Near-Linear-Time Deterministic Plane Steiner Spanners and TSP Approximation for Well-Spaced Point Sets

We describe an algorithm that takes as input n points in the plane and a parameter {\epsilon}, and produces as output an embedded planar graph having the given points as a subset of its vertices in which the graph distances are a (1 + {\epsilon})-approximation to the geometric distances between the given points. For point sets in which the Delaunay triangulation has bounded sharpest angle, our algorithm's output has O(n) vertices, its weight is O(1) times the minimum spanning tree weight, and the algorithm's running time is bounded by O(n \sqrt{log log n}). We use this result in a similarly fast deterministic approximation scheme for the traveling salesperson problem.Comment: Appear at the 24th Canadian Conference on Computational Geometry. To appear in CGT

On Euclidean Steiner (1+?)-Spanners

Lightness and sparsity are two natural parameters for Euclidean (1+?)-spanners. Classical results show that, when the dimension d ? ? and ? > 0 are constant, every set S of n points in d-space admits an (1+?)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ? > 0 for constant d ? ? have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+?)-spanner. They gave upper bounds of O?(?^{-(d+1)/2}) for the minimum lightness in dimensions d ? 3, and O?(?^{-(d-1))/2}) for the minimum sparsity in d-space for all d ? 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+?)-spanners of lightness O(?^{-1}log?) in the plane, where ? ? ?(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+?)-spanners. Using a new geometric analysis, we establish lower bounds of ?(?^{-d/2}) for the lightness and ?(?^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ? 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+?)-spanners of lightness O(?^{-1}log n) for n points in Euclidean plane

Near-Linear-Time Deterministic Plane Steiner Spanners for Well-Spaced Point Sets

See article for Abstract.This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/computational-geometry/Keywords: Sparse, Euclidean TSP, Spanne

Experimental study of geometric t-spanners : a running time comparison

The construction of t-spanners of a given point set has received a lot of attention, especially from a theoretical perspective. We experimentally study the performance of the most common construction algorithms for points in the Euclidean plane. In a previous paper [10] we considered the properties of the produced graphs from five common algorithms. We consider several additional algorithms and focus on the running times. This is the first time an extensive comparison has been made between the running times of construction algorithms of t-spanners