Deterministic metric 1-median selection with very few queries

01/05/2022

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by   Ching-Lueh Chang, et al.

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Given an n-point metric space (M,d), metric 1-median asks for a point p∈ M minimizing ∑_x∈ M d(p,x). We show that for each computable function fℤ^+→ℤ^+ satisfying f(n)=ω(1), metric 1-median has a deterministic, o(n)-query, o(f(n)·log n)-approximation and nonadaptive algorithm. Previously, no deterministic o(n)-query o(n)-approximation algorithms are known for metric 1-median. On the negative side, we prove each deterministic O(n)-query algorithm for metric 1-median to be not (δlog n)-approximate for a sufficiently small constant δ>0. We also refute the existence of deterministic o(n)-query O(log n)-approximation algorithms.