Minimum Weight Euclidean $$(1+\varepsilon )$$-Spanners

Abstract

AbstractGiven a set S of n points in the plane and a parameter \(\varepsilon >0\), a Euclidean \((1\,+\,\varepsilon )\)-spanner is a geometric graph \(G=(S,E)\) that contains a path of weight at most \((1+\varepsilon )\Vert pq\Vert _2\) for all \(p,q\in S\). We show that the minimum weight of a Euclidean \((1+\varepsilon )\)-spanner for n points in the unit square \([0,1]^2\) is \(O(\varepsilon ^{-3/2}\,\sqrt{n})\), and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It improves upon the baseline \(O(\varepsilon ^{-2}\sqrt{n})\), obtained by combining a tight bound for the weight of an MST and a tight bound for the lightness of Euclidean \((1+\varepsilon )\)-spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to d-space for all \(d\in \mathbb {N}\): The minimum weight of a Euclidean \((1\,+\,\varepsilon )\)-spanner for n points in the unit cube \([0,1]^d\) is \(O_d(\varepsilon ^{(1-d^2)/d}n^{(d-1)/d})\), and this bound is the best possible. For the \(n\times n\) section of the integer lattice, we show that the minimum weight of a Euclidean \((1+\varepsilon )\)-spanner is between \(\varOmega (\varepsilon ^{-3/4}n^2)\) and \(O(\varepsilon ^{-1}\log (\varepsilon ^{-1})\, n^2)\). These bounds become \(\varOmega (\varepsilon ^{-3/4}\sqrt{n})\) and \(O(\varepsilon ^{-1}\log (\varepsilon ^{-1})\sqrt{n})\) when scaled to a grid of n points in \([0,1]^2\).KeywordsGeometric spannerYao-graphFarey sequences