region(n). Each node n is assigned an id consisting of k d bits, i.e., id(n) = a a...a k d, where k is the deth of n in the tree. The first (k ) d bits of id(n) coe fro its arent node and the reaining d bits are a (k ) d+, a (k ) d+,..., a k d where a (k ) d+i = (or ) if the i th diensional range of the region(n) is the first half (or the second half) of its arent s i th diensional range. Only a subset of roducts of P (e.g., reservoir saling []) is used to build the index. Main Idea: As we discuss in Section and Section 3, the liitation of the basic uad tree based indexing schee [] is that its runed regions are largely deendent on the saling ethod and the setting of the slit threshold, ρ. We ay end u having different trees with different runing caabilities for different sales and settings of ρ. Also, we know that the ideal saling ethod and the o-

5 tiu value of ρ are hard to redict. Much of this liitation can be itigated by exloiting a roerty of the runed nodes in the uad-tree. Hence, we resent an advanceent of the basic, called, which extends the runed node regions based on node doinances. Unlike the basic [], the regions of the children of an internal node in ay not be eual in areas. The seudocode of the uad-tree based data indexing schees is given in Algorith. The extension of runed node regions and the node-doinances in for all skylines as well as their arallel coutations are discussed in the following sections.. Index for DSL To construct the uad tree index for DSL(), [] alies the following: a node n is arked as runed iff n QT ree such that n, n and. If such n exists for n, we say doinates n w.r.t. and is denoted by n. We also say n doinates n w.r.t. and is denoted by n n. However, checking this airwise node doinance n n by checking airwise object doinance [] is inefficient. An alternative to decide the node doinance efficiently is given in the following lea: Lea. If n such that dynaically doinates all of the d corners of n w.r.t., then n. Proof. We know that all n is bounded by the d corners of node n. Therefore, n doinates any n w.r.t. iff doinates all of the d corners of n w.r.t., i.e., n. Hence the lea. The children of a runed node are set to null. D: The index for DSL() [], we call it D, of the roducts P given in Fig. (a) and =<, > with sales {,,,, 7,, } is shown in Fig. (a) by setting ρ to. The gray regions are runed nodes. Here, node with id is arked as runed as of node with id and of node with id doinate all of its four corners (arked by green circles) w.r.t.. D: To extend the region of a runed leaf node n in a D further, denoted by region + (n ), we first gather all objects that dynaically doinate the corners of n w.r.t.. Then, we insert these in a in hea H and reeatedly retrieve the root until region(arent(n )) or region(arent(n )). Finally, we readjust the regions of the children of arent(n ) considering (or if not in the sae orthant as of n ) as the new slitting oint. We call the above tree as D here. The D of the roducts P given in Fig. (a) with sales {,,,, 7,, } for =<, > is shown in Fig. (b). The gray atterned regions are the new runed areas. Here, we redistribute the regions of the children of the node with id considering as the new slitting oint. Siilarly, we redistribute the regions of the children of the nodes with ids and considering as the new slitting oint. The aster constructs the D, which is shared by all workers. The node doinances of D is given in Algorith. For D, the lines - of Algorith are not executed and the for loo in line can terinate as soon as an o is found such that o n and n is runed... DSL in Parallel with D To coare two objects for the in hea H, we use the euclidean distances of o and o to the given uery. Algorith : Inut : uery, slit threshold ρ, sales S Outut: begin root new Node( root,, null); // (node tye, center-of-slit, arent node) 3 for each o S do isslitted false; // a boolean flag 5 root insert(o, root, null); // start fro root node if isslitted then 7 testdoinances(root); // check node doinances insert(object o, Node n, Node arent) begin // the current node is not a leaf node if n!=null and!n.gettye().euals( leaf ) then // set uery to coute child s index o.setquery(n.getquery()); index o.getorthant(); // insert the oint in the child node 3 n.children[index] insert(o, n.children[index], n); // the current node is a leaf node else if n!=null and n.gettye().euals( leaf ) then 5 n.add(o); // slit the leaf node if exceeds the threshold if n.size()> ρ and!n.ispruned() then 7 n slit(n); // only leaf node is slitted isslitted true; // triggers node doinance test return n; slit(node n ) begin Object center getcenter(n.getranges()); 3 n new Node( internal, center, n.getparent()); n.setranges(n.getranges()); n.setorthant(n.getorthant()); 5 Object objects[] n.getpoints(); Node children[] = n.getchildren(); // redistribute the objects aong the children 7 for each o objects do o.setquery(center); children[.getorthant()].add(o); 3 return n ; The stes of couting DSL() in arallel are as follows: -() Firstly, the aster divides P into several chunks P j P (such that P j = P ) and then sends these chunks P j as well as the D to its workers. -() A worker does the followings: (a) P j finds its node id in the D, if it is fro the runed node, then it is ignored, otherwise, it is inserted into the in hea H; j (b) initializes DSL j to, reeatedly retrieves the root roduct fro H j and adds it to DSL j iff DSL j such that ; and (c) sends the local DSL j to aster. -(3) Finally, the aster collects all local DSL js and insert the into the in hea H. Then, the aster coutes the global DSL() by following the sae techniue as given in ste (b) for the worker. Unlike [], we do not transfor the objects P into a new sace w.r.t. the given uery, we establish the node doinances and coute DSL() in the original data sace and thereby, scan the roduct dataset P only once. The standard skyline SKL can also be couted in arallel using the D considering the given uery at zero... Correctness of D The following lea roves the correctness of DSL() couted using the D. Lea 3. Assue that n is a runed node in D and region + (n). Then, DSL().

6 D (a) D ' 7 D+ ' ' 7 D+(Otization Heuristics) dataset given in Fig. (a) and =<, > ' Algorith : testdoinancesd Inut : Node root begin N retrievenodes(root); 3 for each n N do if n.ispruned() then 5 continue; // already runed (b) D Figure : The uad tree indices for DSL() of the // coute the objects that doinate node n for each n N do 7 S coutedoinatingobjects(n, n ); // doinating objects are found for n if S.size()!= then n.runed true; // change node status H insert(s); // initialize the in-hea Node arent n.getparent(); while!h.isety() do 3 center H.retrieveRoot(); // root entry center irror(center, root.getquery()); // extend regions with center or center 5 if arent.iscovered(center) then extendregions(arent, center); 7 else if arent.iscovered(irroredcenter) then extendregions(arent, center ); Proof. According to the construction of D, n such that doinates all d corners of region(n) w.r.t. (Lea ). Now, we insert all these into H and select the that has the least distance to and region(arent(n)) (or region(arent(n))). We get region + (n) in D by redistributing the regions of the children of arent(n) including n considering (or ) as the new slitting oint in D. Since doinates all d corners of the region + (n) w.r.t. and is bounded within region+(n), we get. Hence, the Lea.. Index for MRSL The runed regions in the uad tree index for couting the MRSL() are established as er the following []. Lea. A node n is arked as runed iff, such that (a) O ( ) = O (n ); (b) O ( ) = O (n ); (c) n and (d) n. The children of a runed node are set to null. MR: The basic uad tree index for couting MRSL(), we call it MR here, of the dataset P given in Fig. (a) for the idoints of the sales {,,,, 5,, 7,,, } and the uery =<, > is shown in Fig. 5(a) by setting ρ = (ρ as er Lea ). MR: To extend the runed region of a leaf node n further, denoted by region+ (n ), firstly we gather all MR (a) MR MR+ Figure 5: The uad tree indices for MRSL() of the exale dataset given in Fig. (a) and =<, > 3 3 idoints that dynaically doinate the5 corners 5 of n w.r.t. in the sae orthant as of n. Then, we insert these in a in hea H and retrieve to- idoints (b) MR and such that region(arent(n )) and also, region(arent(n )). Finally, we readjust the regions of the children of arent(n ) by considering in, as the new slitting oint. We call it MR here. The MR of the dataset P given in Fig. (a) for the idoints of the sales {,,,, 5,, 7,,, } and =<, > is shown in Fig. 5(b). The gray atterned regions are the new runed areas. Here, we redistribute the regions of the children of the node with id considering in, as the new slitting oint. Siilarly, we redistribute the regions of the children of the node with id considering in 7, as the new slitting oints, resectively. The aster constructs the MR, which is shared by all workers. The node doinances of MR is given in Algorith 3. For MR, the lines - of Algorith 3 are not executed and the for loo in line can terinate as soon as two objects o and o is found in the sae orthant as of n such that o n and o n, finally n is runed... MRSL in Parallel with MR The arallel stes of couting MRSL() with MR are listed as follows: -() Firstly, the aster divides P into several chunks P j P (such that P j = P ) and then sends these chunks P j as well as the MR to its workers. -() A worker does the followings: (a) construct X j =, Pj; (b) o X j finds its node id in the MR, if it is fro the runed node, then it is ignored, otherwise, it is inserted into the in hea H; j (c) initializes MRSL j to, reeatedly retrieves the root o fro H j and adds it to MRSL j iff MRSL j such that O (o ) = O ( ) and o ; and (d) sends MRSL j to the aster. -(3) Finally, the aster does the followings: (i) collects all local MRSL js fro its workers and inserts the into the in hea H ; and (ii) coutes the MRSL() by following the sae techniue as given in ste (c) for the worker. Only MRSL() are reorted as the MRSL of the uery... Correctness of MR The following lea roves the correctness of MRSL() couted using the MR. Lea 5. Assue that n is a runed node in MR and region + (n). Then, MRSL(). A oint that is doinated by both and w.r.t.. The coordinates of in, coe fro and/or.

7 Algorith 3: testdoinancesmr 5 break; // arent region contains o while!h.isety() do 7 o H.retrieveRoot(); if arent.iscovered(o ) then break; // arent region contains o center coutevirtualmin(o, o ); extendregions(arent, center); 7 D Inut : Node root begin N retrievenodes(root); 3 for each n N do if n.ispruned() then 5 continue; // already runed 7 for each n N do 7 S coutedoinatingobjects(n, n ); if S.size() then n.runed true; // changenode status H insert(s); BR+ Node arent n.getparent(); while!h.isety() do 3 o H.retrieveRoot(); if arent.iscovered(o ) then Proof. According to the construction of MR,, in the sae orthant as of n such that and doinate all d corners of region(n) w.r.t. (Lea ). Now, we select the air and that has the least Euclidean distance to by inserting the into H and region(arent(n)), region(arent(n)). We get region+(n) by redistributing the regions of the children of arent(n) including the node n considering in, as the new slitting oint. Since in, doinates all d corners of region + (n) w.r.t. and is bounded within region + (n), we get or, i.e., DSL(). Hence, the Lea..3 Index for BRSL The runed regions in the uad tree index for couting the BRSL() are established as er the following lea: Lea. A node n is arked as runed iff such that (a) O ( ) = O (n ) and (b) n. Proof. We know that if such that n (condition (b)), then doinates all d corners of node n. As all users u n is bounded by its d corners, u, u n. Since, is also in the sae orthant as of n w.r.t. (condition (a)), any user u n cannot be in BRSL(). Hence the lea. The children of a runed node are set to null. BR: The basic index for couting BRSL(), we call it BR, of the dataset P given in Fig. (a) for the idoints of the sales {,,,, 7,, } and uery =<, > is shown in Fig. (a) by setting ρ =. BR: To extend the runed region of a leaf node n, denoted by region+ (n ), firstly we gather all idoints that dynaically doinate the corners of a node n w.r.t. in the sae orthant as of n. Then, we insert these into the in hea H and retrieve the root such that region(arent(n )). Finally, we readjust the regions of the children of arent(n ) by considering as the new slitting oint. We call it BR here. The BR of the dataset P given in Fig. (a) with idoints of the sales {,,,, 7,, } for =<, > is shown in Fig. (b). The regions with gray atterns are the new runed D+(Otization Heuristics) 7 7 BR (a) BR 7 D 7 BR+ Figure : The uad tree indices for BRSL() of the dataset given in Fig. and =<, > (b) BR Algorith : testdoinancesbr Inut : Node root begin N retrievenodes(root); 7 3 for each n N do if n.ispruned() then 5 continue; // already runed for each n N do 7 S coutedoinatingobjects(n, n ); if S.size()!= then n.runed true; // change node status H insert(p); Node arent n.getparent(); while!h.isety() do 3 center H.retrieveRoot(); if arent.iscovered(center) then 5 extendregions(arent, center); areas. Here, we redistribute the regions of the children of the node with id considering as the new slitting oint. Siilarly, we redistribute the regions of the children of the nodes with ids and considering 7 and as the new slitting oints, resectively. The aster constructs the BR, which is shared by all workers. The node doinances of BR is given in Algorith. For BR, the lines -5 of Algorith 3 are not executed and the for loo in line can terinate as soon as an object o is found in the sae orthant as of n such that o n and n is runed..3. BRSL in Parallel with BR The arallel stes of couting BRSL() with BR in two-rounds are listed as follows: -() In the first round, the aster divides P into several chunks P j P (such that P j = P ) and sends these chunks P j as well as the BR to its workers. -() A worker does the followings: (a) P j convert to its idoint, finds the node id of in the BR, if it is fro the runed node, then it is ignored, otherwise, it is inserted into the in hea H j ; (b) initializes MSL j to, reeatedly retrieves the root fro H j and adds it to MSL j iff MSL j such that O ( ) = O ( ) and ; and (c) sends the MSL j to the aster. -(3) Then, the aster does the followings: (i) collects all MSL js fro its workers and insert the into a in hea H ; and (ii) coutes the MSL() by following the sae techniue as given in ste (b) for the worker. -() In the second round, the aster divides U into several chunks U j U (such that U j = U) and then sends these D+(

8 MR(Otiization) MR+(Otiization) D+ ' ' (a) D(Otiization) OtD D+(Otization (b) OtD Heuristics) Figure 7: The otial uad tree indices for DSL() of the dataset given in Fig. (a) and =<, > BR+ chunks U j and themsl() to its workers. -(5) A worker does the followings: (a) u U j finds its node id in the BR, if it is fro the runed node, then it is ignored, otherwise, 7 it is inserted into the local BRSL j 7 iff MSL() such that O (u) = O () and u; and (b) sends the BRSL j to the aster. -() As a final ste, the aster collects all local BRSL 7 js fro its workers into the global BRSL()..3. Correctness of BR The following lea roves the correctness of BRSL() BR+(Otiization) couted using the BR. Lea 7. Assue that n is a runed node in BR and u region + (n). Then, u BRSL(). Proof. According to the construction of BR, in the sae orthant as of n such that doinates all d corners of region(n) w.r.t. (Lea ). Now, we insert all these into H and retrieve the that has the least Euclidean distance to and region(arent(n )). We get region + (n) in BR by redistributing the regions of the children of arent(n) including the node n considering as the new slitting oint. Since u is bounded within region+(n) and doinates all d corners of region+(n), we get u, i.e., DSL(u). Hence, the Lea. 5. OPTIMIZATION HEURISTICS This section resents index secific heuristics to irove the erforance of all of the aforeentioned indexing schees. 5. Aggressive Partitioning In and indexing schees, we sto artitioning a node if the nuber of sales in it is below the threshold, ρ. However, the sale objects selected by the adoted saling ethod ay not reresent the node sace well. Therefore, the basic stoing criteria ay not rune sufficient nuber of objects to exedite the arallelization of dynaic and reverse skyline ueries. To overcoe this liitation, we aly the following heuristic to reartition a non-ety and unruned node n aggressively as given as: if (n) > δ, where (n) denotes the area of n. For silicity, we roose to reartition a unruned node n only once. For each /, the reartitioning is conducted as: -D/D: We insert all of the node n into a in-hea H, retrieve the root fro H and then, artition n considering as the center of slit. The child node of n doinated by w.r.t. is runed. The otial D and D of the dataset in Fig. (a) and uery =<, > with sale roducts {,,,, 7,, } is shown in Fig MR MR MR(Otiization) 7 (a) OtMR (b) MR+(Otiization) OtMR Figure : The otial uad tree indices for MRSL() of the dataset given in Fig. (a) and =<, > BR 7 BR(Otiization) (a) OtBR BR+ BR+(Otiization) of the dataset given in Fig. (a) and =<, > -MR/MR: We insert all of n into a 3 in-hea H, retrieve the to- idoint objects 5 and fro H and then, artition n considering in, as the center of slit. The child node of n doinated by in, w.r.t. 7 is runed. The otial MR and MR of the 7 (b) OtBR Figure : The otial uad tree indices for BRSL() exale dataset given in Fig. (a) and uery =<, > with the idoints of the sales {,,,, 5,, 7,,, } is shown in Fig.. -BR/BR: We insert all of n into a in-hea H, retrieve the root fro H and then, artition n considering as the center of slit. The child node of n doinated by w.r.t. is runed. The otial BR and BR of the exale dataset given in Fig. (a) and uery =<, > with idoints of the sales {,,,, 7,, } is shown in Fig.. To aly aggressive artitioning on node n, we need at least one sale object n for D/D, two idoints n and n for MR/MR and one idoint n for BR/BR. Now, we roose two different heuristics for δ as follows: (a) AV G{ (n) n.runed = true, n /} and (b) MIN{ (n) n.runed = true, n /}. The MIN heuristic assues that an unruned node is not reresented well by the saling ethod if its area is greater than the area of a runed node and is suitable for uniforly distributed data sace. On the other hand, the AVG heuristic assues that an unruned node is not reresented well by the saling ethod if its area is greater than the areas of all runed nodes in average and is suitable for skewed data distribution in correlated and anticorrelated data sace. The Table shows the ratio of runed areas of different uad-tree based data indexing schees for arallelizing the dynaic and reverse skylines of the exale dataset given in Fig. (a) and the uery =<, >, where we set δ to AVG.

9 (a) CarDB (b) UN (c) CO (d) AC Figure : Data distribution in tested two-diensional real CarDB and synthetic UN, CO and AC datasets Table : Settings of araeters Paraeter Values Tested Datasets Real (CarDB), Synthetic (UN, CO, AC) Data Cardinality Thousands Millions Diensionality No. of 5 ( thread er rocessor) No. of Sales Objects Slit Threshold 5 Objects 5. Load Balancing Consider the final round of dynaic skyline coutation carried out by the aster rocessor. Assue that the size of the accuulated local dynaic skyline objects in the aster is T. Now, the aster needs to erfor O(T ) airwise doinance checkings in the worst case to finalize the global dynaic skyline. This worst-case tie colexity ay doinate the overall efficiency. To itigate this erforance bottleneck, we roose to arallelize the doinance checkings aong the workers until the size of the accuulated local dynaic skyline objects in the aster becoes below a threshold τ. For reverse skyline ueries, we also roose to arallelize the idoint skyline objects for each orthant.. EXPERIMENTS Here, we coare the efficiencies of different uad-tree based data indexing schees:, Ot, and Ot for arallelizing dynaic skyline, onochroatic reverse skyline and bichroatic reverse skyline ueries.. Datasets, Queries and Environent Datasets: We evaluate the erforance of all indexing schees for arallelizing the dynaic and reverse skylines using real data, naely CarDB 3, consisting of 5 car objects. This is a six-diensional dataset with attributes: ake, odel, year, rice, ileage and location. We consider only the three nuerical attributes year, rice and ileage in our exerients. The dataset is also noralized into the range [, ]. We randoly select half of the car objects as roducts and the rest as the user data for bichroatic reverse skyline. We also resent exeriental results based on synthetic data: unifor (UN), correlated (CO) and anticorrelated (AC), consisting of varying nuber of roducts, users and diensions. The cardinalities of these datasets in roducts and users are thousands (K) illions (M). The diensionality (d) varies in. The data distributions of the above tested datasets for d = are shown in Fig.. Test (Skyline) Queries: For all exerients, we run a nuber of ueries generated (synthetic) and selected (CarDB) randoly by following the distribution of the tested datasets. Couting Environent: We execute all of our algo- as ned Are of Prun Ratio o Ot Ot 5 7 Nuber of Sales (a) Effect of Sales roductcardiensions sales slitthresh Ot Ot Ratio of Pruned Areas R Ot Ot 3 5 Slit Threshold (b) Effect of ρ Figure : Effect of (a) sales and (b) ρ on runed areas for dynaic skyline uery in indexing schees riths in Swinburne HPC syste with 5 rocessors and GB ain eory. The aster-worker is siulated with Java ulti-threading. Table suarizes the values of different araeters used in our exerientation.. Data Indexing Evaluation This section evaluates all indexing schees in ters of the ratios of runed areas. Firstly, we build data indices for skyline ueries (ueries follow the distribution of the dataset) using 5 sales for each dataset, where we set roduct cardinality P to K, diensions d to and slit threshold ρ to. Table 3 shows the average ratios of the runed areas in different data indexing schees. It is evident fro Table 3 that the roosed outerfors the basic indexing schee in ters of the ratios of runed areas. Also, the roosed aggressive runing heuristic otiizes the runed areas for both and indexing schees. Secondly, we conduct two exerients using the sae data settings for DSL ueries: (a) varying #sales = with ρ = and (b) varying ρ= 5 with 5 sales in CarDB dataset. We set δ to AVG for both exerients. The results are shown in Fig.. It is evident fro Fig. that both #sales and ρ lay an iortant role in uad-tree based data indexing schees. We see that Q+ Tree is less suscetible to the settings of #sales and ρ than the. The otiized and data indexing schees are ostly tolerant to the settings of the..3 Efficiency Study Results Here, we resent the efficiency study results of all indexing schees for couting the DSL, MRSL and BRSL ueries in arallel. Firstly, we exerient for skyline ueries using 5 sales for each dataset, where we set roduct cardinality P and user cardinality U to K, diensions d to, slit threshold ρ to and threads to 5. We also set τ to for DSL ueries. Fig. shows the average of the execution ties of all skyline ueries. We see that the roosed indexing schee outerfors the basic indexing schee in arallelizing all tyes of skyline ueries. 3 htts://autos.yahoo.co/ htt://

10 Table 3: Ratios of runed areas in different uad-tree based data indexing schees on different datasets Schee(s) CarDB Unifor (UN) Correlated (CO) Anticorrelated (AC) DSL MRSL BRSL DSL MRSL BRSL DSL MRSL BRSL DSL MRSL BRSL 3.% 7.% 7.% 5.7% 3.% 3.% 3.5%.%.3% 3.5%.%.% Ot(AVG) 5.% 3.% 5.3% 7.% 3.% 3.% 57.35%.5% 3.% 5.3% 3.%.7% Ot(MIN) 5.7%.% 7.%.%.5%.% 5.5% 35.% 3.%.% 3.3% 35.5% 7.% 7.% 77.%.3%.5%.77% 73.% 7.7% 75.33% 7.% 7.% 73.5%.5%.7% 3.3%.3%.%.77% 5.3%.% 7.%.3% 7.5% 5.3% Ot(MIN) 5.%.77%.%.%.5%.% Chart Title 77.% 75.3% 7.%.% 73.% Chart Title 75.% ecs) ution Ti e(illise Execu CarDB UN CO AC (a) DSL Ot(MIN) Ot(AVG)Q+T Ot(MIN) lisecs) ion Ti e (ill Executi E CarDB UN CO AC (b) MRSL Ot(MIN) Ot(MIN) lisecs) tion Ti e (il Execut CarDB UN CO AC (c) BRSL Ot(MIN) Ot(MIN) Figure : Execution ties of all indexing schees for (a) DSL (b) MRSL and (c) BRSL ueries in all datasets (a) DSL Ot(MIN) (b) MRSL Ot(MIN) (c) BRSL Figure 3: CarDB (K): threads vs. efficiency (a) DSL Ot(MIN) (b) MRSL Ot(MIN) Figure : UN (3M): threads vs efficiency (c) BRSL Ot(MIN) Ot(MIN) The aggressive artitioning heuristic iroves the efficiency of both and indexing schees in ost cases for all datasets. However, we do not observe any significant iroveent of alying aggressive artitioning on the efficiencies of arallelizing the skyline ueries in soe cases. This indicates that searching the runed regions of the data objects in the tree is soeties ore costly (due to tree deth) than erforing the airwise doinance check for the. Therefore, we advocate to use indexing only if there is no significant iroveent in the ratios of runed regions after alying aggressive artitioning for the. The following sections study the effect of different araeters..3. Effect of threads This section investigates the effect of #workers, i.e., Java threads, on the execution tie of rocessing skyline ueries in arallel. We run exerients with CarDB and UN datasets. For CarDB dataset, we set P = K, U = K, #sales to 5, ρ =, d = and vary #threads fro to. For UN dataset, we set P = 3M, U = M, sales to, ρ = 5, d = and vary #threads fro to 5. For both CarDB and UN datsets, we also set τ to for balancing loads in DSL ueries. The average results for skyline ueries are shown in Fig. 3 and Fig.. It is evident that indexing offers the best efficiency with less threads than the basic for all skyline ueries. We observe that secs) e (illi on Tie Executio E illisecs) ie (i ution Ti Execu Ot(MIN) Diension (a) CarDB (K) Ot(MIN) Ot(MIN) 3 5 Diensions (b) UN (3M) Figure 5: Chart DSL: Titlediensionality vs efficiency )5 3 3 Diension (a) CarDB (K) Ot(MIN) Ot(MIN) 3 Ot(MIN) 3 5 Diensions (b) UN (3M) Figure : MRSL: diensionality vs efficiency increased #threads ay not irove the efficiency at all as the overhead of aintaining threads also gets increased..3. Effect of diensionality Here, we study the effect of diensionality on the efficiencies of rocessing all tyes of skyline ueries in arallel by exerienting with CarDB and UN datasets. For CarDB dataset, we set P = K, U = K, #sales to 5, ρ =, #threads to 5 and vary the diensionality d fro to 3. For UN dataset, we set P = 3M, U = M, #sales to, ρ = 5, #threads to and vary d fro to. For both CarDB and UN datsets, we also set τ to for balancing loads in DSL ueries. The average results for skyline ueries are shown in Fig. 5, Fig. and Fig. 7. We see that data indexing schee offers better efficiencies than the basic schee for all tyes of skyline ueries. We also observe that the otiization heuristics irove the efficiencies in ost cases for both datasets..3.3 Effect of data cardinality This section exaines the effect of data cardinality on the efficiencies of rocessing all skyline ueries in arallel by exerienting with illion of objects in UN dataset. Here, we set U = M, #sales to, ρ = 5, #threads to, d = and vary P fro M to M. We also set τ = for DSL ueries. The average results for ueries of all

11 isecs) e (illi on Ti Executi E Ot(MIN) Diension (a) CarDB (K) Ot(MIN) Ot(MIN) 3 5 Diensions (b) UN (3M) Figure 7: BRSL: diensionality vs efficiency Ot(MIN) Μ 3Μ 5Μ 7Μ Μ Product Cardinality (a) DSL 3 5 Ot(MIN) Μ 3Μ 5Μ 7Μ Μ Product Cardinality (b) MRSL Ot(MIN) Μ 3Μ 5Μ 7Μ Μ Product Cardinality (c) BRSL Figure : UN: roduct cardinality vs efficiency skyline tyes are shown in Fig.. We see that the roosed schee scales well and outerfors the basic schee for arallelizing all tyes of skyline ueries..3. Effect of sales and slit threshold This section studies the effect of the settings of #sales and the slit threshold, ρ, on the efficiencies of rocessing BRSL ueries in arallel by exerienting with CarDB dataset. Firstly, we set P = K, U = K, #threads to 5, d = and vary #sales fro to. The average results of ueries are shown in Fig. (a). In the second exerient, we set #sales to 5 and vary ρ fro to 5. The average results are shown in Fig. (b). We see that the roosed indexing is ore adative with the settings of both #sales and ρ than the basic indexing. However, the setting of the above araeters as well as #worker rocessors (Java threads) in higher diensions and data cardinalities (i.e., ulti-araeter otiization) is an oen challenge for future research Sales Ot(MIN) 5 7 (a) Sales vs. Efficiency Slit Threshold Ot(MIN) (b) ρ vs. Efficiency Figure : CarDB: effect of sales and ρ on efficiency of bichroatic reverse skyline ueries 7. CONCLUSION This aer resents an efficient uad-tree based data indexing schee, called, for arallelizing the dynaic and reverse skylines. We also resent several otiization heuristics to irove the erforance of the uad-tree based data indexing schees. We conduct extensive exerients with both real and synthetic datasets and deonstrate the efficiency of the roosed data indexing schee by coaring the results with its existing counterart. Acknowledgent: The work is artially suorted by the ARC discovery grant DP7.. REFERENCES [] A. Arvanitis, A. Deligiannakis, and Y. Vassiliou. Efficient influence-based rocessing of arket research ueries. In CIKM, ages 3,. [] S. Börzsönyi, D. Kossann, and K. Stocker. The skyline oerator. In ICDE, ages 3,. [3] E. Dellis and B. Seeger. Efficient coutation of reverse skyline ueries. In VLDB, ages 3, 7. [] P. M. Deshande and D. Padanabhan. Efficient reverse skyline retrieval with arbitrary non-etric siilarity easures. In EDBT, ages 3 33,. [5] M. S. Isla and C. Liu. Know your custoer: couting k-ost roising roducts for targeted arketing. The VLDB J., 5():55 57,. [] M. S. Isla, R. Zhou, and C. Liu. On answering why-not uestions in reverse skyline ueries. In ICDE, ages 73, 3. [7] H. Köhler, J. Yang, and X. Zhou. Efficient arallel skyline rocessing using hyerlane rojections. In SIGMOD, ages 5,. [] C. Li, B. C. Ooi, A. K. H. Tung, and S. Wang. DADA: a data cube for doinant relationshi analysis. In SIGMOD, ages 5 7,. [] X. Lian and L. Chen. Monochroatic and bichroatic reverse skyline search over uncertain databases. In SIGMOD, ages 3,. [] X. Lin, Y. Yuan, Q. Zhang, and Y. Zhang. Selecting stars: The k ost reresentative skyline oerator. In ICDE, ages 5, 7. [] K. Mullesgaard, J. L. Pederseny, H. Lu, and Y. Zhou. Efficient skyline coutation in areduce. In EDBT, ages 37,. [] D. Paadias, Y. Tao, G. Fu, and B. Seeger. An otial and rogressive algorith for skyline ueries. In SIGMOD, ages 7 7, 3. [3] Y. Park, J. Min, and K. Shi. Processing of robabilistic skyline ueries using areduce. PVLDB, (): 7, 5. [] Y. Park, J.-K. Min, and K. Shi. Parallel coutation of skyline and reverse skyline ueries using areduce. PVLDB, (): 3, 3. [5] D. Pertesis and C. Doulkeridis. Efficient skyline uery rocessing in satialhadoo. Inforation Systes, 5:35 335, 5. [] M. Sharifzadeh and C. Shahabi. The satial skyline ueries. In VLDB, ages 75 7,. [7] Y. Tao, L. Ding, X. Lin, and J. Pei. Distance-based reresentative skyline. In ICDE, ages 3,. [] J. S. Vitter. Rando saling with a reservoir. ACM Trans. Math. Softw., ():37 57, 5. [] A. Vlachou, C. Doulkeridis, and Y. Kotidis. Angle-based sace artitioning for efficient arallel skyline coutation. In SIGMOD,. [] G. Wang, J. Xin, L. Chen, and Y. Liu. Energy-efficient reverse skyline uery rocessing over wireless sensor networks. IEEE Trans. Knowl. Data Eng., (7):5 75,. [] T. Wu, D. Xin, Q. Mei, and J. Han. Prootion analysis in ulti-diensional sace. PVLDB, ():,. [] X. Wu, Y. Tao, R. C.-W. Wong, L. Ding, and J. X. Yu. Finding the influence set through skylines. In EDBT, ages 3,. [3] J. Zhang, X. Jiang, W.-S. Ku, and X. Qin. Efficient arallel skyline evaluation using areduce. IEEE Trans. Parallel Distrib. Syst.,. [] S. Zhang, N. Maoulis, and D. W. Cheung. Scalable skyline coutation using object-based sace artitioning. In SIGMOD, ages 3,.