Metric 1-median selection with fewer queries

Let $h\text{\hspace{0.17em}⁣}:{\mathbb{Z}}^{+}\to {\mathbb{Z}}^{+}\setminus \{1\}$ be any function such that $h(n)$ and $\lceil n^{1\mathrm{/}h(n)}\rceil$ are computable from $n$ in $O(h(n)\cdot n^{1+1\mathrm{/}h(n)})$ time. We show that given any $n$-point metric space $(M,d)$, the problem of finding $\mathop{$\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}$}{i\in M},\sum{j\in M},d(i,j)$(breakingtiesarbitrarily)hasadeterministic,$O(h(n)\cdot n^{1+1/h(n)})$-time,$O(n^{1+1/h(n)})$-query,$(2,h(n))$-approximation and nonadaptive algorithm. Our proofs modify those of Chang~\cite{Cha15, Cha15CMCT} with the following improvements: (1) We improve Chang’s~\cite{Cha15} query complexity of $O(h(n)\cdot n^{1+1\mathrm{/}h(n)})$ to $O(n^{1+1\mathrm{/}h(n)})$, everything else being equal. (2) Chang’s~\cite{Cha15CMCT} unpublished work establishes our result only when $n$ is a perfect $(h(n))$th power.