This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

Transcription

1 Thi article appeared in a journal publihed by Elevier. The attached copy i furnihed to the author for internal non-commercial reearch and education ue, including for intruction at the author intitution and haring with colleague. Other ue, including reproduction and ditribution, or elling or licening copie, or poting to peronal, intitutional or third party webite are prohibited. In mot cae author are permitted to pot their verion of the article (e.g. in Word or Tex form) to their peronal webite or intitutional repoitory. Author requiring further information regarding Elevier archiving and manucript policie are encouraged to viit:

2 Computational Geometry 43 (2010) Content lit available at ScienceDirect Computational Geometry: Theory and Application Approximation algorithm for the kinetic robut K -center problem Sorelle A. Friedler,1, David M. Mount 2 Department of Computer Science, Univerity of Maryland, College Park, MD 20742, USA article info abtract Article hitory: Received 19 January 2009 Received in revied form 12 January 2010 Accepted 13 January 2010 Available online 18 January 2010 Communicated by P. Agarwal Keyword: Kinetic data tructure Robut tatitic Clutering Approximation algorithm Two complication frequently arie in real-world application, motion and the contamination of data by outlier. We conider a fundamental clutering problem, the k-center problem, within the context of thee two iue. We are given a finite point et S of ize n and an integer k. Inthetandardk-center problem, the objective i to compute a et of k center point to minimize the maximum ditance from any point of S to it cloet center, or equivalently, the mallet radiu uch that S can be covered by k dikofthiradiu.in the dicrete k-center problem the dik center are drawn from the point of S, andinthe abolute k-center problem the dik center are unretricted. We generalize thi problem in two way. Firt, we aume that point are in continuou motion, and the objective i to maintain a olution over time. Second, we aume that ome given robutne parameter 0 < t 1 i given, and the objective i to compute the mallet radiu uch that there exit k dik of thi radiu that cover at leat tn point of S. We preent a kinetic data tructure (in the KDS framework) that maintain a (3 + ε)- approximation for the robut dicrete k-center problem and a (4 + ε)-approximation for the robut abolute k-center problem, both under the aumption that k i a contant. We alo improve on a previou 8-approximation for the non-robut dicrete kinetic k-center problem, for arbitrary k, and how that our data tructure achieve a (4+ε)-approximation. All thee reult hold in any metric pace of contant doubling dimenion, which include Euclidean pace of contant dimenion Elevier B.V. All right reerved. 1. Introduction In the deign of algorithm for optimization problem in real-world application, it i often neceary to conider the problem in the preence of complicating iue. We conider two uch iue here. The firt i the proceing of kinetic data, that i, object undergoing continuou motion. Application involving kinetic data are omnipreent, including, for example, particle-baed imulation in phyic and monitoring the motion of moving object uch a automobile, cell phone uer, mobile enor, or object carrying RFID tag. The econd confounding iue arie when the data i heterogeneou and the tatitical trend of the majority may be obcured by the deviant behavior of a minority of point, called outlier. The objective i to produce a olution to ome given optimization problem that i robut to corruption due to outlier. Thee two iue have been conidered individually in the context of kinetic data tructure and robut tatitic, repectively, but no publihed work to date ha involved the combination of the two. The combination of thee two iue preent unique challenge. One reaon i the different effect thee two iue have on the tructure of algorithmic o- * Correponding author. addree: orelle@c.umd.edu (S.A. Friedler), mount@c.umd.edu (D.M. Mount). 1 The work of Sorelle Friedler ha been upported in part by the AT&T Lab Fellowhip Program. 2 The work of David Mount ha been upported in part by the National Science Foundation under grant CCR and the Office of Naval Reearch under grant N /$ ee front matter 2010 Elevier B.V. All right reerved. doi: /j.comgeo

3 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) lution. Kinetic data tructure are typically concerned with handling local propertie involving the interaction of a mall number of object. On the other hand, algorithmic olution in robut tatitic involve global characteritic of the data et, uch a identifying which ubet of point contitute the majority. In thi paper we conider a well-known clutering problem, called the k-center problem, in the context of kinetic data and outlier. We refer to it a the kinetic robut k-center problem. Many framework have been propoed for handling kinetic data [26,4,38,39]. We aume a common model for proceing point in motion, called kinetic data tructure (KDS), which wa propoed by Bach, Guiba, and Herhberger [5]. In thi model the motion of each point i given by a piecewie function of contant algebraic degree, called a flight plan. KDStrack pecific propertie of moving point. Thi i done through a et of boolean condition, called certificate, and a correponding et of update rule. Certificate guarantee geometric relation neceary to a particular problem olution, and update rule pecify how to repond whenever a certificate fail. The KDS framework ha become the tandard approach for computing dicrete tructure for kinetic data et becaue it provide a general and flexible framework for the development of algorithmic olution that are both provably correct and efficient. Example include maintaining convex hull [5], Voronoi diagram [2], and minimum panning tree on geometric graph [6]. There are four criteria under which the computational cot of a KDS i evaluated: reponivene, efficiency, compactne, and locality [24,25]. Reponivene meaure the complexity of the cot to repair the olution after a certificate fail. Efficiency meaure the number of certificate failure a compared to the number of required change to the olution a the point move. Compactne meaure the ize of the certificate et. Locality meaure the number of certificate in which each point participate. Guiba provide a more detailed overview of kinetic data tructure in [24,25]. The other iue of interet in thi paper i robutne. The tudy of tatitical etimator that are inenitive to outlier i the domain of robut tatitic [36]. Robut tatitic have been extenively tudied in mathematic, operation reearch, and computer cience. Robutne in the preence of outlier i important, for example, when conidering heterogeneou population that contain iolated and unuual data point. It i alo ueful when conidering buine and public-ervice application. Since cot i an important factor, it i deirable to provide a ervice to a large egment of the population, while limiting expenive ervice cot involving a mall fraction of outlier. Charikar et al. [8] explored the robut facility location problem, which determine the location of tore while minimizing the ditance from cutomer to the tore and the total cot of opening facilitie. In uch a model it i unprofitable to open a new facility to ervice a mall number of iolated cutomer. Recently, Degener et al. [14] gave a determinitic algorithm for the kinetic verion of the non-robut facility location problem and Agarwal and Phillip [1] gave a randomized algorithm for the robut 2-center problem with an O(nk 7 log 3 n) expected execution time. We conider a clutering problem that involve a combination of thee two important element. Clutering i a frequently tudied problem in operation reearch and computer cience. Common formulation include k-center, k-mean, and facility location problem [30,29,17,35]. The (tandard) k-center problem i defined a follow: Given a et of n point, find k center point that minimize the maximum ditance (called the radiu) from any point to it cloet center. In the dicrete verion the center mut be drawn from the original n point. In contrat, in the abolute verion the center may be arbitrary point in pace [30]. Unle otherwie tated, we will aume the dicrete verion of the problem. Kariv and Hakimi [30] proved that the dicrete and abolute verion of the k-center problem are NP-hard in a graphtheoretic context (for arbitrary k). The problem of finding a (2 ε)-approximation for the dicrete k-center problem i NP-complete (alo in a graph-theoretic context) [28,34]. The problem in the Euclidean metric cannot be approximated to within a factor of (auming P NP) [17]. Demaine et al. [15] give algorithm for the k-center problem on planar graph and map graph that achieve time bound exponential in the radiu and k. Since the k-center problem i NP-hard for arbitrary k or exponential in k for fixed k, we conider approximation algorithm. An algorithm provide a c-approximation to the k-center problem if the radiu choen for the k centerinomore than c time the optimal radiu. Feder and Greene [17] gave a 2-approximation for the geometric k-center problem and Hochbaum and Shmoy [29] and Gonzalez [21] gave 2-approximation algorithm for the graph-theoretic verion of the k- center problem, both for arbitrary k. In light of the above lower bound of (2 ε), thee approximation algorithm provide the bet poible approximation bound. The robut k-center problem generalize the k-center problem to handle outlier by allowing flexibility in the number of point that atify the ditance criteria. In our formulation we are given a et of n point, an integer k, and a threhold parameter t, where 0 < t 1. The objective i to compute the mallet radiu r uch that there exit k dik of radiu r that cover at leat tn point. The non-robut verion arie a a pecial cae, when t = 1. Since the robut k-center problem i a generalization of the non-robut verion, the approximation lower bound [17] for the Euclidean context hold for the robut k-center problem a well (auming P NP). Charikar et al. [8] howed that in the graph-theoretic context, the robut k-center problem with forbidden center (in which ome location cannot be choen a center), ha a lower bound of 3 ε. They alo gave a 3-approximation algorithm for the robut k-center problem. Recently, Chen gave a contant factor approximation algorithm for the robut k-median problem [9]. The (non-robut) kinetic k-center problem i a generalization of the tatic verion, o again the approximation lower bound [17] hold. No other lower bound are known for the kinetic problem. Gao, Guiba, and Nguyen [18] give an 8-approximation algorithm for the kinetic dicrete k-center problem. Har-Peled handle the dicrete and abolute kinetic k-center problem with an O(nk) time algorithm, which create a larger tatic et of center that i competitive at any time [27].

4 574 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) The kinetic robut k-center problem ha not been tudied before, but many application domain involving moving point benefit from robut clutering calculation. Thee include the egmentation problem in viion, which attempt to eparate meaningful part of a moving image [7,16,40]; context-aware application, which run on mobile device that are carried by individual and interact with the environment and each other [41]; and traffic detection and management, which we ue a our main motivating example. Traffic detection ha been tudied extenively with tactic that include uing enor [31,23,37], knowledge-baed ytem [12], and individual vehicle monitoring (e.g., car GPS navigation ytem) [19,3,20]. Our model aume individual vehicle monitoring with the aumption of a flight plan provided by the navigation ytem and a deired number, k, of congeted area to monitor. Throughout thi paper, we will aume that the point et S reide in a pace of contant doubling dimenion. Wedefine the dik of radiu r centered at point u to be the et of point of S whoe ditance from u i le than or equal to r. A metric pace i aid to have contant doubling dimenion if any metric dik of radiu r can be covered by at mot a contant number, λ, of dik of radiu r/2. Euclidean pace of contant dimenion i an example. The doubling dimenion d i defined to be log 2 λ [32]. To generalize the concept of a metric pace to a kinetic context, we aume acce to function giving the ditance between two point at a given time and the earliet future time at which two point will be within ome given ditance Contribution A mentioned above, we preent the firt concurrent conideration of two practical domain, robut tatitic and kinetic data tructure, and an approximation algorithm and correponding efficient kinetic data tructure to olve the kinetic robut k-center problem. Our algorithm approximate the tatic k-center, kinetic k-center, and robut k-center problem a well, ince all are pecial cae of our problem. The input conit of a kinetic point et S in a metric pace of contant doubling dimenion d, the number of center k, a robutne threhold 0 < t 1, and an approximation parameter ε > 0. Some of our complexity bound depend on the apect ratio of the point et, which i defined a follow in a kinetic context. Let d min and d max be lower and upper bound, repectively, on the ditance between any two point over the entire motion. The apect ratio, denoted by α, i defined to be d max /d min.weobtaina(3 + ε)-approximation for the tatic and kinetic form of the robut dicrete k-center problem and a (4 + ε)-approximation for the abolute verion of the robut k-center problem. Note that the firt bound improve upon the 8-approximation for the kinetic dicrete k-center problem a given by Gao, Guiba, and Nguyen [18] and generalize it to the robut etting. However, due to complication ariing from the need for robutne, our reult aume that k i contant, while their hold for arbitrary k. We improve their reult for the non-robut kinetic problem for arbitrary k by howing that our data tructure achieve a (4 + ε)-approximation, while maintaining the ame quality bound a their KDS (ee Section 5). To our knowledge, our kinetic robut algorithm i the firt approximation algorithm for the kinetic abolute k-center problem (even ignoring robutne). We give an example in Section to how that our (3 + ε)-approximation for the robut dicrete k-center problem i tight. The KDS ued by our algorithm i efficient. In Section 4.3 we will etablih bound of O((log α)/ε d ) for locality and O(n/ε d+1 ) for compactne. Our reponivene bound i O((log n log α)/ε 2d ), implying that the data tructure can be updated quickly. Our efficiency bound of O(n 2 (log α)/ε) i reaonable ince the combinatorial tructure upon which our kinetic algorithm i baed require Ω(n 2 ) update in the wort cae (even for the non-robut cae) [18], o any approach baed on thi tructure require Ω(n 2 ) update. 2. Weak hierarchical panner Our approach i to extend a panner contruction for kinetic data tructure developed by Gao et al. [18], which they call a deformable panner. Thi kinetic tructure i defined auming a point et S in R d for any fixed d, but the contruction generalize eaily to any metric pace of contant doubling dimenion. The panner i baed on a hierarchical clutering involving a pare ubet of point, called center. To avoid confuion with the ue of the term center a a cluter center, henceforth we ue the term node for a point in the dicrete hierarchy, and the term center when referring to the center of a dik in the olution to the k-center problem. We will ue the term point to refer to an element of S. Each node i aociated with a point of S. Becaue of the cloe relationhip between node and the aociated point, we will ometime blur thi ditinction, for example, by referring both to a node u in the panner and point u in S, or referring to the ditance between two node (by which we mean the ditance between their aociated point). Given a point et S, ahierarchy of dicrete center i a equence of ubet S 0 S 1 S m uch that the following propertie hold for 0 i m: S 0 = S and S m =1. For i 1, each node of S i 1 i within ditance 2 i of ome node in S i,theith level of the hierarchy. For any two node u, v S i, with aociated point u, v S, uv 2 i. By definition, for each node v in level i 1, there exit a node u in level i uch that v i within ditance 2 i of u. One uch node u i elected (arbitrarily) to be v parent (and v i a child of u). We ue other tandard tree relationhip

5 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) including ancetor and decendant (both of which we conider in the improper ene, o that a node i an ancetor and decendant of itelf) and ibling [11]. Some propertie about the deformable panner, which follow immediately from the above propertie or are proven in [18], are given below: The hierarchy ha a height O(log α). Any node in S 0 i within ditance 2 i+1 of it ancetor in level S i. Gao et al. [18] howed that the hierarchy of dicrete center could be ued to define a panner for the point et S. Given a parameter γ 1, called the tretch factor, aγ -panner for a point et i a graph where the point are the vertice and the edge et ha the property that the hortet path length in the graph between any two point i at mot γ time the metric ditance between thee point. Given a uer-upplied parameter c > 4, two node u and v on level i of the hierarchy are aid to be neighbor if they lie within ditance c 2 i of each other. For each pair of neighboring node in the hierarchy, an edge i created between their aociated point. Gao et al. how that the reulting graph i a panner for S, and they etablih a relationhip between the value of c and the reulting tretch factor. Throughout, we will aume that c = 8, which implie that the reulting graph i a 5-panner. Although we will ue the term panner when referring to our tructure, we will not be making ue of panner propertie directly in our reult. The KDS preented in [18] for the deformable panner i hown to be efficient, local, compact, and reponive. The algorithm maintain four type of certificate: parent-child certificate, edge certificate, eparation certificate, and potential neighbor certificate. We will ue thee ame certificate in our algorithm, but with different update rule. There i one additional difference between our tructure and that of Gao et al. In order to obtain our tronger approximation bound, we will need to make a number of copie of thi tructure, each with lightly different parameter etting. The number of copie, which depend on the approximation parameter ε, willbedenotedby(ε), orimply whenever ε i clear from context. It value will be given in the next ection. Recall that the ith level of the hierarchy of dicrete center i naturally aociated with the ditance 2 i (both a the covering radiu and a the eparation ditance between node). The lowet level of the hierarchy i aociated with the ditance 2 0 = 1, which we call the bae ditance of the hierarchy. Each copy in our tructure will employ a different bae ditance. In particular, for 0 p <, letb p = (1 + p ). Oberve that 1 b p < 2. In our tructure, the ith level of the pth panner copy, denoted S i (p), will be aociated with the ditance b p 2 i. The neighbor of a node are defined to be thoe node within ditance c b p 2 i, rather than c 2 i. To implify our algorithm preentation, unle otherwie tated, we will aume that p = 0, and o b p = 1. There i no lo of generality in doing o, becaue an equivalent way of viewing the variation in the bae ditance i to imagine that ditance have been caled. In particular, when dealing with the pth copy, imagine that all ditance have divided by b p, and the hierarchy i then contructed on the caled point uing the default bae ditance of 1. Since the lowet level of the hierarchy, S 0 i required to atify the requirement that the ditance between any two point i at leat 2 0 b p, it will be ueful to aume that ditance have been caled uniformly o that d min = Robut K -center algorithm Gao et al. [18] gave an 8-approximation for the non-robut dicrete verion of the kinetic k-center problem(for arbitrary k). In Section 5 we improve thi to a (4 + ε)-approximation for arbitrary k. In thi ection we preent a (3 + ε)- approximation algorithm for the robut dicrete verion of thi problem and a (4 + ε)-approximation algorithm for the robut abolute verion, both for contant k. Recall that the non-robut verion i a pecial cae of the robut verion (by etting t = 1), o thee algorithm alo apply to the non-robut cae Intuitive explanation For the ake of intuition regarding ome of the more complex technical element of our algorithm we firt preent the algorithm by Charikar et al. [8] for the tatic robut dicrete k-center problem, which i the bai for our algorithm, and we explain why it cannot be applied directly in the kinetic context. Henceforth we refer to it a the expanded-greedy algorithm. The algorithm i given a input a point et S of cardinality n, a number of center k, and a robutne threhold t. Radiu value are choen in a parametric earch o that all potential optimal value are conidered. For a given target radiu r and for each point v S, wedefinethegreedy dik G v to be a dik of radiu r centered at v, and the expanded dik E v to be the dik of radiu 3r centered at v. (We ometime let G v and E v repreent the geometric dik and ometime the ubet of S contained within the dik. It will be clear from context which interpretation i being ued.) Initially all the point of S are labeled a uncovered. The algorithm repeatedly pick the node v uch that the greedy dik G v contain the mot uncovered point, and it then mark all point within the expanded dik E v a covered. If after k iteration it ucceed in covering at leat tn point, it return uccefully, and otherwie it fail. The algorithm i preented in Fig. 1. The analyi of thi algorithm approximation bound (which will be preented later in Section 3.3) ue a charging argument, where each point covered by an optimal dik i charged either to the expanded dik that cover it or to a nonoverlapping greedy dik [8]. The proof relie on two main point. Firt, greedine implie that any optimal dik that doe not overlap any greedy dik cannot cover more point than any greedy dik. Second, if an optimal dik doe overlap ome

6 576 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) expanded-greedy(s, k, t, r) n S C r (C r will hold the et of k center being created for radiu r) V S (V hold the et of candidate center) for each v V contruct G v and E v and compute count G v of uncovered point in S within ditance r of v for j = 1tok, letv j be the v V with larget G v add v j to C r and mark all point in E v a covered for all v V,update G v if at leat tn point are covered, return C r, and otherwie return failure Fig. 1. An overview of the expanded-greedy algorithm [8] for a ingle radiu value r. greedy dik G v, then the expanded dik E v cover all the point of thi optimal dik. Thi implie that optimal dik cannot be repeatedly damaged by greedy dik. To better motivate our kinetic algorithm, it will be helpful to firt conider a very imple tatic algorithm, which doe not achieve the deired approximation bound, but we will then how how to improve it. Thi initial algorithm applie the expanded-greedy algorithm to individual level of the dicrete hierarchy decribed in Section 2. It tart at the highet level of the panner and work down. For each level i, it run the expanded-greedy algorithm with radiu r = 2 i, conidering jut the node at thi level a poible center. It return the et of center aociated with the lowet level that ucceed in covering at leat tn point. There are, however, ome aumption inherent to the expanded-greedy algorithm that do not hold for thi imple tatic algorithm. Let u conider each of thee aumption and our approach for dealing with them All important radii will be conidered The proof of the expanded-greedy algorithm relie on the poibility for the algorithm to pick a node and cover all point within the optimal radiu of that node. However, in thi initial algorithm, if the optimal radiu i lightly larger than 2 i then our algorithm would be forced to chooe the center at the next higher level, nearly doubling the radiu value. A mentioned at the end of Section 2, we olve thi problem by creating multiple panner with bae ditance that vary, thu partitioning the interval between 2 i and 2 i+1 into O(1/ε) ubinterval (ee Section 4). The algorithm i then applied to all panner, and the bet reult over all i choen All point in S are candidate center Our imple tatic algorithm conider only node in level i a poible center, and o point of S that do not reide on level i are excluded from conideration. If ome of thee excluded center are in the optimal olution, then the algorithm might need to ubtitute a node at level i at ditance up to 2 i+1 from an optimal center, which would require the need for a larger radiu. It would be unacceptably low in the kinetic context to conider all the point of S. Our olution intead i to take candidate center from level i l 1, for a uitably choen l (whoe value will depend on ε). We will how that, for each optimal center, there i at leat one candidate point that i cloe enough to enable u to obtain our approximation bound. We are now able to cover all of the point covered by an optimal olution ince the optimal center i a decendant of ome center in thi lower level G v and E v are known exactly For the tatic cae, the algorithm of Charikar et al. [8] accurately count the number of point within the greedy and expanded radii of each node. However, maintaining thee count in a kinetic context would require keeping certificate between each point in S and all potential covering center. Thi would increae the compactne and locality complexitie (preented later in Section 4.3) by an unacceptable amount. The hierarchical panner tructure allow u to count the number of point in a fuzzy greedy dik in which all point within ome inner ditance are guaranteed to be counted and no point outide of ome outer ditance are counted. We call thi range ketching. Due to fuzzine, ome of the counted point may lie outide the greedy radiu, but we can increae the expanded radiu lightly o that any optimal dik affected by thi fuzzine are till fully covered by the expanded dik Precondition In thi ection we decribe the data that we maintain in our kinetic algorithm in order to produce an approximate olution to the robut k-center problem. It will implify the preentation to aume for now that the point are tatic and rely on the decription of the kinetic data tructure in Section 4 for proof that thee value are maintained correctly in the kinetic context. Recall that our contruction involve multiple copie of the panner uing variou bae ditance. From the real parameter ε > 0 we derive two additional integer parameter (ε) = 10/ε and l(ε) = 4 log 2 ε (abbreviated repectively a and l whenever ε i clear). Thee value will be jutified in the analyi appearing in the proof of Theorem 3.2. The value

7 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) repreent the number of panner we maintain, and l determine the number of level of the hierarchy that we will decend at each tep of the algorithm in order to find candidate center. Oberve that = O(1/ε) and l = log 2 (1/ε) + O(1). Our algorithm depend on a number of radiu value, each of which i a function of the level i, the panner copy 0 p < (ε), and l(ε). Givenε, i, and p, wedefinethegreedy radiu and the expanded radiu to be, repectively g i (ε, p) = 2 i ( 1 + p ) ( l ) and e i (ε, p) = 3g i (ε, p). We alo define two lightly maller radii ( g (ε, p) = 2 i 1 + p ) (1 ) i + 2 l and e (ε, p) = 3g (ε, p). i i When ε and p are clear from context, we abbreviate the value a g i, e i, g i, and e i, repectively. Given the important role of level i l 1 in our contruction, when l i clear from context, we define i = max(0, i l 1), and then ue i in thee context. Our algorithm maintain the following information: For each node u at level i of the panner, we maintain: ThepointofS aociated with u, and converely the node aociated with each point of S. The parent, children, and neighbor of u. The number of point of S that are decendant of u. For each node u at level i of the panner, we maintain: The number of point lying approximately within ditance g i of u. The number of point lying approximately within ditance e i of u. (The ene of approximation will be defined formally in Section 3.3.) For each level i in the dicrete hierarchy we maintain: A priority queue aociated with level i toring the node of S i ordered by the count of point within ditance g i (approximately) a decribed above. (The ue of thi priority queue will be clarified in Section 4.) 3.3. The dicrete problem Recall that our algorithm take a input the et S of n point and additional parameter ε (approximation parameter), k (number of center), and t (robutne threhold). Alo recall that α denote S apect ratio bound. The algorithm make ue of two parameter and l, which are both function of ε. It return a et of k center choen from S for a (3 + ε)- approximation of the optimal olution to the kinetic, robut k-center problem Algorithm overview The algorithm i applied to all panner in our tructure. For each panner, it applie a binary earch over it level, to determine the mallet covering radiu. At each level i, the k bet center for that level are calculated uing the per-level ubroutine decribed later in thi ection. If thi algorithm return in failure (meaning that it failed to cover at leat tn point of S), the binary earch continue by conidering higher level of the panner (larger covering radii); otherwie, it continue to earch through the lower level (maller radii). On termination of the binary earch, the k center reulting from the earch are tored a repreentative for that panner. The k center with minimum radiu e i out of all panner i output a the final olution Range ketching In order to maintain the count decribed a neceary precondition, we need to efficiently count the number of point of S within a fuzzy dik, which we call a range-ketch query. We are given a pair of concentric dik (B, B), where B B, and return a count including all point within B and no point outide of B [13]. Given a node v at level i,letg v and G v denote the dik centered at v of radii g i and g i, repectively, and let E v and E v denote the dik centered at v of radii e i and e, repectively. In our algorithm we will apply range-ketch querie of two type, (G i v, G v) and (E v, E v). The anwer to the query (B, B) will be repreented a a collection of panner node, all from the ame level of the panner, where the deired count i the total number of point decended from thee node. Thi collection of node will be denoted by μ(b). See Fig. 2 for an illutration. We anwer a range-ketch query by firt identifying an eaily computable uperet of node covering the query region, and then pruning thi to form the deired et of node. To determine thi uperet of μ(g v ) (or μ(e v ))wefirtdevelop the following lemma. Recall from Section 2 the concept of a node neighbor, and let c = 8 denote the parameter ued in the definition. More formally, given a ubet U of node at ome level of the panner, let N(U) = u U( {u} neighbor(u) ).

8 578 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) Fig. 2. For v S i the range-ketch query return μ(g v ), all the node of S i that lie within G v and no node with any decendant that lie outide of G v. Theenodearecircledanddikarehaded.Thepointdrawna areind(μ(g v )) and are counted for the priority queue. The dahed circle ha radiu (g i + g i )/2. Given a node v and h 0, we define it h-fold neighbor et to be: { {v} if h = 0, N (h) (v) = N(N (h 1) (v)) otherwie. Given a et of node U,letD(U) denote the et of decendant of thoe node, and let D (h) (v) = u N (h) (v) D(u). Lemma 3.1. For any node v in S i and any h 1,allthepointofSthatliewithinditanceh 2 i+3 2 i+1 of v are in D (h) (v). Proof. Under our aumption that c = 8, N(v) contain all the node in level i that are within ditance c 2 i = 2 i+3 of v. (Recall that we conider the cae p = 0.) By induction, N (h) (v) contain all the node in level i that lie within ditance h 2 i+3 of v, and thu, any node u in level i that i not in thi et i at ditance greater than h 2 i+3 from v. Recall that, from baic panner propertie, all of u decendant are within ditance 2 i+1 of u. It follow that D (h) (v) contain all the point of S within ditance h 2 i+3 2 i+1 of v. The above lemma provide a way to determine a et of node that cover all the point of S lying within a given ditance of a given node. Uing thi, we can identify a et of node at level i whoe decendant contain a uperet of the point for the range-ketch querie of interet to u. Lemma 3.2. Let h(m) = m(2 l 2 + 1). Then for any node v in S i : (i) The decendant of N h(1) (v) contain all point within ditance g i of v. (ii) The decendant of N h(3) (v) contain all point within ditance e i of v. (iii) The decendant of N h(4) (v) contain all point within ditance e i + g i of v. Proof. By traightforward manipulation (and under our aumption that p = 0), each ditance can be rewritten a follow: (i) g i = 2 i ( l ) = (2 l 2 + 1)2 (i +3) 2 (i +1). (ii) e i = 3 2 i ( l )<3(2 l 2 + 1)2 (i +3) 2 (i +1). (iii) e i + g i = 4 2 i ( l )<4(2 l 2 + 1)2 (i +3) 2 (i +1). The proof follow from Lemma 3.1 applied at level i. We will apply Lemma 3.2 a a ubroutine in our range-ketching procedure. Uing the lemma, we firt identify a et of node whoe decendant provide a uperet of the point that might contribute to the query reult (depending on the radiu of interet, g i, e i,org i + e i ), and we then prune thi et by eliminating thoe node whoe covering dik lie entirely outide the inner dik. To ee how to perform thi pruning, recall that if u i a node at level i, it decendant all lie within ditance 2 (i +1) = 2 i l of u. (Note that if i = 0, then the only decendant i u itelf.) Thu, if uv >(g + g i i )/2, then (under our aumption that p = 0) it i eay to verify (by the definition of g i and g i ) that every decendant u of u atifie u v > g i + g i 2 i l = g i 2 + g i g i 2 2 i l = g i, and o u may be omitted from the range-ketch reult. Converely, if uv (g i + g i )/2, then it i eay to verify that every decendant u of u atifie u v g i, and o u may contribute to the range-ketch reult. Given thi obervation, the code i given in Fig. 3 for the pecial cae of the greedy dik G v. It return a et μ(g v ) of node whoe decendant

9 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) range-ketch(node v, leveli) um 0 U reult of Lemma 3.2 part (i) applied to v for each u U if uv > g +g i i then U U \ u 2 ele um um + D(u) return (U, um) Fig. 3. The range-ketch and counting ubroutine for the μ(g v ) query. To anwer the μ(e v ) query, all reference to μ(g v ) change to μ(e v ), g i and g i to e i and e i repectively, and Lemma 3.2 part (ii) i ued. We will call the range-ketching routine hown here range-ketch G and the one created through thee ubtitution range-ketch E. update-greedy-dik(node v, leveli) U reult of Lemma 3.2 part (iii) applied to v (μ(e v ), E v ) range-ketch E (v, i) for each u U if uv e i + g i (μ(g u ), G u ) range-ketch G (u, i) for each w μ(e v ) if w i unmarked and w μ(g u ) G u G u D(w) update u poition in the priority queue Fig. 4. Subroutine to update greedy dik count that call the range-ketch ubroutine hown in Fig. 3. per-level-ubroutine(s, k, t, ε, p, level i) n S C p,i (C p,i ={k center being created for panner p and level i}) V S i (p) (V ={candidate center}, S i (p) ={level i of panner p}) for each v V (μ(g v ), G v ) range-ketch G (v, i) (μ(e v ), E v ) range-ketch E (v, i) for j = 1tok let v j be the v V with the larget G v C p,i C p,i {v j } update-greedy-dik(v j, i) for each uncovered w μ(e v ),markw a covered if at leat tn point are covered return (C p,i, e i ) otherwie return failure. Fig. 5. The per-level ubroutine for panner p and level i of the kinetic robut k-center algorithm. Thi ubroutine call the range-ketch and update-greedydik ubroutine hown in Fig. 3 and 4 repectively. atify the requirement of the range ketch. The deired count i the um of the number of decendant of thee node, D(μ(G v )). The procedure return both the et of node and the um. Recall from our earlier decription of the expanded-greedy algorithm, that whenever a center v i added to the olution at level i, the point lying within the expanded dik E v are marked a covered. In a kinetic etting it i too expenive to mark thee point explicitly. Our approach intead will be to modify the count aociated with each greedy dik that overlap E v. In particular, we apply range ketching to determine the node u in level i whoe greedy dik G u overlap E v, and for each unmarked node w in their common interection, we decreae G u by the weight D(w). The procedure i given in Fig. 4. To prevent node from being counted twice, we mark all thee node w after E v ha been proceed (ee Fig. 5) Main ubroutine and analyi We now have the tool needed to introduce the main ubroutine for the algorithm. Recall that the input conit of the point et S and parameter k, t, ε. The quantitie and l depend on ε and repreent the number of panner and the number of level of reolution, repectively. The current panner i indexed by 0 p <, and the current level of the dicrete hierarchy i i. The per-level ubroutine (preented in Fig. 5) calculate the candidate lit of k center for a given panner p and level i. It i called from the algorithm overview preented earlier. Before giving the kinetic verion of the algorithm, we firt decribe the tatic verion, and we focu on jut one tage of the algorithm. In the tatic context thi algorithm require preproceing to create the priority queue and perform range ketching to determine initial count for G v and E v in all level of all panner. We comment that thi can be done in time O((1/ε) d n log n log α) ince there are n point, priority queue inertion take time O(log n), center are calculated for O(log α) level, and range ketching take time O(1/ε d ) a hown by the following lemma. Lemma 3.3. Range-ketch querie for level i involving any of the ditance g i,e i or g i + e i canbeanweredintimeo(1/ε d ). Proof. The running time of a range-ketch query for a node v i dominated by the time needed to compute the et U = N h(m) (v) of node at level i identified by Lemma 3.2. By Lemma 3.1, the ditance from v to any of thee node i at mot

10 580 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) h(4)2 (i +3) 2 (i +1) 4 ( 2 l ) 2 (i +3) 16 2 (i +l). Recall that i = max(0, i l 1). Ifi = 0, the ditance to any of the identified node i O(2 l ).Ifi = i l 1, the ditance i O(2 i ). By definition of the hierarchy, the node of level i are eparated from each other by a ditance of at leat 2 i. By a tandard packing argument, thi implie that the number of uch node within any dik of radiu r i at mot O((1 + r/2 i ) d ), where d i the dimenion. If i = 0, it follow that the number of node identified in Lemma 3.2 i O((1 + 2 l /2 0 ) d ).On the other hand, if i = i l 1, the number of node i O((1 + 2 i /2 i l 1 ) d ). In either cae, the number of node i clearly O(2 l d ).Sincel = log 2 (1/ε) + O(1), it follow that the number of node i O(1/ε d ). The node of U are determined by computing the h(m)-fold neighbor et of v. We can do thi by applying h(m) level of a breadth-firt earch to the graph of neighbor. The time to do thi i proportional to the product of the number of node viited and their degree. Gao et al. [18] how that each node ha degree at mot (1 + 2c) d 1. By our aumption that c = 8 and d i fixed, thi i O(1). Thu, the total range-ketch query time i proportional to U, which i O(1/ε d ). Theorem 3.1. After preproceing ha completed, our algorithm take time O(k(log n log log α)/ε 2d ) per panner, which i O((log n log log α)/ε 2d ) under our aumption that k i a contant. Proof. We firt how that, for a ingle level of a ingle panner, the per-level ubroutine take time O(k(log n)/ε 2d ).Toee thi, oberve that it perform k iteration. During each iteration, it update the count for O(1/ε d ) greedy dik (thoe that overlap with dik E j )baedontheo(1/ε d ) node in level i that are contained in each greedy dik. Each update alo involve adjuting the poition of an entry in a priority queue holding at mot n point, which can be done in O(log n) time. Thi per-level ubroutine i invoked O(log log α) time per panner, ince there are O(log α) level in the dicrete hierarchy, over which the binary earch i performed to determine the bet radiu. Thu, the total time required to update any one panner i O(k(log n log log α)/ε 2d ).Sincek i a contant, our algorithm take time O((log n log log α)/ε 2d ). We will now etablih the approximation bound of 3 + ε. Theorem 3.2. Let r opt be the optimal radiu for the dicrete robut k-center olution for S, and let r apx be the radiu found by our algorithm. Then for any 0 < ε 1,wehaver apx (3 + ε)r opt. Proof. Let v 1,...,v k denote the k optimal center. We may expre the optimal radiu value, r opt,a2 i + x for ome integer i and 0 x < 2 i.letp = x/2 i.ifp =, eti i + 1 and p = 0 (effectively rounding up to the next panner copy). Clearly, 0 p <, and p 1 2 i < x p 2i. We firt how that it i poible, given the information we maintain and the algorithm we ue, for u to cover a many point a are covered by the optimal olution. For 1 j k, leto j denote the optimal dik of radiu r opt centered at v j.let u j denote the ancetor of v j in level i.(ifi = 0, then u j = v j.) By the baic propertie of the dicrete hierarchy we have u j v j 2 (i +1) 2 (i l). The node u j will be conidered by our algorithm (during the proceing of panner copy p and level i). Let G denote j the dik of radiu g (ε, p) centered at u i j. We aert that every point lying within O j will be included in the range-ketch count for u j.toeethi,letw be a point of S lying within O j,thati, v j w r opt. By the triangle inequality we have ( u j w u j v j + v j w 2 (i l) + r opt < 2 (i l) + 2 i + 2 i p ) ( = 2 (2 i l p )) ( 2 i 1 + p ) (1 ) + 2 l = g i (ε, p). Therefore w lie within G, the inner radiu for the range-ketch query, and o it mut be included among the point j counted in the range-ketch query for u j. In ummary, for each optimal dik O j, there i a point u j at level i of panner copy p whoe range ketch cover a uperet of S O j. To etablih the approximation bound, let u 1,...,u k denote the node choen by our algorithm when run at level i of panner copy p. (Thee are generally different from the u j decribed in the previou paragraph.) Let G j and E j denote the dik centered at u j of radii g i (ε, p) and e i (ε, p), repectively. We will how that the expanded dik centered at thee point will cover at leat a many point a the optimal olution, which i a leat tn. Since the algorithm return the mallet expanded radiu that (approximately) cover at leat tn point, thi implie that r apx e i (ε, p). Given a dik D, let D denote the number of point of S contained within it. We will how that, for 0 j k, O 1 O j E 1 E j. Our proof i imilar to that of Charikar et al. [8], and i baed on an argument that charge each

11 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) Fig. 6. An example for which our algorithm give an approximation ratio of 3 ε, wherek = 2and tn =11. point covered in the optimal olution to a point covered by the olution produced by our algorithm. The proof proceed by induction on j. The bai cae i trivial, ince for j = 0, both et are empty. Auming by induction that O 1 O j 1 E 1 E j 1, we conider what happen after the next center u j i added to the approximate olution. We conider two cae: If G j interect any of the k ( j 1) remaining optimal dik, define O j be any uch optimal dik. Any point w of O j i within ditance g i (ε, p) + 2r opt of u j. It i eay to verify that r opt g i (ε, p), and therefore u j w 3g i (ε, p) = e i (ε, p). Thu, E j cover all the point of O j. We charge each point in O j to itelf. If G j doe not interect any of the remaining O j,leto j be the remaining optimal dik covering the greatet number of point of S. By our earlier remark, there exit a node in level i whoe range-ketch count include all the point of O j.sinceg j wa choen greedily to maximize the number of unmarked point it cover, it follow that the number of unmarked point covered by G j i at leat O j. We charge each point in O j to an unmarked point in G j. Each time a point i charged, it i charged either to itelf or to ome point in a greedy dik G j that i dijoint from all remaining optimal dik, and therefore, each point i charged at mot once. It follow that O 1 O k E 1 E k. To complete the analyi of the approximation bound, it uffice to how that r apx /r opt 3 + ε. A oberved earlier, we have r apx e i (ε, p) and r opt = 2 i + x > 2 i ( Thu, we have r apx < e i(ε, p) r opt 2 i (1 + p 1 ) = ( 1 = p p i (1 + p )( l ) 2 i (1 + p 1 ) ) (1 ) l 3( ) ) ( l ). Under our aumption that = 10/ε, l = 4 log 2 ε, and ε 1, it follow eaily that 1 9/ε, and 3 2 l 3ε/16 ε/5. Therefore, we have ( r apx < ε )( 1 + ε ) ( ε ) = 3 + ε, r opt which complete the proof. The aumption that ε 1 in the tatement of the theorem i a technicality. The analyi may be modified to work for any contant value of ε Tightne of the approximation ratio In thi ection, we preent an example that demontrate that our (3 + ε)-approximation ratio for the dicrete k-center problem i nearly tight. In particular, given any ufficiently mall ε > 0, we hall how that our algorithm achieve an approximation ratio of 3 ε on thi example. Let δ = ε/6, and conider the et of 14 point illutrated in Fig. 6. Thi point et conit of a collection of nine cluter placed on the real line, where each cluter contain from one to four point, each lying within ditance δ of ome integer point. Let k = 2 and tn =11. It i eay to verify that the optimal radiu i r opt = 1 + δ, which i achieved by placing center at poition 1 and 5, o that the two dik centered at thee point cover the = 11 point clutered about {0, 1, 2} and {4, 5, 6}, repectively. We will etablih our bound under the mot favorable aumption for the approximation algorithm. In particular, we aume that the approximation algorithm i free to elect any point a a candidate center (not jut the node in level i ). We aume that the bae ditance for the hierarchy of dicrete center ha been choen o that any deired radiu arie a the value of the greedy radiu, g i (ε, p), for ome level i of ome panner copy p. Finally, we aume that the count returned

12 582 S.A. Friedler, D.M. Mount / Computational Geometry 43 (2010) by range-ketching algorithm are exact. Thu, relaxing any of thee retriction in our algorithm will not ignificantly affect the tightne of the approximation bound. We aert that, even under thee favorable aumption, r apx 3(1 δ). To ee thi, uppoe not. Letting p and i denote, repectively, the panner copy and level that produce thi value, we have e i (ε, p) = r apx < 3(1 δ). For all ufficiently mall δ, thi i le than 4 2δ. Thi implie that each expanded dik cover at mot the ingle cluter containing it center and the cluter about the three conecutive integer point on either ide of it. We alo have g i (ε, p)<g i (ε, p) = e i(ε, p) 3 < 1 δ. Since two point from different cluter are eparated by a ditance of at leat 1 δ, each greedy dik cover only a ingle cluter. Since the cluter near 0 ha the mot point (four), ome point of thi cluter will be choen firt. The expanded dik centered here cover only the even point in the cluter near 0, 1, and 2. The next center to be choen i the cluter near 10 having three point. The expanded dik i not large enough to include any other point. Thu, the algorithm ucceed in covering only = 10 point, and therefore it fail. Since r apx 3(1 δ), we ee that the approximation ratio i a deired. r apx 3(1 δ) = 3 6δ 3 6δ 3 ε, r opt 1 + δ 1 + δ 3.4. The abolute problem Recall that in the abolute formulation the center may be any point in pace. In thi ection we preent a (4 + ε)- approximation algorithm for the abolute, robut k-center problem. The algorithm i the ame a for the dicrete problem, except that we modify the value of the radii upon which the algorithm i baed. In particular, in place of g i (ε, p) and e i (ε, p), wedefine: ĝ i (ε, p) = 2g i and ê i (ε, p) = 2ĝ i (ε, p). We alo define two lightly maller radii for the range-ketch querie: ĝ i (ε, p) = 2g i (ε, p) and ê i (ε, p) = 2ĝ i (ε, p). Since the value of thee radii have increaed, we alo increae the value ued in variou part of Lemma 3.2 to h(2), h(4), and h(6), repectively. By a traightforward modification of the analyi of the approximation bound given in the proof of Theorem 3.2 for the dicrete cae, we have the following. Theorem 3.3. Let r opt be the optimal radiu for the abolute robut k-center olution for S, and let r apx be the radiu found by thi abolute algorithm. Then for any 0 < ε 1,wehaver apx (4 + ε)r opt. Proof. The proof of Theorem 3.2 make ue of two key fact about the greedy and expanded dik. The firt i that there exit a panner copy p and a level i uch that, for any optimal dik, there exit a node u in level i uch that the aociated greedy dik G u of radiu g i contain thi optimal dik. In the abolute cae, the center of an optimal dik O may be at an arbitrary point of pace, but by chooing any point of S that i covered by O and centering a dik O of radiu 2r opt at thi point, we ee that O i contained within O. Therefore, in our modified algorithm, there exit a node u in level i uch that the greedy dik Ĝ u of radiu ĝ i = 2g i contain O, and hence contain O a well. The econd key fact ued in our analyi of the dicrete algorithm i that, if any optimal dik overlap a greedy dik G u, then the correponding expanded dik E u contain the optimal dik. In the abolute cae, if any optimal dik O overlap a greedy dik Ĝ u,theneverypointofo lie within ditance ĝ i + 2r opt 2g i + 2g i = 4g i = ê i of u. Therefore, O Ê u.given thee two key fact, the remainder of the proof i the ame a that of Theorem 3.2, but with an appropriate adjutment of the pecific value of (ε) and l(ε) and with the fact that the expanded radiu ha increaed by a factor of ê i /e i = 4/3. 4. Kinetic panner maintenance and quality 4.1. Certificate The KDS algorithm of Gao, Guiba, and Nguyen [18] for the deformable panner maintain four type of certificate. Thee are applied to all level i of each panner. A parent-child certificate guarantee that a node in level i i within ditance 2 i+1 of it parent. An edge certificate guarantee that a pair of neighboring node in level i lie within ditance c 2 i of each other (where we chooe c = 8). A eparation certificate guarantee that any two ditinct neighboring node in level i are eparated by a ditance of at leat 2 i.apotential neighbor certificate guarantee that two non-neighboring level-i node