Towards Plane Spanners of Degree 3

TL;DR

This paper introduces algorithms for constructing plane spanners with maximum degree 3 for points in convex position, rectangular lattices, and general position, achieving specific stretch factors and linear Steiner points.

Contribution

It provides new algorithms for degree-3 plane spanners with explicit stretch factors for various point configurations, including convex, lattice, and general positions.

Findings

Constructed a plane (3+4π)/3-spanner for convex points with degree 3.

Developed a plane 3√2-spanner for rectangular lattice points with degree 3.

Presented methods to compute degree-3 spanners with linear Steiner points for general position points.

Abstract

Let $S$ be a finite set of points in the plane that are in convex position. We present an algorithm that constructs a plane $\frac{3+4\pi}{3}$-spanner of $S$ whose vertex degree is at most 3. Let $\Lambda$ be the vertex set of a finite non-uniform rectangular lattice in the plane. We present an algorithm that constructs a plane $3\sqrt{2}$-spanner for $\Lambda$ whose vertex degree is at most 3. For points that are in the plane and in general position, we show how to compute plane degree-3 spanners with a linear number of Steiner points.