Does a Robot Path Have Clearance C?

Abstract

Most path planning problems among polygonal obstacles ask to find a path that avoids the obstacles and is optimal with respect to some measure or a combination of measures, for example an u-to-v shortest path of clearance at least c, where u and v are points in the free space and c is a positive constant. In practical applications, such as emergency interventions/evacuations and medical treatment planning, a number of u-to-v paths are suggested by experts and the question is whether such paths satisfy specific requirements, such as a given clearance from the obstacles. We address the following path query problem: Given a set S of m disjoint simple polygons in the plane, with a total of n vertices, preprocess them so that for a query consisting of a positive constant c and a simple polygonal path $\pi$ with k vertices, from a point u to a point v in free space, where k is much smaller than n, one can quickly decide whether $\pi$ has clearance at least c (that is, there is no polygonal obstacle within distance c of $\pi$). To do so, we show how to solve the following related problem: Given a set S of m simple polygons in $\mathfrak {R}^{2}$, preprocess S into a data structure so that the polygon in S closest to a query line segment s can be reported quickly. We present an $O(t \log n)$ time, O(t) space preprocessing, $O((n / \sqrt{t}) \log ^{7/2} n)$ query time solution for this problem, for any $n ^{1 + \epsilon } \le t \le n^{2}$. For a path with k segments, this results in $O((n k / \sqrt{t}) \log ^{7/2} n)$ query time, which is a significant improvement over algorithms that can be derived from existing computational geometry methods when k is small.