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Given an $n$-point metric space $(M,d)$, {\sc metric $1$-median} asks for apoint $p\in M$ minimizing $\sum_{x\in M}\,d(p,x)$. We show that for eachcomputable function $f\colon \mathbb{Z}^+\to\mathbb{Z}^+$ satisfying$f(n)=\omega(1)$, {\sc metric $1$-median} has a deterministic, $o(n)$-query,$o(f(n)\cdot\log n)$-approximation and nonadaptive algorithm. Previously, nodeterministic $o(n)$-query $o(n)$-approximation algorithms are known for {\scmetric $1$-median}. On the negative side, we prove each deterministic$O(n)$-query algorithm for {\sc metric $1$-median} to be not $(\delta\logn)$-approximate for a sufficiently small constant $\delta>0$. We also refutethe existence of deterministic $o(n)$-query $O(\log n)$-approximationalgorithms.

Deterministic metric $1$-median selection with very few queries