# American Institute of Mathematical Sciences

April  2019, 15(2): 565-576. doi: 10.3934/jimo.2018057

## An adaptive probabilistic algorithm for online k-center clustering

 1 Department of Information and Operations Research, College of Applied Sciences, Beijing University of Technology, Beijing 100124, China 2 School of Computer Science and Technology, Shandong Jianzhu University, Jinan 250101, China

* Corresponding author: Dongmei Zhang

Received  August 2017 Revised  December 2017 Published  April 2018

The $k$-center clustering is one of the well-studied clustering problems in computer science. We are given a set of data points $P\subseteq R^d$, where $R^d$ is $d$ dimensional Euclidean space. We need to select $k≤ |P|$ points as centers and partition the set $P$ into $k$ clusters with each point connecting to its nearest center. The goal is to minimize the maximum radius. We consider the so-called online $k$-center clustering model where the data points in $R^d$ arrive over time. We present the bi-criteria $(\frac{n}{k}, (\log\frac{U^*}{L^*})^2)$-competitive algorithm and $(\frac{n}{k}, \logγ\log\frac{nγ}{k})$-competitive algorithm for semi-online and fully-online $k$-center clustering respectively, where $U^*$ is the maximum cluster radius of optimal solution, $L^*$ is the minimum distance of two distinct points of $P$, $γ$ is the ratio of the maximum distance of two distinct points and the minimum distance of two distinct points of $P$ and $n$ is the number of points that will arrive in total.

Citation: Ruiqi Yang, Dachuan Xu, Yicheng Xu, Dongmei Zhang. An adaptive probabilistic algorithm for online k-center clustering. Journal of Industrial & Management Optimization, 2019, 15 (2) : 565-576. doi: 10.3934/jimo.2018057
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An example of a sequence of points $1, x^2, ..., x^{2t}$ coming over time. The circles with dotted lines are the clusters produced by an online algorithm. Solid ones are those produced by the optimal off-line algorithm
An illustration of partition of $S^*_i$ for any given $i\in[k]$
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