Revisiting Visibility in the Plane

We consider two closely related problems: computing the region visible from a point amid simple polygonal obstacles and computing the lower envelope of a set of disjoint segments. Visibility problems such as these were proposed and promptly solved in the late ’80s and early ’90s before the widespread popularity of the word RAM model. All previously published algorithms thus run in Ω(n log n) time, although they can be sped up in the word RAM model to some extent by substituting appropriate word RAM data structures and algorithms. Our main result is a deterministic linear-time algorithm for the case in which endpoints are presorted. Using the fastest known integer sorting algorithm of Han and Thorup [17], we thus obtain an algorithm for the general case that runs in O(n √ log log n) time. We expect our algorithm for the presorted case to be practically efficient. Our algorithm actually solves the more general problem of computing the lower envelope of a set of non-intersecting partial functions defined over intervals. We obtain our results via a novel approach that considers a partial lower envelope problem in which we need only find the lowest partial function at q ≤ 2n x-coordinates. We combine this algorithm with another that exploits the bounded precision of x-coordinates of endpoints and otherwise performs only comparisons to determine the vertical ordering of partial functions. In addition to our word RAM result, we obtain a better understanding of the inherent complexity of our visibility problems, resulting in a pointer machine algorithm that runs in O(nα(n)) time as long as endpoints are appropriately presorted.