Towards Plane Spanners of Degree 3

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

A plane 1.88-spanner for points in convex position

Let $S$ be a set of $n$ points in the plane that is in convex position. For a real number t>1, we say that a point $p$ in $S$ is $t$-good if for every point $q$ of $S$, the shortest-path distance between $p$ and $q$ along the boundary of the convex hull of $S$ is at most $t$ times the Euclidean distance between $p$ and $q$. We prove that any point that is part of (an approximation to) the diameter of $S$ is $1.88$-good. Using this, we show how to compute a plane $1.88$-spanner of $S$ in $O(n)$ time, assuming that the points of $S$ are given in sorted order along their convex hull. Previously, the best known stretch factor for plane spanners was $1.998$ (which, in fact, holds for any point set, i.e., even if it is not in convex position).</p