Metric 1-median selection with fewer queries

TL;DR

This paper presents a deterministic, efficient algorithm for selecting the 1-median in metric spaces with fewer queries, improving previous methods by reducing query complexity and removing restrictions on the size of the metric space.

Contribution

It introduces a new algorithm that achieves near-optimal query complexity for 1-median selection in metric spaces, extending and improving upon Chang's prior work.

Findings

Reduces query complexity from Chang's $O(h(n) imes n^{1+1/h(n)})$ to $O(n^{1+1/h(n)})$

Provides a deterministic, nonadaptive algorithm with $O(h(n) imes n^{1+1/h(n)})$ runtime

Removes the restriction that $n$ must be a perfect $h(n)$th power

Abstract

Let $h\colon\mathbb{Z}^+\to\mathbb{Z}^+\setminus\{1\}$ be any function such that $h(n)$ and $\lceil n^{1/h(n)}\rceil$ are computable from $n$ in $O(h(n)\cdot n^{1+1/h(n)})$ time. We show that given any $n$-point metric space $(M,d)$, the problem of finding $\mathop{\mathrm{argmin}}_{i\in M}\,\sum_{j\in M}\,d(i,j)$ (breaking ties arbitrarily) has a deterministic, $O(h(n)\cdot n^{1+1/h(n)})$-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 $O(h(n)\cdot n^{1+1/h(n)})$ to $O(n^{1+1/h(n)})$, everything else being equal. (2) Chang's~\cite{Cha15CMCT} unpublished work establishes our result only when $n$ is a perfect $(h(n))$th power.