The Online Median Problem

Transcription

1 The Online Median Problem Ramgopal R. Mettu C. Greg Plaxton November 1999 Abstract We introduce a natural variant of the (metric uncapacitated) -median problem that we call the online median problem. Whereas the -median problem involves optimizing the simultaneous placement of facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time; a facility, once placed, cannot be moved; the total number of facilities to be placed,, is not known in advance. The objective of an online median algorithm is to minimize competitive ratio, that is, the worst-case ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a linear-time constant-competitive algorithm for the online median problem. In addition, we present a related, though substantially simpler, linear-time constant-factor approximation algorithm for the (metric uncapacitated) facility location problem. The latter algorithm is similar in spirit to the recent primal-dual-based facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time. Department of Computer Science, University of Texas at Austin, Austin, TX This research was supported by NSF Grant CCR ramgopal, plaxton@cs.utexas.edu.

2 1 Introduction Recently the first constant-factor approximation algorithm was discovered for the -median problem by Charikar et al. [3]; in this paper, we ask whether a constant competitive ratio can be achieved for a natural online extension of the -median problem. Let Í be a nonempty set of Ò points and let be a metric distance function on Í. The -median problem is concerned with marking points such that the sum over all points Ü of the weight of Ü times the distance from Ü to the closest marked point is minimized. For the online median problem, we wish to find an ordering of the Ò points such that for all, ¼ Ò, the first points provide a good solution (e.g. constant factor approximation) to the -median problem. An obvious approach to the online median problem is to iteratively choose the point that minimizes the objective function. Greedy strategies of this kind are commonly applied in the design of online algorithms [1, 9]. It turns out, however, that for the online median problem, the simple strategy suggested above has an unbounded competitive ratio. We show that a modification of this strategy that we call hierarchically greedy can be used to obtain a constant-competitive linear-time algorithm for the online median problem. We develop this strategy by first considering a simple greedy algorithm for facility location. 1.1 Problem Definitions Without loss of generality, throughout this paper we consider a fixed set of points Í with an associated distance function Í Í ÁÊ and nonnegative functions Û Í ÁÊ. We are primarily interested in the case where the function is a metric, that is, where is nonnegative, symmetric, satisfies the triangle inequality, and Ü Ýµ ¼ iff Ü Ý. For the online median problem, it will prove to be useful to consider a slightly more general class of distance functions in which the triangle inequality is relaxed to the following -approximate È triangle inequality, where ½: For any sequence of points Ü ¼ Ü in Í, Ü ¼ Ü µ ¼ Ü Ü ½ µ. We refer to such a distance function as a -approximate metric. We let Ò Í, and define a subset of Í to be a configuration iff it is nonempty. For any point Ü and configuration, we define Ü µ as ÑÒ Ý¾ Ü Ýµ. We consider three computational problems: -median, online median, and facility location. For the -median and online median problems, the cost of a configuration, denoted Ó Ø µ, is defined to be È Ü¾Í Ü µ Û Üµ. The input to the -median problem is Í µ, Û, and an integer, ¼ Ò. The output is a minimum-cost configuration of size. The input to the online median problem is Í µ and Û. The output is a total order on Í. We define the competitive ratio of such an ordering as the maximum over all, ¼ Ò, of the ratio of the cost of the configuration given by the first points in the ordering to that of an optimal -median configuration. We define the competitive ratio of an online median algorithm as the supremum, over all possible choices of the input instance Í µ and Û, of the competitive ratio of the ordering produced by the algorithm. For the facility location problem, the cost of a configuration, denoted Ó Ø µ, is defined as the sum of È Ü¾ ܵ and È Ü¾Í Ü µ Û Üµ. The input to the facility location problem is Í µ,, and Û. The output is a minimum-cost configuration. 1.2 Previous Work There has been much prior work on the facility location and -median problems; here we focus on the work that is most relevant to our results. The first constant-factor approximation algorithm for facility location is due to Shmoys et al. [17] and is based on rounding the (fractional) solution to a linear program. Chudak [4] gives an LP-based ½ ¾µ-approximation algorithm for facility location. This was the best constant factor known until the recent work of Charikar and Guha [2], which establishes a slightly lower approximation ratio of ½¾. The first constant-factor approximation for the -median problem was recently given by 1

3 Charikar et al. [3] and is also LP-based. That work follows a sequence of bicriteria results utilizing LPbased techniques [14, 15]. Jain and Vazirani [10] give the first nearly linear-time combinatorial algorithms for the facility location and -median problems, achieving approximation ratios of and, respectively. While the latter algorithms are combinatorial, the primal-dual approach used in their analysis is based on linear programming theory. (See [6] for an excellent introduction to the primal-dual method.) Strategies based on local search and greedy techniques for facility location and the -median problem have been previously studied. The work of Korupolu et al. [11] shows that a simple local search heuristic proposed Kuehn and Hamburger [13] yields both a constant-factor approximation for the facility location problem and a bicriteria approximation for the -median problem [11]. Guha and Khuller [7] showed that greedy improvement can be used as a postprocessing step to improve the approximation guarantee of certain facility location algorithms. Guha and Khuller also provide the best lower bound known of ½ on the approximation ratio for this problem. More recently, Charikar and Guha [2] achieved the best approximation ratio known for facility location by combining a local search heuristic with the best LP-based algorithm known. Charikar and Guha also give a -approximation for the -median problem by building on the techniques of Jain and Vazirani [10]. To the best of our knowledge, the online median problem has not been previously studied. Note that any constant-competitive algorithm for the online median problem is also a constant-factor approximation algorithm for the -median problem, but the converse does not hold. In particular, constant-factor approximation algorithms for the -median problem known prior to this work [2, 3, 10] seem to rely heavily on the knowledge of. As such it is unclear whether any of these algorithms can be easily modified to obtain a constant-competitive online median algorithm. 1.3 Contributions Algorithms for problems in discrete location theory arise in many practical applications; see [5, 16], for example, for numerous pointers to the literature. Given that many of these problems are NP-hard, it is desirable to develop fast approximation algorithms. As mentioned above, it is not uncommon for approximation algorithms to be based on a greedy approach. In this paper, we show that greedy strategies yield a fast constant-factor approximation algorithm for the facility location problem and a fast constant-competitive algorithm for the online median problem. We give a linear-time algorithm for the facility location problem that achieves an approximation ratio of. The main idea of the algorithm is to compute and use the value of balls about every point in the metric space. In retrospect, the idea of value is implicit in the work of Jain and Vazirani [10]. We make this idea explicit and use the values of balls to make greedy choices. Additionally, our algorithm is faster than the Jain-Vazirani algorithm by a logarithmic factor. While a simple greedy algorithm yields a constant-factor approximation bound for the facility location problem, it appears that a more sophisticated approach is needed to obtain a constant-factor approximation guarantee for the -median problem, let alone a constant-competitiveness result for the online median problem. For example, in Section 3 we show that perhaps the most natural greedy approach to the -median (resp., online median) problem leads to an unbounded approximation (resp., competitive) ratio. Our main result is a linear-time constant competitive algorithm for the online median problem. We achieve this result using a hierarchically greedy approach. The basic idea behind this approach is as follows: Rather than selecting a point based on a single greedy criterion, we greedily choose a region (the set of points lying within some ball) and then recursively select a point within that region. Thus, the choice of point is influenced by a sequence of greedy criteria addressing successively finer levels of granularity. 2

4 1.4 Outline The rest of this paper is organized as follows. In Section 2, we present our facility location algorithm and prove that it achieves a constant approximation ratio. In Section 3, we present our online median algorithm and prove that it is constant-competitive. Section 4 offers some concluding remarks. 2 Facility Location The following definitions are used throughout the present section as well as Section 3. For any nonnegative integer, let denote the set ¼. A ball is a pair Ü Öµ, where the center Ü of, denoted ÒØÖ µ, belongs to Í, and the radius Ö of, denoted ÖÙ µ, is a nonnegative real. Given a ball Ü Öµ, we let ÈÓÒØ µ denote the set Ý ¾ Í Ü Ýµ Ö. However, for the sake of brevity, we tend to write instead of ÈÓÒØ µ. For example, we write Ü ¾ and instead of Ü ¾ ÈÓÒØ µ and ÈÓÒØ µ ÈÓÒØ µ, respectively. È The value of a ball Ü Öµ, denoted ÚÐÙ µ, is ݾ Ö Ü Ýµµ Û Ýµ. For any ball Ü Öµ and any nonnegative real, we define as the ball Ü Öµ. 2.1 Algorithm In the first step of the following algorithm, we assume for the sake of convenience that there is at least one point Ü such that Û Üµ ¼. (The problem is trivial otherwise.) The output of the algorithm is the configuration Ò, which we also refer to as. Remark: The indexing of the sets has been introduced solely to facilitate the analysis. For each point Ü, determine an associated ball Ü Ü Ö Ü µ such that ÚÐÙ Ü µ ܵ. Determine a bijection ³ Ò Í such that Ö ³ ½µ Ö ³ µ, ¼ Ò. Let Ü Ö µ denote the ball ³ µ, ¼ Ò. Let ¼. For ¼ to Ò ½: If ¾ then let ½ Ü ; otherwise, let ½. We now sketch a simple linear-time implementation of the above algorithm. For each point Ü, the associated radius Ö Ü can be computed in Ç Òµ time. (This is essentially a weighted selection problem.) Thus the first step requires Ç Ò ¾ µ time. The second step involves sorting Ò values and can be accomplished in Ç Ò ÐÓ Òµ time. The running time for the third step is negligible. Each iteration of the fourth step can be easily implemented in Ç Òµ time, for a total of Ç Ò ¾ µ time. 2.2 Approximation Ratio In this section we establish the following theorem. Theorem 1 For any configuration, Ó Ø µ Ó Ø µ. Immediate from Lemmas 2.3 and 2.7 below. 3

5 Lemma 2.1 For any point Ü, there exists a point Ü in such that and Ü Ü µ ¾Ö. If there is no such point Ü with, then ¾ is empty, and so Ü belongs to. Lemma 2.2 Let Ü and Ü be distinct points in. Then Ü Ü µ ¾ ÑÜÖ Ö. Assume without loss of generality that. Thus Ö Ö. Furthermore, Ü Ü µ ¾Ö since Ü belongs to and ¾ is empty. For any point Ü and any configuration, let charge Ü µ Ü µ Ü ¾ Ñܼ Ö Ü Üµ È Lemma 2.3 For any configuration, Ü¾Í Ö Ü µ Û Üµ Ó Ø µ. Note that Ü¾Í Ö Ü µ Û Üµ Ü ¾ Ü ¾ which is equal to Ó Ø µ since ÚÐÙ µ Ü µ. ܾ Ö Ü Üµµ Û Üµ ÚÐÙ µ Ü¾Í Ü¾Í Ü µ Û Üµ Ü µ Û Üµ Lemma 2.4 Let Ü be a point, let be a configuration, and let Ü belong to. If Ü Ü µ Ü µ then Ö Ü µ ÑÜÖ Ü Ü µ. If Ü does not belong to, then Ö Ü µ Ü Ü µ Ö. Otherwise, Ö Ü µ Ö Ü Ü µµ Ü Ü µ Ö Ü Ü µ. Lemma 2.5 Let Ü be a point and let Ü belong to. If Ü belongs to, then Ö Ü µ Ö. By Lemma 2.2, there is no point Ü in such that and Ü belongs to. The claim now follows from the definition of Ö Ü µ, since Ü µ Ü Ü µ. Lemma 2.6 Let Ü be a point and let Ü belong to. If Ü does not belong to, then Ö Ü µ Ü Ü µ. The claim is immediate unless there is a point Ü in such that Ü belongs to. If such a point Ü exists, then Lemmas 2.2 and 2.5 imply Ü Ü µ ¾ ÑÜÖ Ö and Ö Ü µ Ö, respectively. The claim now follows since Ü Ü µ Ü Ü µ Ü Ü µ ¾Ö Ö Ö. Lemma 2.7 For any point Ü and configuration, Ö Ü µ Ö Ü µ. Let Ü be some point in such that Ü Ü µ Ü µ. By Lemma 2.1, there exists a point Ü in such that and Ü Ü µ ¾Ö. If Ü belongs to, then Ö Ü µ Ö by Lemma 2.5. The claim follows since implies Ö Ö and Lemma 2.4 implies Ö Ü µ Ö. If Ü does not belong to, then Ö Ü µ Ü Ü µ by Lemma 2.6. Thus Ö Ü µ Ü Ü µ Ü Ü µ Ü Ü µ ¾Ö. The claim now follows by Lemma 2.4, since the ratio of Ü Ü µ ¾Ö to ÑÜÖ Ü Ü µ is at most 3. 4

6 3 Online Median Placement In the previous section, we found that a simple greedy algorithm yields interesting results for the facility location problem. Ideally, we would like to formulate a similar algorithm for the online median problem. The most obvious greedy algorithm is to select as the next point in the ordering the one that minimizes the objective function. Unfortunately, this algorithm gives an unbounded competitive (resp., approximation) ratio for the online median (resp., -median) problem. To see this, consider an instance consisting of Ò points, one red and the rest blue, such that the following conditions are satisfied: the red point has weight ¼; each blue point has weight ½; the distance from the red point to any blue point is ½; the distance between any pair of distinct blue points is ¾. The aforementioned greedy algorithm chooses the red point first in the ordering, since that gives a cost of Ò ½ while choosing any other point gives a cost of ¾Ò. But then the ratio for a configuration of size Ò ½ is unbounded since the greedy cost is ½ and the optimal cost is ¼. (This example also shows that no online median algorithm can achieve a competitive ratio below ¾ ¾.) Ò ½ We show that a more careful choice of the point, which we call hierarchically greedy, works well. Let (resp., Æ) denote the largest (resp., smallest) distance between two distinct points in the metric space. We define a certain ball about each point, and select a ball of maximum value. But rather than simply choosing the center of ball as the next point in the ordering, we apply the approach recursively to select a point within. At each successive level of recursion, we consider geometrically smaller balls about the remaining candidate points. Within Ç ÐÓ µ levels of recursion, we arrive at a ball containing only a single Æ point, and we return this point as the next one in the ordering. Note that whereas the greedy algorithm discussed in the previous paragraph makes a single greedy choice to select a point, the hierarchically greedy algorithm makes Ç ÐÓ µ greedy choices per point. Æ Throughout this section, let, «,, and denote real numbers satisfying the following inequalities. ½ (1) «½ (2) «½µ «½ «¾ «««½ (3) (4) The online median algorithm of Section 3.1 below makes use of the following additional definitions. A child of a ball Ü Öµ is any ball Ý Ö µ where Ü Ýµ Ö. For any point Ü, let ÓÐØ Ü µ denote «the ball Ü ÑÜ Ý¾Í Ü Ýµµ. For any point Ü and configuration, let ÓÐØ Ü µ denote the ball Ü Ü µ µ. For any nonempty sequence ±, we let ±µ (resp., ØÐ ±µ) denote the first (resp., last) element of ±. 3.1 Algorithm Let ¼. For ¼ to Ò ½, execute the following steps: Let denote the singleton sequence where is a maximum value ball in ÓÐØ Ü µ Ü ¾ Í Ò. While the ball ØÐ µ has more than one child, append a maximum value child of ØÐ µ to. Let ½ ÒØÖ ØÐ µµ. 5

7 The output of the online median algorithm is a collection of point sets such that, ¼ Ò, and ½, ¼ Ò. Note that it is sufficient for an implementation of the algorithm to maintain the ball ØÐ µ, as opposed to the entire sequence. The sequence has been introduced in order to facilitate the analysis. We discuss two implementations of the online median algorithm in Section 3.4. The first implementation has a slightly superlinear running time. The second implementation runs in linear time, but assumes a (linear) preprocessing phase in which all distances are rounded down to the nearest integral power of. (Note that for the preprocessing phase to be well-defined, we require ½.) If the input distance function is a metric, it is straightforward to see that such rounding produces a -approximate metric. 3.2 Competitive Ratio Before proceeding with the analysis, we introduce a number of additional definitions. Let Þ denote the unique point in ½ Ò, ¼ Ò. For any configuration and set of points, let Ó Ø µ È Ý¾ Ý µ Û Ýµ. For any configuration, we partition Í into sets ÐÐ Ü µ Ü ¾ as follows: For each point Ý in Í, we choose a point Ü in such that Ý µ Ü Ýµ and add Ý to ÐÐ Ü µ. For any configuration, point Ü in, and set of points, we define Ò Ü µ as ÐÐ Ü µ ÓÐØ Ü µ and ÓÙØ Ü µ as ÐÐ Ü µ Ò Ò Ü µ. For any configuration and set of points, we define Ò µ as ܾ Ò Ü µ and ÓÙØ µ as Í Ò Ò µ. In this section we present our main result, Theorem 2 below. In order to minimize the competitive ratio of ¾ ½µ implied by the theorem, we set to ½, set «to approximately and set and to the right-hand sides of Equations (3) and (4), respectively. We thereby establish a competitive ratio of slightly below ¼ for the online median problem. In Section 3.4 we describe a linear-time implementation of the online median algorithm for which the parameter is required to be strictly greater than ½. The degradation in the competitive ratio that results by setting greater than ½ can be made arbitrarily small by choosing sufficiently close to ½. Theorem 2 For any configuration, Ó Ø µ ¾ ½µ Ó Ø µ. Let Ò µ and let ¼ ÓÙØ µ Í Ò. Note that Ó Ø µ Ó Ø µ Ó Ø ¼ µ and Ó Ø µ Ó Ø µ Ó Ø ¼ µ. Thus the theorem follows immediately from Lemmas 3.2, 3.4, and 3.5 below. Lemma 3.1 For any configuration, point Ü in, and point Ý in ÓÙØ Ü µ, Ý µ ½µ Ý µ. Let ÓÐØ Ü µ Ü Öµ. Note that Ü Ýµ Ö. Also, by the definition of ÓÐØ Ü µ, there is a point Þ in such that Ü Þµ Ö. Hence Ý Þµ Ü Ýµ Ü Þµ Ü Ýµ Ö Ü Ýµ Ü Ýµ ½µ Ü Ýµ ½µ Ý µ. The claim follows since Ý Þµ Ý µ. 6

8 Lemma 3.2 For any configuration, cost ÓÙØ µµ ½µ Ó Ø ÓÙØ µµ Summing the inequality of Lemma 3.1 over all Ý in ÓÙØ Ü µ, we obtain cost ÓÙØ Ü µµ ½µ Ó Ø ÓÙØ Ü µµ The claim now follows by summing the above inequality over all Ü in. Lemma 3.3 For any configuration and point Ü in, cost Ò Ü µµ ½µÓ Ø Ò Ü µµ ÚÐÙ ÓÐØ Ü µµ Assume that ÓÐØ Ü µ Ü Öµ. Note that Ü Ýµ Ö for some Ý in. Thus, for any Þ in ÓÐØ Ü µ, Ý Þµ Ý Üµ Ü Þµ ½µÖ. It follows that Ó Ø Ò Ü µµ is at most ½µ times Þ¾Ò Ü µ Ö Û Þµ Þ¾Ò Ü µ Ü Þµ Û Þµ Þ¾ ÓÐØ Ü µ cost Ò Ü µµ ÚÐÙ ÓÐØ Ü µµ Ö Ü Þµµ Û Þµ Lemma 3.4 For any configuration and point Ü in, cost Ò µµ ½µÓ Ø Ò µµ The claim follows by summing the inequality of Lemma 3.3 over all Ü in. Our main technical lemma is stated below. The proof is given in the next subsection. Lemma 3.5 For any configuration, È Ü¾ ÚÐÙ ÓÐØ Ü µµ Ó Ø µ. 3.3 Proof of Lemma 3.5 In this section we establish our main technical lemma, Lemma 3.5. Lemma 3.6 Let Ü Öµ belong to. Then Ü µ Ö. ܾ ÚÐÙ ÓÐØ Ü µµ Let Þ be a point in such that Ü Þµ Ü µ. If µ then ÓÐØ Ü µ and the result is immediate. Otherwise, let Ý µ denote the predecessor of in and assume inductively that Ý µ. Note that Ü Ýµ and «Ö. Thus Ü µ Ü Þµ Ý Þµ Ü Ýµ µ«ö Ö, where the last step follows from Equation (4). Lemma 3.7 Let Ü Öµ belong to and let Ý µ belong to. If and Ü Ýµ Ö, then the following claims hold: (i) ÖÙ µµ Ö «; (ii) ØÐ µ; (iii) the successor of in, call it, satisfies ÚÐÙ µ ÚÐÙ µµ. 7

9 Let µ Ý ¼ ¼ µ. For part (i), we know that Ý ¼ Þ µ ¼ by Lemma 3.6. Also, we have Ý ¼ Þ µ Ý ¼ ݵ Ý Üµ Ü Þ µ ¼ ¼ ««½ Ö ¼ µ Ö ««Ö Ö Ö «Combining the two inequalities and applying Equation (4), we obtain «¾ «««¼ «½ «½ Ö ¼ µ Ö Multiplying through by «½µ and rearranging, we get Ö «¾ «¾ «««½ ¼ «¼, establishing the claim. For part (ii), note that Ü Ýµ Ö Ö Ö by part (i) and Equation (3). Thus has at least two «children; the claim follows. For part (iii), we use Equations (2) and (3) and part (i) to observe that Ü Ý ¼ µ Ü Ýµ Ý Ý ¼ µ Ö Ö ««½ ¼ Ö «Ö «½ ««½ ½ ««¾ ¼ which is at most Ö by Equation (3). It follows that µ is contained in a child of. Thus ÚÐÙ µ ÚÐÙ µµ. For ease of notation, throughout the remainder of this section we fix a configuration, and let denote. We now describe a pruning procedure that takes as input the sequences, ¼, and produces as output sequences, ¼. The sequence is initialized to, ¼. The (nondeterministic) pruning procedure then performs a number of iterations. In a general iteration, the pruning procedure checks whether there exist two balls Ü Öµ and Ý µ in distinct sequences and, respectively, such that and Ü Ýµ Ö. If not, the pruning procedure terminates. If so, the sequence is redefined as the proper suffix of (the current) beginning at the successor of. Note that part (ii) of Lemma 3.7 ensures that the pruning procedure is well-defined. Furthermore, the procedure is guaranteed to terminate since each iteration reduces the length of some sequence. Lemma 3.8 Let Ü Öµ belong to and let Ý µ belong to. If then Ü Ýµ Ö. Ö Immediate from the definition of the pruning procedure. Lemma 3.9 Each sequence is nonempty. Immediate from part (ii) of Lemma 3.7 and the definition of the pruning procedure. 8

10 Lemma 3.10 Let Ü be a point and assume that ¼ Ò. Then ÚÐÙ ÓÐØ Ü µµ ÚÐÙ ÓÐØ Ü µµ Since, ÖÙ ÓÐØ Ü µµ ÖÙ ÓÐØ Ü µµ. The claim follows. Lemma 3.11 Let Ü be a point and assume that ¼. Then ÚÐÙ µµ ÚÐÙ ÓÐØ Ü µµ If Ü belongs to, then ÖÙ ÓÐØ Ü µµ ¼, so ÚÐÙ ÓÐØ Ü µµ ¼ and there is nothing to prove. Otherwise, ÚÐÙ µµ ÚÐÙ ÓÐØ Ü µµ by the definition of the online median algorithm, and the claim follows by Lemma Lemma 3.12 Let Ü be a point and assume that ¼. Then ÚÐÙ µµ ÚÐÙ ÓÐØ Ü µµ We prove that the claim holds before and after each iteration of the pruning procedure. Initially, and the claim holds by Lemma If the claim holds before an iteration of the pruning procedure, then it holds after the iteration by part (iii) of Lemma 3.7. A ball Ü Öµ is defined to be covered iff Ü µ Ö. A ball is uncovered iff it is not covered. Lemma 3.13 For any uncovered ball Ü Öµ, Ó Ø µ ÚÐÙ µ. È È Note that Ó Ø µ ݾ Ý µ Û Ýµ ݾ Ö Ý Üµµ Û Ýµ ÚÐÙ µ. Let Á denote the set of all indices in such that some ball in is covered. We now construct a matching between the sets and as follows. First, for each in Á, we match with a point Ü in that belongs to the last covered ball in the sequence. (Note that such a point Ü is guaranteed to exist by the definition of Á. Furthermore, Lemma 3.8 ensures that we do not match the same point with more than one index.) Second, for each in Ò Á in turn, we match with an arbitrary unmatched point Ü in. We now construct a function ³ mapping each point Ü in to an uncovered ball. For each Ü in that is matched with an index in Ò Á, we set ³ ܵ to µ. For each Ü in that is matched with an index in Á, we set ³ ܵ to the successor of the last covered ball in unless ØÐ µ is covered, in which case we set ³ ܵ to the ball Ü ¼µ. Lemma 3.14 For any pair of distinct points Ü and Ý in, ³ ܵ ³ ݵ. Immediate from Lemma 3.8 and the fact that the ball Ü ¼µ is contained in ØÐ µ. Lemma 3.15 For any point Ü in, ÚÐÙ ³ ܵµ ÚÐÙ ÓÐØ Ü µµ. If Ü is matched with an index in Ò Á, the claim follows by Lemma If Ü is matched with an index in Á, we consider two cases. If ØÐ µ is covered, then Ü Þ since ØÐ µ has exactly one child. The claim follows since ³ ܵ ÓÐØ Ü µ Ü ¼µ. If ØÐ µ is uncovered, then the predecessor of ³ ܵ in, call it Ý Öµ, exists and contains Ü. It follows that ÚÐÙ ³ ܵµ ÚÐÙ µ, where Ü Ö«µ is the child of centered at Ü. Let Ü µ denote the ball ÓÐØ Ü µ. 9

11 Below we complete the proof of the claim by showing that Ö«, which implies that and hence ÚÐÙ µ ÚÐÙ µ. It remains to prove that Ö«in the final case considered above. We have Ü Þ µ Ü Ýµ Ý Þ µ Ö Ö Ö «½ «Ö «½ which is less than Ö«by Equation (4). The desired inequality follows since Ü Þ µ by the definition of. Lemmas 3.13, 3.14, and 3.15 together yield a proof of Lemma Time Complexity In this section we describe two implementations of the online median algorithm given in Section 3.1. Throughout this section, let denote the quantity ÐÓ. The first implementation runs in Ç Ò µ Ò ÐÓ Òµ Æ time. The second implementation runs in Ç Ò ¾ Òµ time and assumes an Ç Ò ¾ µ-time preprocessing phase in which all distances are rounded down to the nearest integral power of. To analyze the running time of the implementations given below, we make use of the following lemma. Lemma 3.16 Let Ü Öµ be a child of a ball in sequence and let ¼ Ü Ö ¼ µ be a child of a ball ¼ in sequence. If then Ö «½µÖ ¼. First, note that Ü Þ µ Ö Ö«µ «Ö «½µ. By Lemma 3.6, Ö ¼ Ü µ Ü Þ µ. Combining these inequalities and using Equation (4), we obtain Ö «½µ Ö ¼ ««½ «¾ «Ö ¼ ««½ «½µÖ ¼ In the first implementation, for each point Ü in Í, we sort the remaining points by their distance from Ü. The total sorting time is Ç Ò ¾ ÐÓ Òµ. Using these sorted arrays, we can compute the value of any given ball in Ç ÐÓ Òµ time. We also maintain the distance from Ü to the nearest point in. Note that Ü ½ µ can be determined in constant time given Ü µ and Þ. The total time to maintain such distances is thus Ç Ò ¾ µ. It follows that the first step of each iteration can be implemented in Ç Òµ time. The total time for the second step is Ç ÐÓ Òµ times the sum over all balls appearing in some sequence, ¼ Ò, of the number of children of. By Lemma 3.16, it is straightforward to see that the latter sum is Ç Òµ, and thus the total time for the second step is Ç Ò ÐÓ Òµ. The running time of the third step is negligible. Thus the running time of the first implementation is Ç Ò µ Ò ÐÓ Òµ, as claimed above. For the second implementation, note that after the preprocessing phase, there are Ç µ distinct distances. Thus, for each point Ü, Ç Ò µ time is sufficient to construct an Ç µ-sized table that can be used to compute the value of any ball Ü Öµ in Ç ½µ time. It follows that the total time for the second step can be improved to Ç Òµ. The running time of the second implementation is therefore Ç Ò ¾ Òµ, which is linear in the size of the input (in bits). 10

12 4 Concluding Remarks We plan to investigate whether the ideas presented above can be applied to other problems. The work of Indyk [8] gives a technique to achieve sublinear time bounds for various location problems through random sampling of the distance function; we would like to see if application of these techniques to our algorithms yield sublinear time bounds. Korupolu et al. [12] give an algorithm and an efficient distributed implementation for hierarchical cooperative caching in which the distance function is an ultrametric. We would like to see if the hierarchical greedy strategy can be used or extended to solve the problem for an arbitrary metric space. It would also be interesting to see if the hierarchical greedy strategy admits an efficient distributed implementation for this problem. A nice feature of our online median algorithm is its simplicity. Although we deal with a harder problem, the algorithm is actually simpler to specify than existing constant-factor approximation algorithms for the -median problem. It would be interesting to see whether our approach could be simplified to yield a fast -median algorithm achieving a small approximation ratio. References [1] A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, Cambridge, UK, [2] M. Charikar and S. Guha. Improved combinatorial algorithms for facility location and -median problems. In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, October To appear. [3] M. Charikar, S. Guha, É. Tardos, and D. B. Shmoys. A constant-factor approximation algorithm for the -median problem. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pages 1 10, May [4] F. A. Chudak. Improved approximation algorithms for uncapacitated facility location. In R. E. Bixby, E. A. Boyd, and R. Z. Ríos-Mercado, editors, Integer Programming and Combinatorial Optimization, volume 1412 of Lecture Notes in Computer Science, pages , Berlin, Springer. [5] W. Domschke and A. Drexl. Location and Layout Planning: An International Bibliography, volume 238 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, [6] M. X. Goemans and D. P. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, MA, [7] S. Guha and S. Khuller. Greedy strikes back: Improved facility location algorithms. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages , January [8] P. Indyk. Sublinear time algorithms for metric space problems. In Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pages , May [9] S. Irani and Karlin A. R. Online computation. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, MA, [10] K. Jain and V. V. Vazirani. Primal-dual approximation algorithms for metric facility location and - median problems. In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, October To appear. 11

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