Dynamic Clustering to Minimize the Sum of Radii

Henzinger, Monika; Leniowski, Dariusz; Mathieu, Claire

doi:10.1007/s00453-020-00721-7

Cited by 5 publications

(8 citation statements)

References 26 publications

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“…Similar Hierarchical structures have been recently employed for solving the the dynamic sum-of-radii clustering problem [14] and the dynamic facility location problem [10]. In comparison to our result that achieves a (2+ϵ)-approximation, the first work proves an approximation factor that has exponential dependency on the doubling dimension while the second one achieves a very large constant.…”

Section: Introductionsupporting

confidence: 56%

Fully Dynamic k-Center Clustering in Doubling Metrics

Goranci,

Henzinger,

Leniowski

et al. 2019

Preprint

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In the k-center clustering problem, we are given a set of n points in a metric space and a parameter k ≤ n. The goal is to select k designated points, referred to as centers, such that the maximum distance of any point to its closest center is minimized. This notion of clustering is of fundamental importance and has been extensively studied.We study a dynamic variant of the k-center clustering problem, where the goal is to maintain a clustering with small approximation ratio while supporting an intermixed update sequence of insertions and deletions of points with small update time. Moreover, the data-structure should be able to support the following queries for any given point: (1) report whether this point is a center or (2) determine the cluster this point is assigned to.We present a deterministic dynamic algorithms for the k-center clustering problem that achieves a (2 + ϵ)-approximation with O(2 O (κ) log ∆ log log ∆ • ϵ −1 ln ϵ −1 ) update time and O(log ∆) query time, where κ bounds the doubling dimension of the metric and ∆ is the aspect ratio. Our running and query times are independent of the number of centers k, and are poly-logarithmic when the metric has constant doubling dimension and the aspect ratio is bounded by a polynomial. * The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 340506.

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Section: Introductionsupporting

confidence: 56%

Fully Dynamic k-Center Clustering in Doubling Metrics

Goranci,

Henzinger,

Leniowski

et al. 2019

Preprint

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“…The k-sum-of-diameters problem is NP-hard to approximate with a factor better than 2 [11]. Henzinger et al [20] recently developed a fully dynamic algorithm for a variant of the sum-of-radii problem which does not limit the number of centers. Instead there is a set F of facilities and a set C of clients and the algorithm must assign a radius r i to each facility i such that every client is within distance at most r i for some facility i.…”

Section: A Related Workmentioning

confidence: 99%

On fully dynamic constant-factor approximation algorithms for clustering problems

Fichtenberger¹,

Henzinger²,

Wiese³

2021

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Clustering is an important task with applications in many fields of computer science. We study the fully dynamic setting in which we want to maintain good clusters efficiently when input points (from a metric space) can be inserted and deleted. Many clustering problems are APX-hard but admit polynomial time O(1)-approximation algorithms. Thus, it is a natural question whether we can maintain O(1)approximate solutions for them in subpolynomial update time, against adaptive and oblivious adversaries. Only a few results are known that give partial answers to this question. There are dynamic algorithms for k-center, k-means, and k-median that maintain constant factor approximations in expected Õ(k 2 ) update time against an oblivious adversary. However, for these problems there are no algorithms known with an update time that is subpolynomial in k, and against an adaptive adversary there are even no (non-trivial) dynamic algorithms known at all. Also, for the k-sum-of-radii and the k-sum-of-diameters problems it is open whether there is any dynamic algorithm, against either type of adversary.In this paper, we complete the picture of the question above for all these clustering problems.• We show that there is no fully dynamic O(1)-approximation algorithm for any of the classic clustering problems above with an update time in n o(1) h(k) against an adaptive adversary, for an arbitrary function h. So in particular, there are no such deterministic algorithms. • We give a lower bound of Ω(k) on the update time for each of the above problems, even against an oblivious adversary. This rules out update times with subpolynomial dependence on k. • We give the first O(1)-approximate fully dynamic algorithms for k-sum-of-radii and for k-sum-ofdiameters with expected update time of Õ(k O(1) ) against an oblivious adversary. • Finally, for k-center we present a fully dynamic (6 + ǫ)-approximation algorithm with an expected update time of Õ(k) against an oblivious adversary. This is the first dynamic O(1)-approximation algorithm for any of the clustering problems above whose update time is Õ(k) and in particular the first whose update time is asymptotically optimal in k.

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“…Technical contribution. From a technical point of view we modify and significantly extend a hierarchical partition of a subset of the facilities that was recently introduced for a related problem, called the dynamic sum-of-radii clustering problem [14]. In that work, a set of facilities J and a dynamically changing clients C are given and the goal is to output a set J ′ ⊆ J together with a radius R j , for each j ∈ J ′ , such that the J ′ covers C and the function j∈J ′ (f j + R j ) is minimized.…”

Section: Introductionmentioning

confidence: 99%

“…In that work, a set of facilities J and a dynamically changing clients C are given and the goal is to output a set J ′ ⊆ J together with a radius R j , for each j ∈ J ′ , such that the J ′ covers C and the function j∈J ′ (f j + R j ) is minimized. In [14] a O(2 2κ )-approximation algorithm with time Õ(2 6κ ) per client insertion or deletion is presented for metrics with doubling dimension κ. Note that the function that is minimized is different from the function minimized in the facility location problem, as the term i∈C min j∈J ′ d(i, j) is replaced by j∈J ′ R j .…”

Section: Introductionmentioning

confidence: 99%

A Tree Structure For Dynamic Facility Location

Goranci¹,

Henzinger²,

Leniowski

2019

Preprint

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We study the metric facility location problem with client insertions and deletions. This setting differs from the classic dynamic facility location problem, where the set of clients remains the same, but the metric space can change over time. We show a deterministic algorithm that maintains a constant factor approximation to the optimal solution in worst-case time Õ(2 O(κ 2 ) ) per client insertion or deletion in metric spaces while answering queries about the cost in O(1) time, where κ denotes the doubling dimension of the metric. For metric spaces with bounded doubling dimension, the update time is polylogarithmic in the parameters of the problem.

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Dynamic Clustering to Minimize the Sum of Radii

Cited by 5 publications

References 26 publications

Fully Dynamic k-Center Clustering in Doubling Metrics

Fully Dynamic k-Center Clustering in Doubling Metrics

On fully dynamic constant-factor approximation algorithms for clustering problems

A Tree Structure For Dynamic Facility Location

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