Streaming Algorithms for Smallest Intersecting Ball of Disjoint Balls

Abstract

In this paper, we propose streaming algorithms for approximating the smallest intersecting ball of a set of disjoint balls in $\mathbb {R}^d$. This problem is a generalization of the $1$-center problem, one of the most fundamental problems in computational geometry. We consider the single-pass streaming model; only one-pass over the input stream is allowed and a limited amount of information can be stored in memory. We introduce three approximation algorithms: one is an algorithm for the problem in arbitrarily dimensions, but in the other two we assume $d$ is a constant. The first algorithm guarantees a $(2+\sqrt{2}+\varepsilon ^*)$-factor approximation using $O(d^2)$ space and $O(d)$ update time where $\varepsilon ^*$ is an arbitrarily small positive constant. The second algorithm guarantees an approximation factor $3$ using $O(1)$ space and $O(1)$ update time (assuming constant $d$). The third one is a $(1+\varepsilon )$-approximation algorithm that uses $O(1/\varepsilon ^{d})$ space and $O(1/\varepsilon ^{(d-1)/2})$ amortized update time. They are the first approximation algorithms for the problem, and also the first results in the streaming model.

MADALGO—Center for Massive Data Algorithmics, a center of the Danish National Research Foundation.