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A Polynomial Coreset for Furthest Neighbor in Planar Metrics

Kacper Kluk, Hung Le, Wojciech Nadara, Marcin Pilipczuk, Hector Tierno, Vinayak

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.

A Polynomial Coreset for Furthest Neighbor in Planar Metrics

Abstract

A furthest neighbor data structure on a metric space and a set answers the following query: given , output maximizing ; in the approximate version, it is allowed to report any with for an accuracy parameter . A particular type of approximate furthest neighbor data structure is an -coreset: a small subset such that for every query there is a feasible answer . Our main result is that in planar metrics there always exists an -coreset for furthest neighbors of size bounded polynomially in . 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 -coreset for furthest neighbors and an invariant of a metric space that we call an -comatching index -- a sibling of -(semi-)ladder index, a.k.a, -scatter dimension, as defined by Abbasi et al [FOCS'23]. While the -(semi-)ladder index of planar metrics admits an exponential lower bound, we show that the -comatching index of planar metrics is polynomial, all in . The exponential separation between -(semi-)ladder and -comatching is rather surprising, and the proof is the main technical contribution of our work.

Paper Structure

This paper contains 69 sections, 54 theorems, 90 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Furthest neighbor.
  3. Structural invariants of metric spaces.
  4. Our main results
  5. Other results
  6. Graphs of bounded genus
  7. Beyond surface-embedded graphs: lower bounds.
  8. Coreset for $k$-center
  9. The lower bound: $(k,d)$-comatching.
  10. The upper bound: $(k,\varepsilon)$-comatching.
  11. Organization of the paper
  12. Overview of the Proof of Theorem \ref{['thm:comatching-bound']}
  13. Approximate distance profiles.
  14. Separation lemma.
  15. Phase one: reduction to a local instance.
  16. ...and 54 more sections

Key Result

Theorem 1.2

Given a point set $P$ in a planar metric $(V,\mathsf{dist})$, and a parameter $\varepsilon\in(0,1)$, there exists a furthest neighbor $\varepsilon$-coreset of size $\operatorname{poly}(1/\varepsilon)$ for $P$ that can be constructed in polynomial time.

Figures (14)

  • Figure 1: The dotted line indicates distance $\geq R$, and the solid line indicates distance $\leq(1-\varepsilon)R$.
  • Figure 2: $\varepsilon$-double ladder. Again, the dotted line indicates distance $\geq R$, and the solid line indicates distance $\leq(1-\varepsilon)R$.
  • Figure 3: Illustration for the proof of the separation lemma.
  • Figure 4: The plane drawing in (a) depicts a local instance $G$, and the local property is illustrated by the pair $(p, q)$. Here the edges of $G$ which belong to paths $\llbracket p_{j_1}q_{j_2} \rrbracket$ for $p_{j_1}, q_{j_2} \neq p, q$ are drawn as solid lines and any other edges as dotted lines. We can draw a link (the dashed red curve) in the plane connecting $p$ and $q$ without intersecting any of the solid lines, so $(p, q)$ does indeed have the local property. The linked non-crossing instance is shown in (b) with respect to pairs $(p_1, q_1)$ and $(p_2, q_2)$. For each pair, the link represented by the dashed line does not intersect the solid lines of the same color to show the local property analogously to a). Furthermore the links cannot intersect each other (while maintaining the local property) since the regions in which links can lie are disjoint, as represented, which guarantees the non-crossing property.
  • Figure 5: The contradiction in the proof of Lemma \ref{['lem:over:ex']}.
  • ...and 9 more figures

Theorems & Definitions (159)

  • Definition 1.1: Furthest Neighbor $\varepsilon$-Coreset
  • Theorem 1.2
  • Definition 1.3: metric ladders
  • Theorem 1.4: Coreset to Comatching
  • Theorem 1.4: Comatching in Planar Metrics
  • proof : Proof of \ref{['thm:1center']}
  • Theorem 1.5
  • Theorem 1.5: Lower bound construction for furthest neighbor and comatchings
  • Corollary 1.6
  • Definition 1.7: $k$-Center $\varepsilon$-Coreset
  • ...and 149 more