Lazy Updates: An Efficient Technique to Continuously Monitoring Reverse knn

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1 Lazy Updates: An Effiient Tehniue to Continuously onitoring Reverse k uhammad Aamir Cheema, Xuemin Lin, Ying Zhang, Wei Wang, Wenjie Zhang The University of ew South Wales, Australia ICTA, Australia {maheema, lxue, yingz, weiw, zhangw}@se.unsw.edu.au ABSTRACT In this paper, we study the problem of ontinuous monitoring of reverse k nearest neighbor ueries. Existing ontinuous reverse nearest neighbor monitoring tehniues are sensitive towards objets and ueries movement. For example, the results of a uery are to be reomputed whenever the uery hanges its loation. We present a framework for ontinuous reverse k nearest neighbor ueries by assigning eah objet and uery with a retangular safe region suh that the expensive reomputation is not reuired as long as the uery and objets remain in their respetive safe regions. This signifiantly improves the omputation ost. As a by-produt, our framework also redues the ommuniation ost in lient-server arhitetures beause an objet does not report its loation to the server unless it leaves its safe region or the server sends a loation update reuest. We also ondut a rigid ost analysis to guide an effetive seletion of suh retangular safe regions. The extensive experiments demonstrate that our tehniues outperform the existing tehniues by an order of magnitude in terms of omputation ost and ommuniation ost. 1. ITRODUCTIO Given a uery point, a reverse k nearest neighbor (Rk) uery retrieves all the data points that have as one of their k nearest neighbors. Throughout this paper, we use R ueries to refer to Rk ueries for whih k = 1. We give a more formal definition of the Rk problem in Setion. Consider the example of Fig. 1 where the nearest neighbor (the losest objet) of is o 1. However, it is not the R of beause the losest point of o 1 is not. The Rs of are o 3 and o. R has reeived onsiderable attention [1, 16, 1, 1, 1, 17, 19, 6,, 1] from database researh ommunity based on the appliations suh as deision support, loation based servie, resoure alloation, profile-based management, et. With the availability of inexpensive mobile devies, position loators and heap wireless networks, loation based Xuemin Lin is supported by Australian Researh Counil Disovery Grants (DP9877, DP8813, DP98773 and DP6668) and Google Researh Award. Wei Wang is supported by ARC Disovery Projets DP98773 and DP Permission to opy without fee all or part of this material is granted provided that the opies are not made or distributed for diret ommerial advantage, the VLDB opyright notie and the title of the publiation and its date appear, and notie is given that opying is by permission of the Very Large Data Base Endowment. To opy otherwise, or to republish, to post on servers or to redistribute to lists, reuires a fee and/or speial permission from the publisher, AC. VLDB 9, August -8, 9, Lyon, Frane Copyright 9 VLDB Endowment, AC ----//. servies are gaining inreasing popularity. An example of suh loation based servies is zhiing 1. Consider that a user needs a taxi and she sends her loation to a taxi ompany s dispath enter. The ompany notifies to a taxi for whih she is the losest passenger (the taxi is R of the user). o o o 1 o 3 Figure 1: o 3 and o are Rs of Other examples of loation based servies inlude loation based games, traffi monitoring, loation based SS advertising, enhaned 911 servies and army strategi planning et. These appliations may reuire ontinuous monitoring of reverse nearest moving objets. For instane, in reality games (e.g., BotFighters, Swordfish), players with mobile devies searh for other mobile devies in neighborhood. For example, in the award winning game BotFighters, a player gets points by shooting other nearby players via mobiles. In suh an appliation, some players may want to ontinuously monitor her reverse nearest neighbors in order to avoid being shot by other players. In the sea-battlefield, a warship may register a ontinuous R uery to monitor other warships that might seek assistanes from it and then may ontat them from time to time. Driven by suh appliations, the ontinuous monitoring of reverse nearest neighbors has been investigated and several tehniues have been proposed reently [1, 9,, ] in the light of loation-based servies. The existing ontinuous monitoring tehniues [1, 9,, ] adopt two frameworks based on different appliations. In [1], the veloity of eah objet is assumed to be expliitly expressed while [9,, ] deal with a general situation where the objet veloities may be impossible to be expliitly expressed. In this paper, our researh is based on the general situation; that is, objet veloities are not expliitly expressible. The tehniues in [9,, ] adopt a two-phase omputation. In the filtering phase, objets are pruned by using the existing pruning paradigms from [16, 17] and the remaining objets are onsidered as the andidate objets. In the verifiation phase, every andidate objet for whih the uery is its losest point is reported as the R. To update the results, at eah time-stamp, if the set of andidate objets is deteted to be unhanged then only the verifiation phase is alled to verify the results. evertheless, both the filtering and verifiation phases are reuired if one of the andidate objets hanges its loation or other objets 1

2 move into the andidate region. Similarly, a set of andidate objets is needed to be re-omputed (reall filtering) if the uery hanges its loation. Previous tehniues [9,, ] reuire expensive filtering if a uery or any andidate objet hanges its loation. Our initial experiment results show that the ost of verifiation phase is muh lower than the ost of filtering phase. In our tehniue, we assign eah uery and objet a safe region (a retangular area). The filtering phase for a uery is not reuired as long as the uery and its andidate objets remain in their orresponding safe regions. This signifiantly redues the omputation time of ontinuously monitoring Rk ueries. As a by-produt, our proposed framework also signifiantly redues the ommuniation ost in a lient-server arhiteture. In the existing tehniues, every objet reports its loation to the server at every time-stamp regardless whether uery results will be affeted or not. Conseuently, suh a omputation model reuires transmission of a large number of loation updates; doing this has a diret impat on the wireless ommuniation ost and power onsumption - the most preious resoures in mobile environment [7]. In our framework, eah moving objet reports its loation update only when it leaves the region. This signifiantly saves the ommuniation osts. Sine we hoose a simple safe region shape (a retangle), our framework an be easily integrated with the work in [7] to develop a safe region based system that supports Rk, range and k ueries Contributions. Below, we summarize our ontributions: 1. We present a framework for ontinuously monitoring R together with a novel set of effetive pruning and effiient inrement omputation tehniues. It not only redues the total omputation ost of the system but also redues the ommuniation ost.. We extend our algorithm for the ontinuous monitoring of Rk. Our algorithm an be used to monitor both mono-hromati and bihromati Rk (to be formally defined in Setion.1). 3. We provide a rigid analysis of the relationship between the omputation/ommuniation osts and the safe regions. This also guides us to effetively selet the safe regions.. Our extensive experiments demonstrate that the developed tehniues outperform the previous algorithms by an order of magnitude in terms of omputation ost and ommuniation ost. The remaining of the paper is organized as follows. In Setion, we give the problem statement, related work and motivation. Setion 3 presents the framework of our tehniues and a set of novel pruning tehniues. Setion presents our tehniues for ontinuously monitoring R ueries, as well as a rigid ost analysis. Setion gives the extension of our tehniues to multidimensional spae, to Rk, and to Bihromati Rk. The experiment results are reported in Setion 6. Setion 7 onludes the paper.. BACKGROUD IFORATIO.1 Problem Definition There are two types of Rk ueries [1] namely, monohromati and bihromati Rk ueries. Below we define both. onohromati Rk uery: Given a set of multidimensional points P and a point P, a monohromati Rk uery retrieves points p P, dist(p,) dist(p,p k ) where dist is a distane metri that is assumed to be Eulidean distane in this paper, and p k is the kth nearest point to p in P {}. ote that, in suh ueries, both the data objets and the uery objets belong to the same lass of objets. Consider an example of the reality game BotFighters, where a player issues a uery to find other players for whom she is the losest person. Bihromati Rk uery: Given two sets O and P eah ontaining different types of objets, a bihromati Rk uery for a point O is to retrieve every objet p P suh that dist(p,) dist(p, o k ) where o k is the kth nearest point of p in O {}. In ontrast to monohromati ueries, the uery and data objets belong to two different lasses. Consider the example of a battlefield where a medial unit might issue a bihromati R uery to find the wounded soldiers for whom it is the losest medial unit.. Related Work First, we present pruning tehniues for snapshot R ueries. Snapshot R ueries report the results only one and do not reuire ontinuous monitoring. Snapshot R Queries: Korn et al. [1] were first to study R ueries. They answer R uery by prealulating a irle for eah data objet p suh that the nearest neighbor of p lies on the perimeter of the irle. R of a uery are the points that ontain in its irle. Tehniues to improve their work were proposed in [, 1]. g d S 3 e 6 o b S 6o a S 1 S S S 6 f g d S 3 e b S a S 1 S S S 6 Figure : Six-regions Figure 3: Filtering and pruning verifiation First work that does not need any pre-omputation was presented by Stanoi et al. [16]. They solve R ueries by partitioning the whole spae entred at the uery into six eual regions of 6 eah (S 1 to S 6 in Fig. ). It an be proved that only the nearest point to in eah partition an possibly be the R. This also means that, in twodimensional spae, there are at most six possible Rs of a uery. Consider the region S 3 where is the nearest objet to and d annot be the R beause its distane to is smaller than its distane to. This an be proved by the triangle d where d 6 and d 6, hene dist(d,) dist(d,). Fig. 3 shows the area (shown shaded) that annot ontain R of. In filtering phase, the andidate R objets (a, b,, e and f in our example) are seleted by issuing nearest neighbor ueries in eah region. In verifiation phase, any andidate objet for whih is its nearest neighbor is reported as R (a and f). In this paper, we all this approah six-regions pruning approah. Tao et al. [17] use the property of perpendiular bisetors to answer Rk ueries. Consider the example of Fig., where a bisetor between and is shown that divides the f

3 spae into two half-spaes (the shaded half-spae and the white half-spae). Any point that lies in the shaded halfspae H : is always loser to than to and annot be the R for this reason. Their algorithm prunes the spae by the half-spaes drawn between and its neighbors in the unpruned region. Fig. shows the example where half-spaes between and a, and f (H a:, H : and H f:, respetively) are shown and the shaded area is pruned. Then, the andidate objets (a, and f) are verified as Rs if is their losest objet. We all this approah half-spae pruning approah. It is shown in [17] that the half-spae pruning is more powerful than the six-regions pruning and it prunes larger area (ompare the shaded areas of Fig. 3 and Fig. ). d g e H : b f a d H : g e b f a H a: Figure : Half-spae Figure : Filtering and pruning verifiation Wu et. al [1] propose an algorithm for Rk ueries in d-spae. Instead of using bisetors to prune the objets, they use a onvex polygon obtained from the intersetion of the bisetors. Any objet that lies outside the polygon an be pruned. Continuous R Queries: Computation-effiient monitoring of ontinuous range ueries [, 11], nearest neighbor ueries [13, 7, 3, 8, 18] and reverse nearest neighbor ueries [1,, 9, ] has reeived signifiant attention. Although there exists work on ommuniation-effiient monitoring of range ueries [7] and nearest neighbor ueries [7, 1], there is no prior work that redues the ommuniation ost for ontinuous R ueries. Below, we briefly desribe the R monitoring algorithms that improve the omputation ost. Benetis et al. [1] presented the first ontinuous R monitoring algorithm. However, they assume that veloities of the objets are known. First work that does not assume any knowledge of objets motion patterns was presented by Xia et al. []. Their proposed solution is based on the six-regions approah. Kang et al. [9] proposed a ontinuous monitoring R algorithm based on the half-spae pruning approah. Consider the examples of Fig. 3 and Fig., the results of the R uery may hange in any of the following three senarios: H f: 1. the uery or one of the andidate objets hanges its loation. the nearest neighbor of a andidate objet is hanged (an objet enters or leaves the irles shown in Fig. 3 and Fig. ) 3. an objet moves into the unpruned region (the areas shown in white in Fig. 3 and Fig. ) Xia et al. [] use this observation and propose a solution for ontinuous R ueries based on the six-regions approah. They answer R ueries by monitoring six pieregions (the white areas in Fig. 3) and the irles around the andidate objets that over their nearest neighbors. Kang et al. [9] use the same observation and propose a solution based on the half-spae pruning approah. They ontinuously monitor R ueries by monitoring the unpruned region (white area in Fig. ) and the irles around the andidate objets that over their nearest neighbors. Both the approahes use a grid struture to store the loations of the objets and ueries. They mark the ells of the grid that lie or overlap with the area to be monitored. Any objet movement in these ells triggers the update of the results. To the best of our knowledge, there exists only one solution for ontinuous monitoring of Rk ueries [] whih is similar to the six-regions based R monitoring presented in []. Wu et al. [] issue k nearest neighbor (k) ueries in eah region instead of single nearest neighbor ueries. The ks in eah region are the andidate objets and they are verified if is one of their k losest objets. To monitor the results, for eah andidate objet, they ontinuously monitor the irle around it that ontains k nearest neighbors. It is important to note that the problem of Rk ueries is different from all-nearest neighbor ueries [] where nearest neighbors of every objet in a given dataset is to be found from another dataset..3 otivation Both the six-regions and the half-spae based solutions have two major limitations. 1. As illustrated in the three senarios presented in the previous setion, the existing tehniues are sensitive to objet movement. If a uery or any of its andidate objets hanges its loation, filtering phase is alled again whih is omputationally expensive. For example, if a uery is ontinuously moving, at eah timestamp both of the approahes will have to ompute the results from srath. For example, in the half-spae based approah, the half-spaes between and its previous andidates are redrawn and the pruning area is adjusted. In our initial experiments, we find that the ost of redrawing the half-spaes (and marking and unmarking the relevant ells) is omputationally almost as expensive as the initial omputation of the results.. The previous tehniues reuire every objet to report its exat loation to the server at every timestamp regardless whether it affets the uery result or not. This has a diret impat on the two most preious resoures in mobile environment, wireless ommuniation ost and power onsumption. Ideally, only the objets that affet the uery results should report their loations. For example, in Fig., as long as objets d, e and g do not enter into the white region or the three irles, they do not affet the results of the uery. otivated by these, we present a framework that provides a omputation and ommuniation effiient solution. ote that, in some appliations, the lients may have to periodially report their loations to the server for other types of ueries. In this ase, saving the ommuniation ost is not possible. evertheless, our framework signifiantly redues the omputation osts for suh appliations. 3. FRAEWORK Eah moving objet and uery is assigned a safe region of a retangular shape. Although other simple shapes (e.g., In rest of the paper, we present our tehniue assuming that the lients send their loations only for the Rk uery. For the ase when the lients periodially send their loations for other types of ueries, our tehniues an be easily applied. The only hange is that the safe regions are stored on the server whih ignores the loation updates from the objets that are still in their safe regions. Experiments show superiority of our approah for both of the ases.

4 irles) ould be used as safe regions, we hoose the safe region of a retangular shape mainly beause defining effetive pruning rules is easier for the retangular safe regions. The lients may use their motion patterns to assign themselves better safe regions. However, we assume that suh information is not utilized by the lients or the server beause we do not assume any knowledge about the motion pattern of the objets. In our framework, the server reommends the side lengths of the safe regions (a system parameter) to the lients. A lient assigns itself a new safe region suh that it lies at the enter of the safe region. An objet reports its loation to the server only when it moves out of its safe region. Suh updates issued by the lients (objets) are alled soure-initiated updates [7]. In order to update the results, the server might need to know the exat loation of an objet that is still in its safe region. The server sends a reuest to suh objet and updates the results after reeiving its exat loation. Suh updates are alled server-initiated updates [7]. If an objet stops moving (e.g., a ar is parked), it notifies the server and the server redues its safe region to a point until it starts moving again. In the previous approahes [, 9], the pruned area beomes invalid if the uery point hanges its loation. On the other hand, in our framework, the uery is also assigned with a safe region and the pruned area remains valid as long as the uery and its andidate objets remain in their respetive safe regions and no other objet enters in the unpruned region. Although the uery is also assigned with a safe region, it reports its loation at every timestamp. This is beause its loation is important to ompute the exat results and a server-initiated update would be reuired (in most of the ases) if it does not report its loation itself. oreover, the number of ueries in the system is usually muh smaller than the number of objets. Hene, the loation updates by the ueries do not have signifiant effet on the total ommuniation ost. Table 1 defines the notations used throughout this paper. otation Definition B x: a perpendiular bisetor between point x and H x: a half-spae defined by B x: ontaining point x H :x a half-spae defined by B x: ontaining point H a:b H :d intersetion of the two half-spaes A[i] value of a point A in the i th dimension maxdist(x, y) maximum distane between x and y (eah of x and y is either a point or a retangle) mindist(x, y) minimum distane between x and y (eah of x and y is either a point or a retangle) R fil, R nd, R retangular region of the filtering objet, andidate objet and uery, respetively R H [i] highest oordinate value of a retangle R in i th dimension R L [i] lowest oordinate value of a retangle R in i th dimension Table 1: otations Like existing work on ontinuous spatial ueries [13, 9, ], we assume that the errors due to the measuring euipments are insignifiant and an be ignored. Our ontinuous monitoring algorithm onsists of the following two phases. Initial omputation: When a new uery is issued, the server first omputes the set of andidate objets by applying pruning rules presented in Setion 3.1. This phase is alled filtering phase. Then, for eah andidate objet, the server verifies it as Rk if the uery is one of its k losest points. This phase is alled verifiation phase. Continuous monitoring: The server maintains the set of andidate objets throughout the life of a uery. Upon reeiving loation updates, the server updates the andidate set if it is affeted by some loation updates. Otherwise, the server alls verifiation module to verify the andidate objets and reports the results. 3.1 Pruning Rules To the best of our knowledge, we are first to present novel pruning rules for R ueries that an be applied when loations of the objets are unknown within their retangular regions. These pruning rules an also be applied on the minimum bounding retangles of the spatial objets that have irregular shapes (in ontrast to the assumption that the spatial objets are points). In Setion, we extend the pruning rules for Rk ueries. In this setion, an objet that is used for pruning other objets is alled a filtering objet and the objet that is being onsidered for pruning is alled a andidate objet Half-spae Pruning First, we present the hallenges in defining this pruning rule by giving an example of a simpler ase where the exat loation of a filtering objet p is known but the exat loation of is not known on a line (shown in Fig. 6). Any objet x annot be the R of if mindist(x, ) dist(x, p) where mindist(x, ) is the minimum distane of x from the line. Hene, the boundary that defines the pruned area onsists of every point x that satisfies mindist(x, ) = dist(x, p). ote that for any point x in the spae on the right side of the line L, mindist(x, ) = dist(x,). Hene, in the spae on the right side of the line L, the bisetor between p and the point satisfies the euation of the boundary (beause for any point x on this bisetor dist(x, ) = dist(x, p)). Similarly, on the left side of L, the bisetor between p and satisfies the ondition. In the area between L and L, a parabola (shown in Fig. 6) satisfies the euation of the boundary. Hene the shaded area defined by the two half-spaes and the parabola an be pruned. ote that the intersetion of half-spaes H p: and H p: does not define the area orretly. As shown in Fig. 6, a point p lying in this area may be loser to than to the point p. H p: L parabola p' L H p: Figure 6: The exat loation of the point on line is not known p H p: L B L A p H p: Figure 7: Approximation of parabola by a line Unfortunately, the pruning of the shaded area may be expensive due to presene of the parabola. One solution is to approximate the parabola by a line AB where A is the intersetion of H p: and L and B is the intersetion of H p: and L. Fig. 7 shows the line AB and the pruned area is shown shaded. Another solution is to move the half-spaes H p: and H p: suh that both pass through a point that satisfies mindist(, ) dist(, p) (e.g., any point lying in the shaded area of Fig. 6). This approximation of the pruning area is tighter if the point lies on the boundary. Fig. 8 shows the half-spaes H p: and H p: moved to suh point.

5 A half-spae that is moved is alled normalized half-spae and a half-spae H p: that is moved is denoted as H p:. Fig. 8 shows the normalized half-spaes H p: and H p: and their intersetion an be pruned (the shaded area). Among the two possible solutions disussed above, we hoose normalized half-spaes in developing our pruning rules for the following reason. In our relatively simple example, the number of half-spaes reuired to prune the area by using the normalized half-spaes is two (in ontrast to three lines for the other solution). The differene between this number beomes signifiant when both the uery and the filtering objet are represented by retangles espeially in multidimensional spae. This makes the pruning by normalized half-spaes a less expensive hoie. ow, we present our pruning rule that defines the pruned area by using two half spaes in two dimensional spae and d half-spaes for d-dimensional spae when d >. This pruning rule uses the normalized half-spaes between d seleted orners of the two retangles to prune the spae. Below, we give a formal desription of our pruning rule in d dimensional spae. Then, we briefly desribe the reason of its orretness in two dimensional spae. First, we define the following onepts: Antipodal Corners Let C be a orner of retangle R1 and C be a orner in R, the two orners are alled antipodal orners 3 if for every dimension i where C[i] = R1 L[i] then C [i] = R H[i] and for every dimension j where C[j] = R1 H[j] then C [j] = R L[j]. For example, in two dimensional spae, a lower-left orner of R1 is the antipodal orner of the upper-right orner of R. Fig. 9 shows two retangles R1 and R. The orners B and are two antipodal orners. Similarly, other pairs of antipodal orners are (D, O), (C, ) and (A, P). Antipodal Half-Spae A half-spae that is defined by the bisetor between two antipodal orners is alled antipodal half-spae. Fig. 9 shows two antipodal half-spaes H :B and H O:D. H p: H p: H p: p H p: Figure 8: Defining pruned region by moving half-spaes C D B H :B R 1 A P O H :B R H O:D H O:D Figure 9: Antipodal orners and normalized half-spaes ormalized Half-Spae Let B and be two points in hyper-retangles R1 and R, respetively. The normalized half-spae H :B is a spae defined by the bisetor between and B that passes through a point suh that [i] = (R1 L[i]+R L[i])/ for all dimensions i for whih B[i] > [i] and [j] = (R1 H[i]+R H[j])/ for all dimensions j for whih B[j] [j]. Fig. 9 shows two normalized (antipodal) halfspaes H :B and H O:D. The point for the two half-spae is also shown. The ineualities (1) and () define the half-spae H :B and its normalized half-spae H :B, respetively. dx (B[i] [i]) x[i] < i=1 dx i=1 (B[i] [i])(b[i] + [i]) 3 R L[i] (R H[i]) is the lowest (highest) oordinate of a hyperretangle R in i th dimension (1) dx (B[i] [i]) x[i] < i=1 8 (R1 L[i] + R L[i]) dx ><, if B[i] > [i] (B[i] [i]) i=1 >: (R1 H[i] + R H[i]), otherwise () ote that the right hand side of the Euation (1) annot be smaller than the right hand side of Euation (). For this reason H B H B. ow, we present our pruning rule. Pruning Rule 1 : Let R and R fil be the retangular regions of the uery and a filtering objet p, respetively. For any point p that lies in T d i=1 H C i :C i, mindist(p, R ) > maxdist(p, R fil ) where H C i :C i is normalized half-spae between C i (the i th orner of the retangle R fil ) and its antipodal orner C i in R. Hene p an be pruned. C D B R H :C A H :B P O R fil H P:A H O:D Figure 1: Pruning area of half-spae pruning and dominane pruning A D R H P:A B C 1 R fil H :B H :C O P H O:D Figure 11: Any point in shaded area annot be R of Fig. 1 shows an example of the half-spae pruning where the four normalized antipodal half-spaes define the pruned region (the area shown shaded). The proof of orretness is non-trivial and is given in our tehnial report (Lemma ) [3]. Below, we present the intuitive justifiation of the proof. Intuitively (as in example of Fig. 8), if we draw all possible half-spaes between all points of R and R fil and move them to a point for whih mindist(, R ) maxdist(,r fil ), then the intersetion of these half-spaes orretly approximates the pruned region. Also note that in two dimensional spae, at most two normalized spaes define suh area. Consider the example of Fig. 1, where only H O:D and H :B define the pruned region (the reason is that these two have largest and smallest slopes among all other possible halfspaes). In fat, the antipodal orners are defined suh that the half-spaes having largest and smallest slopes are among the four antipodal half-spaes. oreover, the point shown in Fig. 1 satisfies mindist(, R ) = maxdist(,r fil ) beause normalized half-spaes are defined suh that lies at the middle of the line that joins the orners A and. Hene the four normalized antipodal half-spaes orretly approximate the pruned region. For ease of explanation, in Fig. 1, we have shown an example where the two retangles R and R fil do not overlap eah other in any dimension. If the two retangles overlap eah other in any dimension (as in Fig. 11), the four half-spaes do not meet at the same point. In Fig. 11, H O:D and H P:A are moved to 1 and H :C and H :B are moved to point. However, it an be verified by alulating the intersetion that the half-spaes that define the pruned region (H :B and H P:A) meet at a point that satisfies mindist(, R ) maxdist(,r fil ).

6 3.1. Dominane Pruning We first give the intuition behind this pruning rule. Consider the example of Fig. 1 again. The normalized halfspaes are defined suh that if R fil and R do not overlap eah other in any dimension then all the normalized antipodal half-spaes meet at same point. We also observe that the angle between the half-spaes that define the pruned area (shown in grey) is always greater than 9. Based on these observations, it an be verified that the spae dominated by (the dotted-shaded area) an be pruned. Formal proof is given in our tehnial report (Lemma 6) [3]. 1 R n 3 F p R fil f R nd R R R nd Figure 1: Shaded areas Figure 13: R nd an be an be pruned pruned by R 1 and R Let R be the retangular region of. We an obtain the d regions as shown in Fig. 1. Let R fil be the retangular region of a filtering objet that lies ompletely in one of the d regions. Let f be the furthest orner of R fil from R and n be the nearest orner of R from f (as shown in region 1 of Fig. 1). A point F p that lies at the entre of the line joining f and n is alled a frontier point. Pruning Rule : Any andidate objet p that is dominated by the frontier point F p of a filtering objet annot be R of. Fig. 1 shows four examples of dominane pruning (one in eah region). In eah partition, the shaded area is dominated by the frontier point of that partition and an be pruned. ote that if R fil overlaps R in any dimension, we annot use this pruning rule beause the normalized antipodal half-spaes in this ase do not meet at the same point. For example, the four normalized antipodal half-spaes interset at two points in Fig. 11. In general, the pruning power of this rule is less than that of the half-spae pruning. Fig. 1 shows the area pruned by the half-spae pruning (the shaded area) and dominane pruning (the dotted area). The main advantage of this pruning rule is that the pruning proedure is omputationally more effiient than the half-spae pruning, as heking the dominane relationship is easier etri Based Pruning Pruning Rule 3 : A andidate objet an be pruned if maxdist(r nd, R fil ) < mindist(r nd, R ) where R nd is the retangular region of the andidate objet. This pruning approah is the least expensive beause it reuires a simple distane omparison. Reall that the halfspae (or the dominane) pruning defines a region suh that any point p that lies in it is always loser to the filtering objet than to. etri based pruning heks this by a simple distane omparison. However, this does not mean that the metri based pruning has at least as muh pruning power as half-spae or dominane pruning. This is beause the half-spae and dominane pruning an trim the retangular region of a andidate objet that lies in the pruned R 1 region. It may lead to pruning of a andidate objet when more than one filtering objets are onsidered. Consider the example of Fig. 13, where two retangles R 1 and R of two filtering objets are shown. The retangle R nd annot be pruned when half-spae pruning is applied on R 1 or R alone. However, the retangle R nd an be pruned when both R 1 and R are onsidered. As in [17], we use loose trimming of the retangle by using trimming algorithm [6]. The trimming algorithm trims a part of the retangle that annot be pruned. First, R nd is pruned by the half-spaes of R 1 and the trimming algorithm trims the retangle that lies in the pruned region. The unpruned retangle R nd (shown with dotted shaded area) is returned. This remaining retangle ompletely lies in the area pruned by R so the andidate objet is pruned. ote that metri based pruning annot prune R nd. Also note that if the exat loation of a andidate objet is known (R nd is a point) and metri based pruning fails to prune the objet then half-spae pruning and dominane pruning also fail to prune the objet. Hene, half-spae pruning and dominane pruning are applied only when the exat loation of a andidate objet is not known Pruning if exat loation of uery is known If the exat loation of the uery or a filtering objet is known, previous pruning rules an be applied by reduing the retangles to points. However, a tighter pruning is possible if the exat loation of the uery is known. Below, we present a tighter pruning rule for suh ase. Pruning Rule : Let R fil be a hyper-retangle and be a uery point. For any point p that lies in T d i=1 HC i: (C i is the i th orner of R fil ), dist(p,) > maxdist(p,r fil ) and thus p annot be the R of. Proof. aximum distane between a retangle R fil and any point p is the maximum of distanes between p and the four orners, i.e., maxdist(p,r fil ) = max(dist(p,c i)) where C i is the i th orner of R fil. Any point p that lies in a half-spae H Ci : satisfies dist(p,) > dist(p, C i) for the orner C i of R fil. Hene a point p lying in T d i=1 HC i:, satisfies dist(p,) > maxdist(p,r fil ). H P: H : p' R fil O P H : H O: Figure 1: Half-spae pruning when exat loation of uery is known Consider the example of Fig. 1 that shows the half-spaes between and the orners of R fil. Any point that lies in the shaded area is loser to every point in retangle R fil than to. It is easy to prove that the pruned area is tight. In other words, any point p that lies outside the shaded area may possibly be the R of. Fig. 1 shows suh point p. Sine it does not lie in H P: it is loser to than to the orner P. Hene it may be the R of if the exat loation of the filtering objet is at orner P.

7 3.1. Integrating the pruning rules Algorithm 1 is the implementation of all the pruning rules. Speifially, we apply pruning rules in inreasing order of their omputational osts (i.e., metri based pruning, dominane pruning and then half-spae pruning). While simple pruning rules are not as restriting as more expensive ones, they an uikly disard many non-promising andidate objets and save the overall omputational time. Algorithm 1 : Prune(R, S fil, R nd ) Input: R : retangular region of ; S fil : a set of filtering objets ; R nd : the retangular region of andidate objet Output: returns true if R nd is pruned; otherwise, returns false Desription: 1: for eah R fil in S fil do : if maxdist(r nd, R fil ) < mindist(r, R nd ) then // Pruning rule 3 3: : return true if mindist(r nd, R fil ) > maxdist(r, R nd ) then : S fil = S fil R fil // R fil annot prune R nd 6: if exat loation of nd is known then 7: return false // the objet annot be pruned 8: for eah R fil in S fil do 9: if R fil is fully dominated by R in a partition P then // Pruning rule 1: trim the part of R nd that is dominated by F p 11: return true if R nd is pruned 1: return 13: for eah R fil in S fil do 1: Trim using half-spae pruning // Pruning rule 1 1: return true if R nd is pruned 16: return false Three subtle optimizations in the algorithm are: 1. As stated in Setion 3.1.3, if the exat loation of the andidate objet is known then only metri based pruning is reuired. So, we do not onsider dominane and half-spae pruning for suh andidates (line 7).. If mindist(r nd, R fil ) > maxdist(r, R nd ) for a given BR R fil, then R fil annot prune any part of R nd. Hene suh R fil is not onsidered for dominane and half-spae pruning (lines -). 3. If the frontier point F p1 of a filtering objet R fil1 is dominated by the frontier point F p of another filtering objet R fil, then F p1 an be removed from S fil beause the area pruned by F p1 an also be pruned by F p. However, note that a frontier point annot be used to prune its own retangle. Therefore, before deleting F p1, we use it to prune the retangle belonging to F p. This optimization redues the ost of dominane pruning.. COTIUOUS R OITORIG In this setion, we present our R monitoring algorithm alled (Swift And Cheap) due to its omputational effiieny and ommuniation ost saving..1 Data Struture Our system has an objet table and a uery table. Objet table (uery table) stores the id and the retangular region for eah objet (uery). In addition, the uery table stores a set of andidate objets S nd for eah uery. ain-memory omputation is the main paradigm in online/real-time uery proessing [13, 9, ]. Grid struture is preferred when updates are intensive [13] beause omplex data strutures (e.g., R-tree, Quad-tree) are expensive to update. For this reason, we hoose grid-based data struture to store the loations and retangular regions of moving objets and ueries. Eah ell ontains two lists: 1) objet list; ) influene list. Objet list of a ell ontains objet id of every objet whose retangular region overlaps the ell. This list is used to identify the objets that may be loated in this ell. Influene list of a ell ontains uery ids of all ueries for whih this ell lies in (or overlaps with) the unpruned region. The intuition is that if an objet moves into this ell, we know that the ueries in the influene list of this ell are affeted. Range ueries and onstrained ueries (nearest neighbors in onstrained region) are issued to ompute Rs of a uery (e.g., six onstrained nearest neighbor ueries are issued in the six-regions based approah). In our algorithm, we also need an algorithm to searh the nearby objets in a onstrained area (the unpruned region). Several ontinuous nearest neighbors algorithms [7, 13, 3] based on grid-based index have been proposed. However, the extension of these grid-aess methods for ueries on onstrained area beomes ineffiient. i.e., the ells around ueries are retrieved even if they lie in the pruned region. To effiiently searh nearest neighbors in a onstrained area, we propose a grid-based aess method where the grid is treated as a oneptual tree. Intermediate entries Grid ells root Figure 1: Coneptual Figure 16: Illustration grid-tree of a grid of filtering phase Fig. 1 shows an example of the oneptual grid-tree of a grid. For a grid-based struture ontaining n n ells where n, the root of our oneptual grid-tree is a retangle that ontains all n n ells. Eah entry at l-th level of this grid-tree ontains (n l) (n l) ells (root being at level ). An entry at l-th level is divided into four eual non-overlapping retangles suh that eah suh retangle ontains (n l 1) (n l 1) ells. Any n-th level entry of the tree orresponds to one ell of the grid struture. Fig. 1 shows root entry, intermediate entries and the ells of grid. ote that the grid-tree does not exist physially, it is just a oneptual visualisation of the grid. The spatial ueries algorithms that an be applied on R- tree an easily be applied on the oneptual grid tree. The advantage of using this grid-tree over previously used gridbased aess methods is that if an intermediate entry of the tree lies in the pruned region, none of the ells inside it are aessed.. Initial Computation The initial omputation onsists of two phases namely filtering and verifiation. Below we disuss them in detail...1 Filtering In this phase (Algorithm ), the grid-tree is traversed to selet the andidate objets and these objets are stored in S nd. These andidate objets are also used to prune other objets. Initially, root entry of the grid-tree is inserted in a min-heap H. We try to prune every de-heaped entry e (line 6) by using the pruning rules presented in the previous setion. If e is a ell and annot be pruned, we insert the objets into heap that are in its objet list. Otherwise, if e o o R o1 o 6 o 3 o

8 is an intermediate entry of the grid-tree, we insert its four hildren into the heap H with key mindist(, R ). If e is an objet and is not pruned, we insert it into S nd. The algorithm stops when the heap beomes empty. Algorithm : Filtering 1: for eah uery do : S nd = φ 3: Initialize a min-heap H with root entry of Grid-Tree : while H is not empty do : de-heap an entry e 6: if (not Pruned(R, S nd, e)) then // Algorithm 1 7: if e is a ell in Grid then 8: for eah objet o in objet list of e do 9: insert o into H if not already inserted 1: else if e is an intermediate entry of grid-tree then 11: for eah of its four hildren do 1: insert into H with key mindist(, R ) 13: else if e is an objet then 1: S nd = S nd {e} Fig. 16 shows an example of the filtering phase. For better illustration, the grid is not shown. Objets are numbered in order of their proximity to. Algorithm iteratively finds the nearest objets and prunes the spae aordingly. In the example of Fig. 16, the algorithm first finds o 1 and prunes the spae. Sine the next losest objet o lies in the pruned spae, it is not onsidered and o 3 is seleted instead. The algorithm ontinues and retrieves o and o and the shaded area is pruned. The algorithm stops beause there is no other objet in the unpruned area (the white area). The retangles of the pruned objets are shown in broken lines. One important note is that in this phase, the all to pruning algorithm at line 6 does not onsider the exat loations of any objet or uery for pruning even if the exat loation is known. This is beause we want to find a set of andidate objets S nd suh that as long as all of them remain in their retangular regions and no other objet enters in the unpruned area, the set of andidate objets is not affeted. For example, the set of andidate objets {o 1, o 3, o, o } will not hange unless or any andidate objet moves out of its retangular region or any of the remaining objets (o and o 6) moves in the unpruned area (the white area). arking the ells in unpruned area: To uikly identify that an objet has moved into the unpruned area of a uery, eah ell that lies in the unpruned area is marked. ore speifially, is added in the influene list of suh ell. We mark these ells in a hierarhial way by using the grid-tree. For example, if an entry ompletely lies in the unpruned region, all the ells ontained by it are marked. The ells are unmarked similarly... Verifiation At this stage, we have a set of andidate objets S nd for eah uery. ow, we proeed to verify the objets. Sine every uery reports its loation to the server at every timestamp, we an use its loation to further refine its S nd. ore speifially, any objet o S nd annot be the R of for whih mindist(o, ) maxdist(o,o ) for any other o S nd. If the objet annot be pruned by this distane based pruning, we try to prune it by using pruning rule. For every uery, its andidate objets that annot be pruned are stored in a list S global. The server sends messages to every objet in S global for whih the exat loation is not known. The objets send their exat loations in response. For eah uery, the list of andidate objets is further refined by using these exat loations. As noted in [16], at this stage, the number of andidate objets for a uery annot be greater than six in two dimensional spae. We verify these andidate objets as follows. Algorithm 3 : Verifiation 1: Refine S nd using the exat loation of : Reuest objets in S nd to send their exat loations 3: Selet andidate objets based on exat loation of the objets : Verify andidate objets (at most six) by issuing boolean range ueries For a andidate objet o, we issue a boolean range uery [1] entered at o with range dist(o, ). In ontrast to the onventional range ueries, a boolean range uery does not return all the objets in the range. It returns true if an objet is found within the range, otherwise it returns false. Fig. 17 shows an example, where andidate objets are o 1 to o. The objet o 3 annot be the R beause o 6 (for whih we know the exat loation) is found within the range. Similarly, o annot be the R beause the retangular region of o 6 ompletely lies within the range. The objet o is onfirmed as R beause no objet is found within the range. The only andidate objet for whih the results is undeided is o 1 beause we do not know the exat loation of objet o whih may or may not lie within the range. The server needs its exat loation in order to verify o 1. For eah uery, the server ollets all suh objets. Then, it sends messages to all these objets and verifies all undeided andidate objets upon reeiving the exat loations. o 7 o 6 o 3 o o 6 Figure 17: Illustration of verifiation phase ote that, to ompute the results of all ueries, the server issues at most two reuest streams and reeives at most two response streams..3 Continuous onitoring The set of andidate objets S nd of a uery hanges only when the uery or one of the andidate objets leaves its retangular region or when any other objet enters into the unpruned region. If S nd is not affeted, we simply all the verifiation phase to update the results. Otherwise, we have to update S nd. One approah to update S nd is to prune the area using urrent retangular regions of and its andidate objets. Any objet that is found in the unpruned region is inluded in S nd. The ells that orrespond to the old unpruned regions are unmarked and the ells that lie in or overlap with the new region are marked. In our experiments, we found that this update of S nd and grid ells is almost as expensive as omputing the S nd from srath. Below, we show that if we hoose to ompute S nd from srath, we may save omputation ost in next timestamps. When a uery or one of its andidate objets leaves its retangular region, other andidate objets may also have moved and are likely to leave their regions in next few timestamps whih will trigger the expensive filtering phase again. o 1 o o

9 Sine we have to ommuniate with these andidate objets in verifiation phase anyway, we ask them to not only send their exat loations but also their new retangles. After reeiving these new retangular regions, we ompute the results of ueries as in initial omputation. ow all the andidate objets have new safe regions and the set of andidate objets is expeted to remain unhanged for longer. Suppose that an objet o is andidate for two ueries 1 and and S nd of 1 is affeted by a loation update of any other objet o. We annot ask o to update its retangular region beause it will affet S nd of uery as well. Hene, the server only asks an objet to update its retangular region if it does not affet other ueries.. Cost Analysis In this setion, we analyse the omputation and ommuniation ost for our proposed solution. First, we present a pruning rule based on six-regions approah and ompute the ommuniation ost. Then, we show that the pruning rules used in our tehniue are superior. Hene the ommuniation ost gives an upper bound. Then, we analyse the omputation ost. Assumptions: We assume that the system ontains objets in a unit spae (extent of the spae on both dimensions is from to 1). Eah retangular region is a suare and width of eah side is w. The enters of all retangular regions are uniformly distributed. Communiation ost: Consider the example of Fig. 18 where a 6 region bounded by the angle EC is shown in thik lines. Suppose that we find a filtering objet whose retangular region R fil is fully ontained in the region. Any objet o an be pruned if dist(o, ) maxdist(r fil, ). In other words, the possible andidates may lie only in the spae defined by EC where EC is an ar and C = E = maxdist(r fil, ). Let r be the distane between and the enter of R fil. Then, maxdist(r fil, ) r + w/ where w/ is the half of the diagonal length of R fil. Sine, all objets are represented by retangular regions, any objet is possible R andidate that has its entre at a distane not greater than w/ from the region EC. So, the range beomes (r + w). Total number of andidates that overlap or lie within the region EC is π(r + w) 6 Let R be the maximum of r of all six regions, the total number of andidate objets is bounded by S nd = π(r + w) (3) The server sends reuest to all these andidate objets and reeives their exat loations. So the total number of messages 1 at this stage is bounded by 1 = π(r + w) () After reeiving the updates, the server eliminates the andidate objets that annot be the R (based on their exat loations). As proved in [16], the number of andidate objets annot be greater than six. Hene, the server needs to verify those six andidate objets. In order to verify a andidate objet o, the server issues a range uery of distane dist(o, ) entered at o. In worst ase, all the objets that lie within this range must report their exat loations. Total number of objets that overlap or lie within the range is π(dist(o,) + w/ ) Sine these andidate objets belong to the nearest neighbors in eah region, dist(o, ) orresponds to the distane of losest objet in the region. For all six regions, the maximum of dist(o, ) is the distane of sixth nearest neighbor from (assuming uniform distribution). So the maximum range is the radius of a irle around that ontains six objets. As we assume a unit spae, the radius of suh irle 6 that ontains six objets is. So the maximum number π of messages reuired to verify all six andidate objets is r 6 = 6 π( π + w/ ) 1 + are the messages reuired to retrieve the serverinitiated updates. Let 3 be the number of soure-initiated updates (the objets that leave their retangular regions). Let v be the average speed of objets. An objet starting at enter of the suare of width w and moving with speed v will take at least w/v time to leave the region. So, total number of updates 3 at eah timestamp is 3 = min( v w,1) ote that the euation bounds the number of soureinitiated updates by. The total ommuniation ost per timestamp ( ) where 1 denotes the loation update of the uery. ote that if w is small, the number of soure-initiated updates 3 inreases and if w is large, the number of server-initiated updates ( 1 + ) inreases. ow, we find R. ote that to use the pruning of Fig. 18, we had assumed that R fil ompletely lies in the 6 degree region EC. Hene r in Euation (3) orresponds to the distane of the losest objet in eah region that ompletely lies in it. Similarly, R is the maximum of r of eah region. D C B E H D G K L E R fil C r' H J H D: A B F w A G R fil r F Figure 18: Half-spae pruning vs six-regions based pruning Figure 19: An objet ompletely lying in the 6 degree region Fig. 19 shows a region DE and a retangular region R fil of a filtering objet (shown in broken line). ote that any retangular region of side length w with enter lying in ABC (the shaded area) will ompletely lie in the region DE. In other words, r orresponds to the losest objet of in the region that has enter lying in ABC. Let r = H = J as shown in Fig. 19. Let the radius belonging to area A be r. The radius r an be omputed as r = r A where A = G + GA = G + w/. The length of G =.866w whih an be found by the triangle FG where FG = w/ and GF = 6. Hene r = r 1.366w. It an be verified that when r = 6 π w, then π(r ) = 6. In other words when radius is r, one objet in eah region will be found suh that it ompletely lies in the region. So 1 an be rewritten as