Time-Optimal Visibility-Related Algorithms on Meshes with Multiple Broadcasting

Given a collection of objects in the plane along with a viewpoint !, the visibility problem involves determining the portion of each object that is visible to an observer positioned at !. The visibility problem is central to various application areas including computer graphics , image processing, VLSI design, and robot navigation, among many others. The main contribution of this work is to provide time-optimal solutions to this problem for several classes of objects, namely ordered line segments, disks, and iso-oriented rectangles in the plane. In addition, our visibility algorithm for line segments is at the heart of time-optimal solutions for determining, for each element in a given sequence of real numbers, the position of the nearest larger element within that sequence, triangulating a set of points in the plane, determining the visibility pairs among a set of vertical line segments, and constructing the dominance and visibility graphs of a set of iso-oriented rectangles in the plane. All the algorithms in this paper involve an input of size n and run in O(log n) time on a mesh with multiple broadcasting of size n n. This is the rst instance of time-optimal solutions for these problems on this architecture.