On Complexity of 1-Center in Various Metrics
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We consider the classic 1-center problem: Given a set P of n points in a metric space find the point in P that minimizes the maximum distance to the other points of P. We study the complexity of this problem in d-dimensional ℓ_p-metrics and in edit and Ulam metrics over strings of length d. Our results for the 1-center problem may be classified based on d as follows. ∙ Small d: We provide the first linear-time algorithm for 1-center problem in fixed-dimensional ℓ_1 metrics. On the other hand, assuming the hitting set conjecture (HSC), we show that when d=ω(log n), no subquadratic algorithm can solve 1-center problem in any of the ℓ_p-metrics, or in edit or Ulam metrics. ∙ Large d. When d=Ω(n), we extend our conditional lower bound to rule out sub quartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a (1+ϵ)-approximation for 1-center in Ulam metric with running time Õ_̃ϵ̃(nd+n^2√(d)). We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension d, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of n strings each of length n, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.
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