Shortest path in a polygon using sublinear space

Abstract

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 $P$ with $n$ vertices in a read only memory, and additional working memory of size $m$, the new algorithm computes the shortest path (in $P$) in $O( n^2 / m )$ expected time, assuming $m = O(n/\log^2 n)$. This requires several new tools, which we believe to be of independent interest. Specifically, we show that violator space problems, an abstraction of low dimensional linear-programming (and LP-type problems), can be solved using constant space and expected linear time, by modifying Seidel's linear programming algorithm and using pseudo-random sequences.