Online Sum-Radii Clustering

Abstract

Abstract. In Online Sum-Radii Clustering, n demand points arrive online and must be irrevocably assigned to a cluster upon arrival. The cost of each cluster is the sum of a fixed opening cost and its radius, and the objective is to minimize the total cost of the clusters opened by the algorithm. We show that the deterministic competitive ratio of Online Sum-Radii Clustering for general metric spaces is Θ(log n), where the upper bound follows from a primal-dual algorithm and holds for general metric spaces, and the lower bound is valid for ternary Hierarchically Well-Separated Trees (HSTs) and for the Euclidean plane. Combined with the results of (Csirik et al., MFCS 2010), this result demonstrates that the deterministic competitive ratio of Online Sum-Radii Clustering changes abruptly, from constant to logarithmic, when we move from the line to the plane. We also show that Online Sum-Radii Clustering in metric spaces induced by HSTs is closely related to the Parking Permit problem introduced by (Meyerson, FOCS 2005). Exploiting the relation to Parking Permit, we obtain a lower bound of Ω(log log n) on the randomized competitive ratio of Online Sum-Radii Clustering in tree metrics. Moreover, we present a simple randomized O(log n)-competitive algorithm, and a deterministic O(log log n)-competitive algorithm for the fractional version of the problem.

An incremental version of the k-center problem on boundary of a convex polygon

“…Conversely, incremental algorithms make partial decisions as known data arrive. There have been many research papers relative to online clustering and online facility location problems, see e.g., Csirik et al (2013), Ehmsen and Larsen (2013), Fotakis (2008), Fotakis and Koutris (2014), Meyerson (2001).…”

Online VQ Codebook Generation using a Triangle Inequality

“…Fokatis 등은 트리(Tree)구조를 활용한 Online Sum-Radii Clustering을 제안했다 [10]. Barbakh 등은 온라인 군집화 시 GLA의 초기값에 따른 민감도를 극복하기 위한 방안을 제시했다 [11].…”

Online algorithms for clustering problems

“…In [35] the multidimensional extension of the problem is studied with linear cost function, where again the cost of a cluster is the sum of a fixed setup cost and its radius. In this paper an O(log n)-competitive algorithm is given for arbitrary metric spaces, and it is also proved that no online algorithm exists with smaller competitive ratio than Ω(log n).…”

“…This conjecture is proved in [35] for p = 1 and d = 2 but it seems to be very hard to extend the lower bound proof to the general case. A further interesting question could be to investigate the flexible model where the algorithm is allowed to change the size and location of the cluster with the cost function defined in this section.…”