The Spatial Skyline Queries

Transcription

1 The Satial Skyline Queries Mehdi Sharifzadeh Comuter Science Deartment University of Southern California Los Angeles, CA Cyrus Shahabi Comuter Science Deartment University of Southern California Los Angeles, CA ABSTRACT In this aer, for the first time, we introduce the concet of Satial Skyline Queries (SSQ). Given a set of data oints P and a set of uery oints Q, each data oint has a number of derived satial attributes each of which is the oint s distance to a uery oint. An SSQ retrieves those oints of P which are not dominated by any other oint in P considering their derived satial attributes. The main difference with the regular skyline uery is that this satial domination deends on the location of the uery oints Q. SSQ has alication in several domains such as emergency resonse and online mas. The main intuition and novelty behind our aroaches is that we exloit the geometric roerties of the SSQ roblem sace to avoid the exhaustive examination of all the oint airs in P and Q. Conseuently, we reduce the comlexity of SSQ search from O( P Q ) to O( S C + P ), where S and C are the solution size and the number of vertices of the convex hull of Q, resectively. We roose two algorithms, B S and VS, for static uery oints and one algorithm, VCS, for streaming Q whose oints change location over time (e.g., are mobile). VCS exloits the attern of change in Q to avoid unnecessary re-comutation of the skyline and hence efficiently erform udates. Our extensive exeriments using real-world datasets verify that both R-tree-based B S and Voronoibased VS outerform the best cometitor aroach in terms of rocessing time by a wide margin (- times better in most cases).. INTRODUCTION Assume that the members of a multidiscilinary task force This research has been funded in art by NSF grants EEC- 99 (IMSC ERC), IIS-080 (PECASE), IIS-09 (ITR), and unrestricted cash gifts from Google and Microsoft. Any oinions, findings, and conclusions or recommendations exressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Permission to coy without fee all or art of this material is granted rovided that the coies are not made or distributed for direct commercial advantage, the VLDB coyright notice and the title of the ublication and its date aear, and notice is given that coying is by ermission of the Very Large Data Base Endowment. To coy otherwise, or to reublish, to ost on servers or to redistribute to lists, reuires a fee and/or secial ermission from the ublisher, ACM. VLDB 0, Setember -, 00, Seoul, Korea. Coyright 00 VLDB Endowment, ACM /0/09. team located at different (fixed) offices want to ut together a list of restaurants for their weekly lunch meetings. These meeting locations must be interesting in terms of traveling distances for all the team members; for each restaurant r in the list, no other restaurant is closer to all members than r. That is, there exists no better restaurant of choice than an interesting restaurant in terms of all of their comarable attributes (here, distances to team members). Generating this list becomes even more challenging when the team members are mobile and change location over time. Suose the information about all restaurants are stored in a database as objects with static attributes (e.g., rating of the restaurant). The database literature refers to the interesting objects with resect to their static attributes as skyline objects of the database; those that are not dominated by any other object. These static skyline objects deend only on the database itself (and the definition of the interesting object). Even though the interesting restaurants to our team members can also be considered as the skyline of the database of restaurants, there are two major distinctions: ) The restaurants distance attributes based on which the domination is defined are dynamically calculated based on the user s uery (i.e., location of the team members). Conseuently, the result deends on both data and the given uery. ) As these attributes are satial, there is a uniue corresonding geometric interretation of satial skyline in the sace of the objects. The uery retrieving the set of interesting restaurants in our motivating examle belongs to a broader novel tye of satial ueries. In this aer, for the first time, we introduce the concet of these Satial Skyline Queries (SSQ). Given a set of data oints P and a set of uery oints Q in a d-dimensional sace, an SSQ retrieves those oints of P which are not dominated by any other oint in P considering a set of satial derived attributes. For each data oint, these attributes are its distances to uery oints in Q. An interesting variation, which is also studied in this aer, is where the domination is determined with resect to both satial and non-satial attributes of P. Besides online ma services and grou navigation/lanning, SSQ is critical for many alications. In the domain of tri lanning, the satial skyline of hotels with resect to the fixed locations of conference venue, beaches and museums includes all the interesting hotels for lodging during a business/leasure tri. No other hotel is closer than these hotels to all three must-see locations. In crisis management domain, the residential buildings that must be evacuated first in the event of several exlosions/fires are those which are in the satial

2 skyline with resect to the fire locations. The reason is that these laces are either otentially traed in the convex hull of fires or located at the edges of the exanding fire. In defense and intelligence alications, consider the locations of soldiers enetrating into enemy s cams as uery locations and the enemy s guard stations as data oints. The stations in the satial skyline are those from which an attack might be initiated against the latoon of soldiers. Since the introduction of the skyline oerator by Börzsönyi et al. [], several efficient algorithms have been roosed for the general skyline uery. These algorithms utilize techniues such as divide-and-conuer [], nearest neighbor search [], sorting [], and index structures [, 0, 8] to answer the general skyline ueries. Several studies have also focused on the skyline uery rocessing in a variety of roblem settings such as data streams [7] and data residing on mobile devices []. However, to the best of our knowledge, no study has addressed the satial skyline ueries. The most relevant work is the algorithm roosed by Paadias et al. [8], which can be utilized to address SSQ by considering it as a secial case of dynamic skyline uery. However, since is addressing a more general roblem, it overlooks the geometric roerties of SSQ and hence its erformance is not otimal in satial domain. We comare our techniues with in Section 7. Otimal rocessing of SSQs is more challenging than the general skyline ueries as each dominance check here reuires the comutation of the dynamic distance attributes. Notice that the algorithms for Grou or Aggregate Nearest Neighbor ueries [9] are related but not alicable to SSQ as they only find the otimal (best) object based on a fixed reference function. Hence, they reuire no intermediate dominance checks that are vital in rocessing SSQs. In this aer, we aroach the roblem of rocessing SSQs as a uery in satial domain from a geometric ersective. First, we analytically exloit the geometric roerties of the roblem and solution saces. We theoretically rove that unlike general skyline there are data oints in P that are definitely in the satial skyline of P with resect to Q indeendent of the rest of P. Likewise, there are uery oints that have no imact on the inclusion/exclusion of any data oint in/from the skyline. Utilizing this theoretical foundation, we roose two algorithms, B S and VS, for static uery oints and one algorithm, VCS, for streaming Q whose oints change location over time (e.g., are mobile). The R-tree-based B S is a customization of [8] for SSQ that benefits from our theoretical foundation by exloiting the geometric roerties of the roblem sace. B S is more efficient than as it not only avoids exensive dominance checks for definite skyline oints but also runes unnecessary uery oints to reduce the cost of each examination. VS, however, emloys the Voronoi diagram and Delaunay grah of the data oints as a roadma to find the first skyline oint whose local neighborhood contains all other oints of the skyline. The algorithm traverses the nodes of the grah (i.e., data oints of P ) in the order secified by a monotone function over their distances to uery oints. Similar to B S, VS also efficiently exloits the geometric roerties of the roblem sace while traversing the Delaunay grah. VS reduces the comlexity of the naive SSQ search from O( P Q ) to O( S C + P ), where S and C are the solution size and the number of vertices of the convex hull of Q, resectively. The P factor can be reduced further to O(log P ) if an index structure is used. We comrehensively study the scenario where the uery oints are the locations of multile moving objects with unredefined trajectories. This occurs in our motivating alications when the team members are mobile agents or the soldiers are moving. Here, we freuently receive the latest uery locations as satial data streams. For each attern of movement, we extract the locus of all oints whose dominance change and hence trigger an udate to the old satial skyline. Conseuently, to address continuous SSQ we roose VCS that exloits the attern of change in Q to avoid unnecessary re-comutation of the skyline and hence efficiently erform udates. Furthermore, we study a more general case of SSQ where one intends to find the skyline with resect to both static non-satial attributes of the data oints and their distances to uery oints. For instance, the best restaurant in LA might be dominated in terms of distance to our team members but it is still in the skyline because of its rating. We show that B S, VS, and VCS all can suort this variation of SSQ as well. Finally, through extensive exeriments with both realworld and synthetic datasets, we show that B S, VS, and VCS can efficiently answer an SSQ uery. Both R-treebased B S and Voronoi-based VS outerform the best cometitor aroach in terms of rocessing time by a wide margin (u to times better). Our exerimental results with synthetically moving objects verify that on average for -0 uery oints only less than % of movements reuire the entire skyline to be recomuted. For the other 7% of movements, VCS outerforms VS by a factor of which makes it the suerior algorithm for continuous SSQ. The remainder of this aer is organized as follows. We first formally define the roblem of SSQs and the terms we use throughout the aer in Section. In Section, we establish the theoretical foundation of our roosed algorithms. In Section, we discuss our alternative solutions for SSQ. We address continuous SSQ for moving objects and incororating non-satial attributes in SSQ in Sections and, resectively. The erformance of our roosed algorithms is evaluated in Section 7. The related work to SSQ and similar nearest neighbor ueries are resented in Section 8. Finally, we conclude the aer and discuss our future work in Section 9.. FORMAL PROBLEM DEFINITION Assume that we have a database of N objects. Each database object with d real-valued attributes can be concetualized as a d-dimensional oint (,..., d ) R d where i is the i-th attribute of. We use P to refer to the set of all these oints. Figure a illustrates a database of six objects P = {a, b, c, d, e, f} each reresenting the descrition of a hotel with two attributes: distance to beach and rice. Figure b shows the corresonding oints in the -dimensional sace where x and y axes corresond to the range of attributes distance and rice, resectively. Throughout the aer, we use oint and object interchangeably to refer to each database object. In the following sections, we first define the general skyline uery using the above database concetualization. Then, we introduce our satial skyline uery based on the definition of the general skyline uery.. General Skyline Query Given the two oints =(,..., d ) and =(,..., d)

3 object distance rice (mile) ($) a b 0 c. d e. 00 f 7 (a) rice a e b c f d distance to beach (b) Figure : A -dimensional database of six objects in R d, dominates iff we have i i for i d and j < j for some j d. To illustrate, in Figure b the oint f=(, 7) dominates the oint d=(, ). Now, given a set of oints P, the skyline of P is the set of those oints of P which are not dominated by any other oint in P. The skyline of the oints shown in Figure b is the set S = {a, c, e}. The Skyline Query is to find the skyline set of the given database P considering attributes of the objects in P as dimensions of the sace. Notice that every oint of the skyline does not need to dominate a oint of P. For instance in Figure b, while the oints c and e each dominate two other oints, the oint a dominates no oint.. Satial Skyline Query Let the set P contain oints in the d-dimensional sace R d, and D(.,.) be a distance metric defined in R d where D(.,.) obeys the triangular ineuality. Given a set of d- dimensional uery oints Q={,..., n} and the two oints and in R d, satially dominates with resect to Q iff we have D(, i ) D(, i ) for all i Q and D(, j ) < D(, j) for some j Q. To be secific, satially dominates iff every i is closer to than to. Figure shows a set of nine -d oints and two uery oints and. With Euclidean distance metric, the oint satially dominates the oint as both and are closer to than to. Note that if we draw the erendicular bisector line of the line segment,,, and will be located on the same side of the bisector line (where is not; see Figure ). We use this geometric interretation of the satial dominance relation between two oints in Section. to justify the foundation of our roosed algorithms. For each oint, consider circles C( i, ) centered at the uery oint i with radius D( i, ). Obviously, i is closer to any oint inside C( i, ) than to. Therefore, by the above definition any oint such as which is inside the intersection of all C( i, ) for all i Q satially dominates. We call this intersection area which otentially includes all the oints which satially dominate as the dominator region of. Similarly, the locus of all oints such as which are satially dominated by is the intersection of outside of all circles C( i, ) (the grey region in Figure ). For a oint, we refer to this region as the dominance region of. Given the two sets P of data oints and Q of uery oints, the satial skyline of P with resect to Q is the set of those oints in P which are not satially dominated by any other oint of P ; the oints which are not inside the dominance region of any other oint. The oint P is in the satial Assume that the large rectangle shows the universe sace of the data oints. Dominator region of ' ''' '' C(, ) Bisector line of ' Dominance region of Figure : The satial skyline of a set of nine oints Symbol Meaning P set of database oints N P, cardinality of P D(.,.) distance function in P Q set of uery oints {,..., n} n Q, cardinality of Q C( i, ) circle centered at i with radius D( i, ) V C() Voronoi cell of CH(Q) convex hull of Q CH v(q) set of vertices of CH(Q) S(Q) satial skyline oints of P with resect to Q Table : Summary of notations skyline of P with resect to Q iff for any oint P there is a uery oint i Q for which we have D(, i ) D(, i ). That is, is in the satial skyline iff we have: P,, i Q s.t. D(, i) D(, i) () We use satial skyline oint and skyline oint interchangeably to denote any oint in the satial skyline. Considering the above definitions, the Satial Skyline Query (SSQ) is to find the satial skyline oints of the given set P with resect to the uery set Q. The naive brute-force search algorithm for finding the satial skyline of P given a uery set Q reuires to examine all oints in P against each other. For each oint, Q distances D(, i) are comuted and comared against the corresonding distances of other oints. If no oint satially dominates, then is added to the solution. The time comlexity of this naive algorithm is O( P Q ) as it exhaustively examines all data oints against each other. However, an otimal SSQ algorithm must examine each oint against only those oints which are inside the dominator region of. In Section, we establish the theoretical foundation of our efficient SSQ algorithms which allows us to avoid a significant number of distance comutations of the naive aroach.. FOUNDATION We first identify the geometric structure of the solution sace corresonding to the SSQ roblem. In articular, we intend to understand how the satial skyline oints of a given uery are related to the data and uery oints. These relationshis constitute the foundation of our roosed solutions to SSQ. To start, we first describe three geometric structures utilized by our solutions, namely Voronoi diagram, Delaunay grah, and convex hull. Notice that while throughout the aer we use Euclidean distance function in -d sace, our results hold in general for any dimension d.. Preliminaries.. Voronoi Diagram and Delaunay Grah The Voronoi diagram of a given set P of oints in R d artitions the sace into regions each including all oints with a common closest oint in P according to a distance

4 V. vertex of VC() V. neighbor of V. edge of ' ' Convex oint (a) (b) (c) Figure : a) Voronoi diagram, b) Delaunay grah, and c) Convex hull metric D(.,.) [].The region corresonding to the oint P contains all the oints x R d for which we have P,, D(x, ) D(x, ) () The euality holds for the oints on the borders of s and s regions. Figure a shows the Voronoi diagram of nine oints in R where the distance metric is Euclidean. We refer to the region V C() containing the oint as its Voronoi cell. For Euclidean distance in R, V C() is a convex olygon. Each edge of this olygon is a segment of the erendicular bisector line of the line segment connecting to another oint of the set P. We call each of these edges a Voronoi edge and each of its end-oints (vertices of the olygon) a Voronoi vertex of the oint. For each Voronoi edge of the oint, we refer to the corresonding oint in the set P as a Voronoi neighbor of. Now consider an undirected grah G(V, E) with the set of vertices V = P. For each two oints and in V, there is an edge connecting and in G iff is a Voronoi neighbor of in the Voronoi diagram of P. The grah G is called the Delaunay Grah of oints in P. Figure b illustrates the Delaunay grah corresonding to oints of Figure a. The Delaunay grah of any set of oints is a connected lanar grah. In Section., we traverse the Delaunay grah of the database oints to find the set of skyline oints... Convex Hull The convex hull of oints in P R d, is the uniue smallest convex olytoe (olygon when d = ) which contains all the oints in P []. Figure c shows the convex hull of the oints of Figure a as a hexagon. The set of vertices of convex hull of oints in P is a subset of P. We use convex oint to denote any of these vertices and non-convex oint to refer to all other oints of P. In Figure c, and are convex and non-convex oints, resectively. We also use CH(P ) and CH v (P ) to refer to the convex hull of P and the set of its vertices, resectively. It is clear that the shae of the convex hull of a set P only deends on the convex oints in P. Conseuently, the location of any non-convex oint P does not affect the shae of CH(P ).. Theories We assume that the set of data oints P and uery oints Q are given. In this section, we exloit the geometric roerties of satial skyline oints of P with resect to Q. To reduce our search sace in finding satial skyline oints, we rove one lemma () and two theorems ( & ) that would hel us to immediately identify definite skyline oints and one theorem () to eliminate some of the uery oints not contributing to the search. The first roerty holds for both general and satial skylines. Convex Hull Lemma. For each i Q, the closest oint to i in P is a skyline oint. Proof. If is the closest oint to i in P, we have D(, i ) < D(, i ) for all P ( ). By definition, no oint in P satially dominates. Hence, is a skyline oint. A restated form of Lemma is valid for the general skyline where all the oints with the minimum value in one dimension are in the skyline. Lemma shows that the inclusion of some data oints in the satial skyline of P does not deend on the location of any other oint of P. For examle in Figure, the oint is a skyline oint as it is the closest oint to the uery oint. It is a skyline oint only because of s location regardless of where the other data oints of P are located. In Section., we utilize Lemma to start our search for skyline oints from a definite skyline oint such as. The following theorem also shows that secific oints in P are skyline oints indeendent of the locations of other oints of P. Theorem. Any oint P which is inside the convex hull of Q is a skyline oint. Proof. The roof is by contradiction. Assume that which is inside the convex hull CH(Q) is not a skyline oint (see Figure a). Then, there is a oint P which satially dominates. Therefore, if we draw the erendicular bisector line of the line segment, all the uery oints i will be located on the same side of the line where is located. Hence, CH(Q) is also on the same side as. That is, the erendicular bisector line of searates CH(Q) from. This contradicts our assumtion that is inside CH(Q) and roves that is a skyline oint. Theorem enables our SSQ algorithms to efficiently retrieve a large subset of skyline oints only by examining them against the uery oints. This theorem is the basis of our roosed SSQ algorithms in Section. The next theorem imroves the alication of Theorem by roving that even some of the uery oints have no imact on the final satial skyline. Hence, our SSQ algorithms can easily ignore them. We first rove the following lemma: Lemma. Given two uery sets Q Q, if a oint P is a skyline oint with resect to Q, then is also a skyline oint with resect to Q. Proof. As is a skyline oint with resect to Q, for any oint P there is a uery oint i Q for which we have D(, i) D(, i) (see Euation ). As Q is a subset of Q, we have i Q. Therefore, according to Euation the oint is a skyline oint with resect to Q.

5 Bisector line of' ' ' ' Bisector line of ' ' (a) (b) (c) Figure : a) Theorem, b) Theorem, and c) Theorem Lemma holds for both general and satial skylines. In the general case, if a oint is in the skyline considering only a subset of its coordinates (i.e., attributes), it remains in the final skyline when all coordinates of are considered. Theorem. The set of skyline oints of P does not deend on any non-convex uery oint Q. Proof. Assume that the uery oint is not a convex oint (i.e., / CH v (Q)) (see Figure b). Consider the set Q = Q {}. We rove that the satial skylines of Q and Q are eual (i.e., S(Q ) = S(Q)). First, assume that is a skyline oint with resect to Q (i.e., S(Q )). According to Lemma, as we have Q Q, is a skyline oint with resect to Q. Now, we assume that we have S(Q). We show that S(Q ). The roof is by contradiction. Assume that is not in S(Q ). Then, there is a oint such as P that satially dominates with resect to Q. By definition, the uery oints in Q and the oint are on different sides of the erendicular bisector line of line segment ; Q is on the same side as. Therefore, the convex hull of Q, CH(Q ), and are searated by the bisector line. Moreover, as is not a convex oint of Q, we have CH(Q) = CH(Q ). Hence, CH(Q) is also searated from by the bisector line. That is, satially dominates with resect to both Q and Q and so we get / S(Q) which contradicts our assumtion. Therefore, we showed that S(Q ). Combining all the above, we roved that S(Q ) = S(Q) and hence the inclusion/exclusion of the single non-convex uery oint in/from Q does not change the set of skyline oints of P. To illustrate, consider the oint inside the convex hull of uery oints shown in Figure b. Theorem imlies that both dominance and dominator regions of any oint is indeendent from. The intuition here is that the circle C(, ) is comletely inside the union of the circles C( i, ) for i CH v(q). With the result of Theorem, we reduce the time comlexity of our SSQ algorithms by disregarding the distance comutation oerations against the non-convex uery oints such as (see Section ). Finally, the last theorem secifies those skyline oints which are identified by examining only the data oints in a limited local roximity around them. Theorem. Any oint P whose Voronoi cell V C() intersects with the boundaries of convex hull of Q is a skyline oint. Proof. The roof is by contradiction. Assume that the Voronoi cell of intersects with CH(Q) but is not a skyline oint (i.e., / S(Q)) (see Figure c). Hence, a oint such as P satially dominates with resect to Q. Therefore, Algorithm B S (set Q) 0. comute the convex hull CH(Q); 0. set S(Q) = {}; 0. box B = MBR(R); 0. minhea H = {(R, 0)}; 0. while H is not emty 0. remove first entry e from H; 07. if e does not intersect with B, discard e; 08. if e is inside CH(Q) or 09. e is not dominated by any oint in S(Q) 0. if e is a data oint. add to S(Q);. B = B MBR(SR(, Q));. else // e is an intermediate node. for each child node e of e. if e does not intersect with B, discard e ;. if e is inside CH(Q) or 7. e is not dominated by any oint in S(Q) 8. add (e, mindist(e, CH v (Q))) to H; 9. return S(Q); Figure : Pseudo-code of the B S algorithm the erendicular bisector line of line segment searates both and Q from. That is, and the convex hull of Q are on different sides of the bisector line. As V C() intersects with CH(Q), the intersection of V C() and CH(Q) is also searated from by the bisector line. It means that the oints in this intersection region are closer to than to. That is, there are some oints such as x inside V C() for which we have D(x, ) < D(x, ). This ineuality contradicts the definition of V C() given in Euation which states that any oint in V C() is closer to than to any other oint in the set P. Thus, is a skyline oint with resect to Q.. SOLUTIONS In this section, we roose two algorithms to solve the SSQ roblem. Both algorithms are emowered by the foundation established in Section ; they utilize Lemma and Theorems and to eliminate unnecessary dominance checks for the definite skyline oints and Theorem to rune some of uery oints.. B S : Branch-and-Bound Satial Skyline Algorithm Our B S algorithm is an imroved customization of the original algorithm for satial skyline [8]. Similar to [8], we assume that the data oints are indexed by a dataartitioning method such as R-tree. For each data oint, let mindist(, A) be the sum P of distances between and the oints in the set A (i.e., A D(, )). Likewise, we define mindist(e, A) as the sum of minimum P distances between the rectangle e and the oints of A (i.e., A mindist(e, )). Figure shows the seudo-code of B S. We describe the algorithm using the set of data oints P = {,..., } and uery oints Q = {,..., } shown in Figure. Each intermediate entry e i in the corresonding R-tree reresents

6 the minimum bounding box of the node N i. B S starts by comuting the convex hull of Q and determines the set of its vertices CH v(q) (e.g., CH v(q) = {,, }). Subseuently, B S begins to traverse the R-tree from its root R down to the leaves. It maintains a minhea H sorted based on the mindist values of the visited nodes. Table shows the contents of H at each ste. First, B S inserts (e, mindist(e, CH v(q))) and (e 7, mindist(e 7, CH v(q))) corresonding to the entries of the root R into H. Then, e with the minimum mindist is removed from H and its children e, e, and e together with their mindist values are inserted into H. Similarly, e is removed and the children of e are added to H. In the next iteration, the first entry is inside CH(Q) and hence is added to S(Q) as the first skyline oint found. Once the first skyline oint is found (i.e., S(Q) ), any entry e must be checked for dominance before insertion into and after removal from H. If e is dominated by a skyline oint, then B S discards e. This dominance check is done against all oints in S(Q); e is dominated by if e is comletely inside the dominance region of (see Section. for the definition). That is, if e does not intersect with any circle C(, ) for CH v(q). To decrease the use of this costly test, B S first alies two easier tests: ) If the entry e does not intersect with the MBR of the union of all circles C(, ) (termed as SR(, Q)), then dominates e. Similarly, if e does not intersect with the intersection of all such MBRs each corresonding to a current skyline oints in S(Q), then a oint in S(Q) dominates e. ) If e is comletely inside the convex hull CH(Q), e cannot be dominated (see Theorem ). If e does not ass any of the above tests, B S reuires to check e against the entire S(Q). B S maintains the rectangle B corresonding to the intersection area described above and udates it when a new skyline oint is found (see dotted box in Figure ). Considering our examle, B S removes from H. As is inside B and is not dominated by, it is added to the skyline oints and the rectangle B is udated accordingly (lines - in Figure ). The next ste examines e which is not dominated by current skyline oints and. Among e s children, only is inserted to H. is outside B (i.e., dominated by ) and hence is discarded. Then, B S removes e 7 and extracts its children e and e as e 7 is not dominated. Entry e does not intersect with B and e is dominated by. Hence, B S discards both entries. At this ste, is removed and added to the skyline oints. The remaining stes discard both dominated entries and e. Finally, the oints, and consist the final satial skyline... B S Correctness The correctness of B S follows that of as both algorithms use the same aroach to exlore the solution sace. B S benefits from all the roerties of. For instance, B S can also utilize any arbitrary monotone function instead of mindist() to sort the entries of its hea. Conseuently, B S is also able to emloy any monotone reference function to suort ranked skyline ueries [8]. However, B S is more efficient than as it reduces the comlexity of the costly dominance checks. ) B S uses only the uery oints on the convex hull of Q instead of the entire Q (Theorem ) which decreases the number of distance comutations during dominance checks. In most cases, B S reuires CH v (Q) < Q oerations to comute distances of each oint. ) B S utilizes the rectangle B to N R e e e e e N 7 N N N N N e e 7 Figure : Points indexed by an R-tree ste hea contents (entry e: mindist(.,.)) S(Q) (e : 8), (e 7 : ) (e : 8), (e : 9), (e 7 : ), (e : ) ( : 8), ( : ), (e : 9), (e 7 : ), ( : 70), (e : ) (e 7 : ), ( : ), ( : 70), (e : ) {, } ( : ), ( : 70), (e : ) {, } ( : 70), (e : ) {,, } Table : B S for the examle of Figure facilitate the runing of dominated rectangular entries from O( Q (d + S(Q) )) to O(d ). Instead of comuting the minimum distances between e and each Q and comaring them against those of each skyline oint in S(Q), B S first checks whether e and B intersect. ) B S does not reuire any dominance check for oints inside CH(Q) (Theorem ). Moreover, B S utilizes Theorem to recede its exensive dominance checks with an intersection check between the Voronoi cell V C() and CH(Q) when the later is cheaer (e.g., when S(Q) and Q are large comaring to CH v(q) and the number of vertices of V C()).. VS : Voronoi-based Satial Skyline Algorithm According to Theorems and, the oints whose Voronoi cells are inside or intersect with the convex hull of the uery oints are skyline oints. Therefore, our VS algorithm utilizes the Voronoi diagram (i.e., the corresonding Delaunay grah) of the data oints to answer an SSQ roblem. We assume that the R-tree on the data oints does not exist. Instead, the Voronoi neighbors of each data oint is known. To be secific, the adjacency list of the Delaunay grah of the oints in P is stored in a flat file. To reserve locality, oints are organized in ages according to their Hilbert values. VS starts traversing the Delaunay grah from a data oint (e.g., NN( ), the closest oint to ). The traversal order is determined by the monotone function mindist(, CH v(q)). VS maintains two different lists to track the traversal; V isited which contains all visited oints and Extracted which contains those visited oints whose Voronoi neighbors have also been visited (extracted). Similar to B S, VS also maintains the rectangle B which includes all candidate skyline oints. Figure 7 shows the seudo-code of VS. It first comutes the convex hull of uery oints and initializes all the data structures. Then, the closest oint to one of the uery oints 7 7

7 Algorithm VS (set Q) 0. comute the convex hull CH(Q); 0. set S(Q) = {}; 0. minhea H = {(NN( ), mindist(nn( ), CH v(q)))}; 0. set V isited = {NN( )}; set Extracted = {}; 0. box B = MBR(SR(NN( ), Q)); 0. while H is not emty 07. (, key) = first entry of H; 08. if Extracted 09. remove (, key) from H; 0. if is inside CH(Q) or. is not dominated by S(Q). add to S(Q);. B = B MBR(SR(, Q));. else. add to Extracted;. if S(Q) = or a Voronoi neighbor of is in S(Q) 7. for each Voronoi neighbor of such as 8. if V isited, discard ; 9. if is inside B or V C( ) intersects with B 0. add to V isited;. add (, mindist(, CH v (Q))) to H;. return S(Q); Figure 7: Pseudo-code of the VS algorithm Figure 8: Examle oints for VS (e.g., = NN( )) and its corresonding entry (, mindist(, CH v (Q))) are added to V isited and minhea H, resectively. Then, VS iteratively examines (, key), the to entry of H. If all s Voronoi neighbors have been already visited (i.e., Extracted), then (, key) is removed from H. Subseuently, if is not dominated by current S(Q), then is added to S(Q) as a skyline oint. If / Extracted, is added to Extracted. Moreover, if at least one of the skyline oints identified so far is a Voronoi neighbor of, then VS adds each unvisited Voronoi neighbor of to V isited and H iff a) is inside current rectangle B or b) s Voronoi cell V C( ) intersects with B. Subseuently, B is udated accordingly and the entry (, mindist(, CH v(q))) is added to the hea H. Finally, when H becomes emty, VS returns S(Q) as the result. We describe VS using the examle shown in Figure 8. The three uery oints i and the data oints are shown as white and black dots, resectively. Table shows the contents of hea H. First, VS adds (, midist(, CH v (Q))) to H and marks as visited. B is also initialized accordingly to the dotted box in Figure 8. The first iteration visits,,,, and 8 as s Voronoi neighbors and adds their corresonding entries to H. It also adds to the Extracted list. The second iteration removes from H as and its neighbors have been already visited. It also adds to S(Q) as is inside CH(Q). The third iteration adds 9, 0, and as the only unvisited neighbors of which are inside B to H. The next two iterations immediately remove and then from H and add them to S(Q) as their neighbors have been already visited. The subseuent 9 iterations add,, and to the skyline and eliminate the remaining entries of H as they are all dominated... VS Correctness Lemma. Given a uery set Q, VS first adds the data oints inside CH(Q) to the skyline S(Q). All other oints are examined and added to S(Q) in ascending order of their mindist() values. Proof. We first show that 0, the first data oint added to S(Q), is inside CH(Q) if there is at least one oint inside CH(Q). The traversal of Delaunay grah once started from oint NN( ) is towards the oint with smaller mindist. The reason is that the mindist of the to entry of H is always decreasing before 0 being added to S(Q). It is clear that the mindist of any oint inside CH(Q) is smaller than those of the oints outside CH(Q). Hence, after a few iterations, the to entry of H becomes a oint inside CH(Q). Then during the next iterations, the oint whose mindist is less than those of all of its neighbors is added as the first oint to S(Q). As mindist is monotone with resect to the distance to each uery oint i, this haens only inside CH(Q). Therefore, 0 is inside CH(Q). If there is no oint inside CH(Q) then V C( 0) intersects with CH(Q). Once 0 is added, all other oints inside CH(Q) are added to S(Q) as their mindist are smaller than those of outside oints. Finally, as the seudo-code of VS shows, subseuent iterations always examine and add the to entry of H which has the minimum mindist to S(Q). Hence, the non-dominated data oints outside CH(Q) are added to S(Q) in the ascending order of their mindist. Lemma. Given a uery set Q, VS identifies all satial skyline oints with resect to Q. Proof. We rovide only the sketch of the roof. It is clear that VS examines all the data oints excet those that either their Voronoi cells are outside the rectangle B or none of their already-visited Voronoi neighbors has a skyline neighbor. The first grou of oints are inside the dominator region of one of the visited oints and hence need to be discarded. The second grou of oints are obviously outside CH(Q) and the Voronoi cell of none of their Voronoi neighbors intersects with CH(Q). These neighbors are all dominated. Voronoi neighbors of these neighbors are also dominated and hence do not intersect with CH(Q). The two layer of dominated Voronoi neighbors around such a oint guarantees that is also dominated by a oint which dominates one of these neighbors. Similar to B S, VS utilizes our foundation described in Section to efficiently reduce the number of dominance checks. The number of data oints visited by VS is much less than those inside rectangle B. The reason is that VS usually stos its traversal earlier than reaching the boundaries of B. The time comlexity of VS is O( S(Q) CH v(q) +Φ( P )) where Φ( P ) is the comlexity of finding the oint from which VS starts visiting inside CH(Q) (e.g., NN( )). If VS utilize an index structure, then Φ( P ) is O(log P ). Otherwise, As the Delaunay grah of P is connected, VS starts from a random oint and kees visiting the neighboring oint closest to until it reaches. For a uniformly distributed P in a suare-shaed area, this takes Φ( P ) = O( P /) stes.

8 ste hea contents (oint : mindist(.,.)) S(Q) ( : ) ( : ), ( : 8), ( : ), ( : ), ( : 8), ( 8 : ) ( : 8), ( : ), ( : ), ( : 8) { } ( 8 : ), ( 9 : 9), ( 0 : 9), ( : ) ( : ), ( : ), ( : 8), ( 8 : ), {, } ( 7 : ), ( 9 : 9), ( 0 : 9), ( : ) 8 ( : ), ( : 8), ( 8 : ), ( 7 : ), {,, } ( 9 : 9), ( 0 : 9), ( : ) {,,,,, } Table : VS for the examle of Figure 8. CONTINUOUS SPATIAL SKYLINE QUERY The algorithms roosed in the revious sections are aroriate for the alications where the uery oints in Q reresent the locations of some stationary objects. Hence, the satial skyline of the data oints P with resect to Q once found is not changing. Now consider the scenario in which each uery oint i reresents the location of a moving object in the sace of R. All moving objects freuently reort their latest locations. Conseuently, we gradually receive the new location of each object as a satial data stream. Arrival of each new location causes an udate to a single oint of Q. We intend to maintain the satial skyline of P with resect to the set of latest locations of all objects (i.e., the current uery set Q). Uon an udate to Q, one can always rerun an SSQ algorithm to return the new set of skyline oints. However, this aroach is exensive secially with the R-tree-based algorithms such as B S and where the entire R-tree must be traversed er each udate. These algorithms cannot tolerate udates for a fast rate data stream. A smarter solution is to udate the set of reviously found skyline oints when a uery oint moves. With B S and, this still reuires a comlete traversal of R-tree and examining all candidate data oints. However, a more efficient aroach must examine only those data oints which may change the satial skyline according to s new location. In the following, we first identify the subset of uery oints on which the dominance of a data oint deends. Then, we rove Lemma which states that affects the dominance of iff is in the line-of-sight of. That is, the line segment does not intersect with the interior of CH(Q). We term visible region of to refer to the locus of all such oints. Finally, in Section. we identify the oints which may change the skyline for different moves of and roose our algorithm for continuous SSQ. The algorithm utilizes Lemma to choose the data oints that must be examined when the location of changes. First, we find the uery oints which may change the dominance of a data oint outside CH(Q). Consider the data oint and uery oints Q = {,..., } in Figure 9. The two tangent lines from to CH(Q), divide the vertices in CH v (Q) into two disjoint sets CH v + (Q) = {,, } and CHv (Q) = { }. The sets CH v + (Q) and CHv (Q) include the vertices on the closer and farther hulls of the convex hull CH(Q) to, resectively. The following lemma roves that only the uery oints in the closer hull to can change the dominance of. Lemma. Given a data oint and a uery set Q, the dominance of only deends on the uery oints in CH + v (Q). Proof. Suose a oint dominates with resect to L Figure 9: Visible region of CH(Q) CH v + (Q). It is clear that the bisector line of which searates and CH v + (Q) intersects with the tangent lines from to CH(Q). Hence, it also searates the entire CH(Q) from. Therefore, is dominated with resect to Q too. The roof of the reverse case is also similar. Therefore, is (not) dominated with resect to Q iff is (not) dominated with resect to CH v + (Q). Hence, the dominance of deends only on CH v + (Q) (not CHv (Q)). Lemma states that the dominator region of does not deend on / CH v + (Q). That is, this region is not changed by the circle C(, ) if CHv (Q). Figure 9 illustrates this case where the dotted circle C(, ) does not contribute to the dominator region of. Utilizing Lemma, we easily find the locus of all oints whose dominance deends on the location of a given uery oint. Consider in the examle shown in Figure 9. The lines L and L ass through the line segments and, resectively ( and are immediate neighbors of on CH(Q)). Each line divides the sace into two half lanes. The figure illustrates the half lanes using arrows. The union of the half lanes which does not contain CH(Q) (visible region of ) is the locus of the data oints whose dominance does deend on. The reason is that for any oint such as in this region, CH v + (Q) does not contain and hence according to Lemma s dominance does not deend on. The general statement of the above result follows. Lemma. The locus of data oints whose dominance deends on CH v(q) is the visible region of.. Voronoi-based Continuous SSQ (VCS ) Assume that the location of the uery oint changes. We use and Q to refer to the new location of and the new set of uery locations, resectively (Q = Q { } {}). In this section, we roose our Voronoi-based Continuous Satial Skyline algorithm (VCS ). We show how VCS uses the visible regions of and to artition the sace into several regions for each a secific udate must be alied to the old skyline. VCS is more efficient than the naive aroach as it avoids excessive unnecessary dominance checks. The algorithm traverses the Delaunay grah of the data oints similar to VS. It first comutes CH(Q ), the convex hull of the latest uery set. Then, it comares CH(Q ) with the old convex hull CH(Q). Deending on how CH(Q ) differs from CH(Q), VCS decides to either traverse only secific ortions of the grah and udate the old skyline S(Q) or rerun VS and generate a new one. With the former case, it tries to examine only those oints on the grah whose dominance changes because of s movement. We now describe the situations where VCS udates the skyline based on the change in CH(Q ). Later, we show how the udate is alied. We first enumerate different ways L

9 ' ' ' ' ++ ' ' + + (a) (b) (c) (d) (e) (f) Figure 0: Change atterns of convex hull of Q when the location of changes to Figure : Examle oints for VCS that the convex hull of uery oints might change when a uery oint moves. Figures 0a-0f show only six change atterns. Each figure illustrates a case where s location changes to (i.e., moves to ). The grey and the thickedged olygons show CH(Q) and CH(Q ), resectively. In general, identifying the aroriate attern by comaring CH(Q) and CH(Q ) is not a straightforward task. Therefore, VCS only tries to recognize secific simle atterns I-V and subseuently udates the skyline accordingly as we exlain later. For all other change atterns such as attern f, it reruns VS. For each of atterns I-V, secific ortions of the Delaunay grah must be traversed by VCS : Pattern I) CH(Q) and CH( ) are eual as both and are inside the convex hull. According to Theorem, the skyline does not change and no grah traversal is reuired. Patterns II-V) The visible regions of CH v(q) and CH v (Q ) together artition the sace into six regions (seven regions for attern V). The intersection region of CH(Q) and CH(Q ) contains data oints which are in skyline with resect to both Q and Q. Therefore, VS does not traverse this ortion of the Delaunay grah. The region labeled by ++ includes the oints inside CH(Q ) and outside CH(Q). According to Theorem, any oint in this region is a skyline oint with resect to Q and hence must be added to the skyline. The oints in the regions labeled by + might be skyline oints and must be examined. The intuition here is that their dominator regions have become smaller because of s movement. The regions labeled by contain the oints whose dominator regions have been exanded and hence might be deleted from the old skyline. The oints in the regions secified by must be examined for inclusion in or exclusion from the skyline as their dominator regions have changed. Finally, neither nor affects the dominance of the oints in the unlabeled white region. The reason is that this region is outside of the visible regions of both and (Lemma ). For each attern I-V, only the data oints in the union of the labeled regions might change the skyline. We use the term candidate region to refer to this region. Once VCS identifies any of atterns I-V, it tries to udate the skyline. The algorithm first assigns the old S(Q) to S(Q ). For attern I, VCS returns the same old skyline as it is still 9 valid. For atterns II-V, VCS starts traversing the Delaunay grah from the closest data oint to. Similar to VS, it initializes the rectangle B and traverses the oints which are inside the intersection of the candidate region and B ordered by their mindist values. At each iteration, if the oint with the minimum mindist has been dominated it is deleted from S(Q ) otherwise it is added to S(Q ). At the end, VCS evaluates the skyline S(Q ) and removes all the oints which are dominated by another oint in S(Q ). The reason behind this final check is to examine the old skyline oints which are not in the traversal range of VCS. We show the effectiveness of VCS using the examle of Figure where shows the new location of in Figure. First, VCS comutes the convex hull of Q = {,, } and comares it with CH(Q). The change attern matches attern V in Figure 0. Therefore, the udate to S(Q) involves both insertion and deletion. Then, VCS initializes S(Q ) to the old S(Q) resulted from alying VS in Section. and rectangle B accordingly. It also adds ( 8, mindist( 8, CH v(q ))) in to H where 8 is the closest oint to, the new location of. Table shows the contents of the minhea H. The second iteration extracts 8 and adds all of its Voronoi neighbors excet in to H as is not in the candidate region of attern V. Similarly, is extracted and all its neighbors excet is added in to H. Next, as is not dominated it remains in S(Q ). The next two iterations add 8 and 0 to S(Q ). The sixth iteration, visits, the only unvisited Voronoi neighbor of 7 in B which is subseuently removed from S(Q ) as it is dominated by S(Q ). The final four iterations of VCS also eliminate remaining oints in H as they are all dominated. No oint in the final skyline set is dominated and hence VCS returns S(Q ) as the result. The above examle shows how VCS avoids dominance checks for oints such as outside the candidate region of the identified change attern and inside the convex hull of uery oints. This roves VCS s sueriority over the naive aroach... VCS Correctness The correctness of VCS follows that of VS. The intuition is that VCS also examines all the candidate skyline oints. It also examines those old skyline oints that are now dominated with resect to the new uery set Q and removes them from the skyline.. NON-SPATIAL ATTRIBUTES One might be interested to find the skyline with resect to both static non-satial attributes of the data oints and their distances to oints of Q. For instance, the best restaurant in LA might be dominated in terms of distance to our team members but it is still in the skyline considering its rating. We show how B S, VS, and VCS can suort this variation of SSQ. Let A be a subset of non-satial at-

10 ste hea contents S(Q ) ( 8 : 9) {,,,,, } ( : ), ( 8 : 9), ( 0 : ), ( 7 : 0), {,,, ( : ), ( : ),, } ( : ), ( 8 : 9), ( 0 : ), ( 7 : 0), {,,, ( 9 : ), ( : ), ( : 0), ( : ),,, } ( : ) ( 8 : 9), ( 0 : ), ( 7 : 0), ( 9 : ), {,,, ( : ), ( : 0), ( : ), ( : ),, } ( 0 : ), ( 7 : 0), ( 9 : ), ( : ), {,,, ( : 0), ( : ), ( : ),, 8} ( 7 : 0), ( 9 : ), ( : ), ( : 0), {,,,, ( : ), ( : ),, 8, 0 } 7 ( : ), ( 7 : 0), ( 9 : ), ( : ), {,,,, ( : 0), ( : ), ( : ),, 8, 0} 8 ( 7 : 0), ( 9 : ), ( : ), ( : 0), {,,,, ( : ), ( : ),, 8, 0 } 9 ( 9 : ), ( : ), ( : 0), ( : ), {,,,, ( : ), 8, 0} 0 ( : ), ( : 0), ( : ), ( : ) {,,,,, 8, 0 } ( : ), ( : 0), ( : ), ( : ), {,,,, ( : ), 8, 0 } Table : VCS for the examle of Figure Points Size Points Size ) Hosital 0.% ) Church % ) Building.% ) School % ) Summit 7% 7) Poulated lace 8% ) Cemetery % 8) Institution % Table : USGS dataset tributes of data oints of P. Assume that S(A) is the set of skyline oints considering only the non-satial attributes in A. Also, let S(A, Q) be the skyline when both attributes in A and distances to uery oints in Q are considered. Conseuently, we have S(A) S(A, Q) and S(Q) S(A, Q). Our algorithms can easily be changed to find S(A, Q) as follows. ) We first use a general skyline algorithm to find S(A). Notice that this is a batch one-time comutation indeendent from the uery (i.e., Q). ) We modify each of our algorithms such that the domination check for each oint inside the search region and outside CH(Q) considers both its nonsatial attributes and its distances to the oints of CH v(q). ) The search region (i.e., rectangle B) is extended to examine all ossible candidate oints. The following lemma secifies the limits of the new search region. Lemma 7. Any oint farther than all oints in S(A) from all i Q, cannot be a skyline oint with resect to attributes in A and uery oints in Q. That is, S(A), i Q, D(, i ) D(, i ) / S(A, Q) Utilizing Lemma 7, we change the rectangle B to the MBR of the oints violating the above euation. This is straightforward as the region is the union of circles C( i, ) where S(A). Therefore, our B S, VS, and VCS algorithms answer SSQs when mixed with non-satial attributes. 7. PERFORMANCE EVALUATION We conducted several exeriments to evaluate the erformance of our roosed aroaches. In real-world alications with a small number of original attributes, the traditional aroach outerforms algorithms such as BNL []. Hence, we only comared our algorithms with. First, we comared both B S and VS with the aroach, with resect to: ) overall uery resonse time, and ) number of dominance checks. Moreover, we comared the disk I/O accesses incurred by the underlying R-tree index CPU cost (sec) number of accesed nodes (K) number of dominance checks (K) BS* BS VS 8 0 Q 0 number of dominance checks (K) BS* BS VS 8 0 Q a) Time b) Dominance Checks BS* BS 8 0 Q 0 CPU cost (sec) BS* BS VS 0.0% 0.0% 0.% 0.% 0.7% MBR(Q) c) I/O d) Time BS BS* VS 0.0% 0.0% 0.% 0.% 0.7% MBR(Q) number of accesed nodes (K) BS* BS 0.0% 0.0% 0.% 0.% 0.7% MBR(Q) e) Dominance Checks f) I/O Figure : Query cost vs. number of uery oints and the area covered by Q structures used by B S with those of. We evaluated all aroaches by investigating the effect of the following arameters on their erformances: ) number of uery oints (i.e., Q ), ) the area covered by MBR of Q, ) cardinality of the dataset, and ) density of the data oints. We also evaluated the erformance of VCS using a synthetic dataset of moving objects. We used a real dataset for our exeriments. The dataset is obtained from the U.S. Geological Survey (USGS) and consists of the locations of 90, 000 different businesses in the entire country. Table shows the characteristics of this dataset which is indexed by an R*-tree index with the age size of K bytes and a maximum of 0 entries in each node (caacity of the node). The index structure is based on the original attributes of the data oints (i.e., latitude and longitude) and is utilized by both and B S aroaches. VS and VCS use a re-built Delaunay grah of the entire dataset. We also used the Graham Scan algorithm [] for convex hull comutation in both VS and VCS. The exeriments were erformed on a DELL Precision 70 with Xeon. GHz rocessor and GB of RAM. We ran 000 SSQ ueries using randomly selected uery oints and reorted the average of the results. In the first set of exeriments, we varied the number of uery oints and studied the erformance of our roosed algorithms. We set the maximum MBR(Q) to 0.% of the entire dataset. Figure a deicts uery resonse time of htt://geonames.usgs.gov/