Rectilinear covering for imprecise input points (Extended abstract)

Abstract

We consider the rectilinear κ-center problem in the presence of impreciseness of input points. We assume that the input is a set S of n unit squares, possibly overlapping each other, each of which is interpreted as a measured point with an identical error bound under the L∞ metric on ℝ2. Our goal, in this work, is to analyze the worst situation with respect to the rectilinear κ-center for a given set S of unit squares. For the purpose, we are interested in a value λ κ(S) that is the minimum side length of κ congruent squares by which any possible true point set from S can be covered. We show that, for κ = 1 or 2, computing λκ(S) is equivalent to the problem of covering the input squares S completely by κ squares, and thus one can solve the problem in linear time. However, for κ ≥ 3, this is not the case, and we present an O(n log n)-time algorithm for computing λ3(S). For structural observations, we introduce a new notion on geometric covering, namely the covering-family, which is of independent interest. © Springer-Verlag 2012.