Streaming with minimum space: An algorithm for covering by two congruent balls

Abstract

In this paper we design a simple streaming algorithm for maintaining two smallest balls (of equal radius) in d-dimension to cover a set of points in an on-line fashion. Different from most of the traditional streaming models, at any step we use the minimum amount of space by only storing the locations and the (common) radius of the balls. Previously, such a geometric algorithm is only investigated for covering with one ball (one-center) by Zarrabi-Zadeh and Chan. We give an analysis of our algorithm, which is significantly different from the one-center algorithm due to the obvious possibility of grouping points wrongly under this streaming model. We obtain upper bounds of 2 and 5.708 for the case of d = 1 and d > 1 respectively. We also present some lower bounds for the corresponding problems. © 2012 Springer-Verlag.