A furthest neighbor data structure on a metric space $(V,\mathrm{dist})$ and a set $P \subseteq V$ answers the following query: given $v \in V$, output $p \in P$ maximizing $\mathrm{dist}(v,p)$; in the approximate version, it is allowed to report any $p \in P$ with $\mathrm{dist}(v,p) \geq (1-\varepsilon)\max_{p' \in P} \mathrm{dist}(v,p')$ for an accuracy parameter $\varepsilon \in (0,1)$. A particular type of approximate furthest neighbor data structure is an $\varepsilon$-coreset: a small subset $Q \subseteq P$ such that for every query $v \in V$ there is a feasible answer $p \in Q$. Our main result is that in planar metrics there always exists an $\varepsilon$-coreset for furthest neighbors of size bounded polynomially in $(1/\varepsilon)$. This improves upon an exponential bound of Bourneuf and Pilipczuk [SODA'25] and resolves an open problem of de Berg and Theocharous [SoCG'24] for the case of polygons with holes. On the technical side, we develop a connection between $\varepsilon$-coreset for furthest neighbors and an invariant of a metric space that we call an $\varepsilon$-comatching index -- a sibling of $\varepsilon$-(semi-)ladder index, a.k.a, $\varepsilon$-scatter dimension, as defined by Abbasi et al [FOCS'23]. While the $\varepsilon$-(semi-)ladder index of planar metrics admits an exponential lower bound, we show that the $\varepsilon$-comatching index of planar metrics is polynomial, all in $1/\varepsilon$. The exponential separation between $\varepsilon$-(semi-)ladder and $\varepsilon$-comatching is rather surprising, and the proof is the main technical contribution of our work.

翻译：关于度量空间 $(V,\mathrm{dist})$ 和集合 $P \subseteq V$ 的最近邻数据结构回答以下查询：给定 $v \in V$，输出最大化 $\mathrm{dist}(v,p)$ 的 $p \in P$；在近似版本中，允许报告任意满足 $\mathrm{dist}(v,p) \geq (1-\varepsilon)\max_{p' \in P} \mathrm{dist}(v,p')$ 的 $p \in P$，其中精度参数 $\varepsilon \in (0,1)$。一类特殊的近似最近邻数据结构是 $\varepsilon$-核集：一个小子集 $Q \subseteq P$，使得对于每个查询 $v \in V$，都存在可行答案 $p \in Q$。我们的主要结果是：在平面度量中，总是存在一个大小为 $(1/\varepsilon)$ 的多项式有界的最近邻 $\varepsilon$-核集。这改进了 Bourneuf 和 Pilipczuk [SODA'25] 的指数界，并解决了 de Berg 和 Theocharous [SoCG'24] 关于带孔多边形情况的公开问题。在技术方面，我们建立了最近邻 $\varepsilon$-核集与度量空间的一个不变量（称为 $\varepsilon$-匹配指数）之间的联系——该指数是 $\varepsilon$-(半)梯子指数（也称 $\varepsilon$-散度维数，由 Abbasi 等人在 [FOCS'23] 中定义）的兄弟概念。尽管平面度量的 $\varepsilon$-(半)梯子指数具有指数下界，但我们证明平面度量的 $\varepsilon$-匹配指数是多项式的（全部关于 $1/\varepsilon$）。$\varepsilon$-(半)梯子指数与 $\varepsilon$-匹配指数之间的指数分离相当令人惊讶，而其证明是我们工作的主要技术贡献。