Shortest Path in a Polygon using Sublinear Space

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon $\Polygon$ with $n$ vertices in a read only memory, and additional working memory of size $\Space$, the new algorithm computes the shortest path (in $\Polygon$) in $O( n^2 /\, \Space )$ expected time. This requires several new tools, which we believe to be of independent interest.