Online Hitting of Unit Balls and Hypercubes in ℝ^d using Points from ℤ^d
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We consider the online hitting set problem for the range space Σ=( X, R), where the point set X is known beforehand, but the set R of geometric objects is not known in advance. Here, geometric objects arrive one by one, the objective is to maintain a hitting set of minimum cardinality by taking irrevocable decisions. In this paper, we have considered the problem when the objects are unit balls or unit hypercubes in ℝ^d, and the points from ℤ^d are used for hitting them. First, we consider the problem for objects (unit balls and unit hypercubes) in lower dimensions. We obtain 4 and 8-competitive deterministic online algorithms for hitting unit hypercubes in ℝ^2 and ℝ^3, respectively. On the other hand, we present 4 and 14-competitive deterministic online algorithms for hitting unit balls in ℝ^2 and ℝ^3, respectively. Next, we consider the problem for objects (unit balls and unit hypercubes) in the higher dimension. For hitting unit hypercubes in ℝ^d, we present a O(d^2)-competitive randomized online algorithm for d≥ 3 and prove the competitive ratio of any deterministic algorithm for the problem is at least d+1 for any d∈ℕ. Then, for hitting unit balls in ℝ^d, we propose a O(d^4)-competitive deterministic algorithm, and for d<4, we establish that the competitive ratio of any deterministic algorithm is at least d+1.
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