A Polynomial Coreset for Furthest Neighbor in Planar Metrics

2026-03-30

AI Summary

This work addresses the construction of efficient coresets for approximate farthest neighbor queries in planar metric spaces. Resolving an open problem posed by de Berg and Theocharous, it establishes for the first time the existence of an ε-coreset of size polynomial in 1/ε. The authors introduce a novel metric invariant—the ε-comatching number—and demonstrate that it can differ exponentially from the previously studied ε-ladder number. By leveraging structural properties of planar graphs together with combinatorial arguments, they tightly relate this new invariant to coreset size, thereby improving the best-known upper bound from exponential to polynomial in 1/ε. This advancement significantly enhances both the theoretical understanding and practical applicability of approximate farthest neighbor search in planar metrics.

Abstract

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.

Problem

• furthest neighbor
• coreset
• planar metrics
• approximation
• metric space

Innovation

• ε-coreset
• furthest neighbor
• planar metrics
• ε-comatching index
• polynomial bound