Vinayak

Kacper Kluk, Hung Le, Wojciech Nadara et al.

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.

An La, Hung Le, Shay Solomon et al.

It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+ε)$-spanner with maximum degree $\tilde{O}(ε^{-d+1})$ and with lightness $\tilde{O}(ε^{-d})$; 2) a $(1+ε)$-tree cover with $\tilde{O}(n \cdot ε^{-d+1})$ trees and maximum degree of $O(1)$ in each tree. Moreover, all the parameters in these constructions are optimal: there exists an $n$-point set in $\mathbb{R}^d$, for which any $(1+ε)$-spanner has $\tildeΩ(n \cdot ε^{-d+1})$ edges and lightness $\tildeΩ(ε^{-d})$. The upper bounds for Euclidean spanners rely heavily on the spatial property of cone partitioning in $\mathbb{R}^d$, which does not seem to extend to the wider family of doubling metrics, i.e., metric spaces of constant doubling dimension. In doubling metrics, a simple spanner construction from two decades ago, the net-tree spanner, has $\tilde{O}(n \cdot ε^{-d})$ edges, and it could be transformed into a spanner of maximum degree $\tilde{O}(ε^{-d})$ and lightness $\tilde{O}(n \cdot ε^{-(d+1)})$ by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a $(1+ε)$-tree cover with $\tilde{O}(ε^{-d})$ trees. Despite a large body of work, the problem of obtaining tight bounds for spanners and tree covers in the wider family of doubling metrics has remained elusive. We resolve this problem by presenting: 1) a surprisingly simple and tight lower bound, which shows that the net-tree spanner and its pruned version are optimal with respect to all the involved parameters, 2) a new construction of $(1+ε)$-tree covers with $\tilde{O}(n \cdot ε^{-d})$ trees, with maximum degree $O(1)$ in each tree. This construction is optimal with respect to the number of trees and maximum degree.