A plane 1.88-spanner for points in convex position

Abstract

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).