Abstract
We consider the minimum line covering problem: given a set S of n points in the plane, we want to find the smallest number l of straight lines needed to cover all n points in S. We show that this problem can be solved in O(n log l) time if l ∈ O(log1 − ε n), and that this is optimal in the algebraic computation tree model (we show that the Ω(nlog l) lower bound holds for all values of l up to \(O(\sqrt n)\)). Furthermore, a O(log l)-factor approximation can be found within the same O(n log l) time bound if \(l \in O(\sqrt[4]{n})\). For the case when l ∈ Ω(log n) we suggest how to improve the time complexity of the exact algorithm by a factor exponential in l.
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Grantson, M., Levcopoulos, C. (2006). Covering a Set of Points with a Minimum Number of Lines. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds) Algorithms and Complexity. CIAC 2006. Lecture Notes in Computer Science, vol 3998. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758471_4
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DOI: https://doi.org/10.1007/11758471_4
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