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

TL;DR

This paper presents a near-linear-time deterministic algorithm for constructing planar graphs that approximate Euclidean distances for well-spaced points, enabling efficient TSP approximation schemes.

Contribution

It introduces a fast deterministic method for creating plane Steiner spanners with near-optimal weight and size, improving TSP approximation for well-spaced point sets.

Findings

Graph distances approximate Euclidean distances within (1 + ε)

Algorithm runs in O(n √log log n) time for certain point sets

Provides a fast deterministic TSP approximation scheme

Abstract

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.