Online Class Cover Problem
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In this paper, we study the online class cover problem where a (finite or infinite) family F of geometric objects and a set P_r of red points in ℝ^d are given a prior, and blue points from ℝ^d arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from F that do not cover any points of P_r. The objective of the problem is to place the minimum number of objects. When F consists of all possible translates of a square in ℝ^2, we prove that the competitive ratio of any deterministic online algorithm is Ω(log | P_r|). On the other hand, when the objects are all possible translates of a rectangle in ℝ^2, we propose an O(log | P_r|)-competitive deterministic algorithm for the problem.
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