### A Randomized Algorithm for Online Unit Clustering

Timothy M. Chan, Hamid Zarrabi-Zadeh
<span title="2007-10-13">2007</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qj6jzeossfhihfshwf7nb7lbcq" style="color: black;">Theory of Computing Systems</a> </i> &nbsp;
In this paper, we consider the online version of the following problem: partition a set of input points into subsets, each enclosable by a unit ball, so as to minimize the number of subsets used. In the onedimensional case, we show that surprisingly the naïve upper bound of 2 on the competitive ratio can be beaten: we present a new randomized 15/8-competitive online algorithm. We also provide some lower bounds and an extension to higher dimensions. Problem 2 (Unit Covering) Given a set of n
more &raquo; ... ts, cover the set by balls of unit radius, so as to minimize the number of balls used. Both problems are NP-hard in the Euclidean plane [10, 19] . In fact, it is NPhard to approximate the two-dimensional k-center problem to within a factor smaller than 2 [9]. Factor-2 algorithms are known for the k-center problem [9, 11] in any dimension, while polynomial-time approximation schemes are known for the unit covering problem [14] in fixed dimensions. Recently, many researchers have considered clustering problems in more practical settings, for example, in the online and data stream models [4, 5, 12] , where the input is given as a sequence of points over time. In the online model, the solution must be constructed as points arrive and decisions made cannot be subsequently revoked; for example, in the unit covering problem, after a ball is opened to cover an incoming point, the ball cannot be removed later. In the related streaming model, the main concern is the amount of working space; as Work of the first author has been supported in part by NSERC.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00224-007-9085-7">doi:10.1007/s00224-007-9085-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zw4zwi4i4vd5bk4wxck75by6vu">fatcat:zw4zwi4i4vd5bk4wxck75by6vu</a> </span>
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