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.

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The main contribution of this paper is to present simple and elegant podality-based algorithms for a variety of computational tasks motivated by, and finding applications to, pattern recognition, computer graphics, computational morphology, image processing, robotics, computer vision, and VLSI design. The problems that we address involve computing the convex hull, the diameter, the width, and the smallest area enclosing rectangle of a set of points in the plane, as well as the problems of finding the maximum Euclidian distance between two planar sets of points, and of constructing the Minkowski sum of two convex polygons. Specifically, we show that once we fix a positive constant e, all instances of size m, n m n 1 2 + £ £ e e j of the problems above, stored in the first m n columns of a mesh with multiple broadcasting of size n n ¥ can be solved time-optimally in Q m n e j time.

Given a sequence of objects in the plane along with a viewpoint 0 , the visibility problem involves determining the portion of each object that is visible to an observer positioned at 0 . The main contribution of this work is to provide tame-optimal solutions to this problem for two classes of objects, namely disks and iso-oriented rectangles in the plane. This problem is of importance in various fields like computer graphics, VLSI design, and robot navigation. Additionally, the visibility algorithm provides the basis for a timeoptimal algorithm to triangulate a set of points in the plane. Specifically, all algorithms an this paper involve an input of size n and run in O(logn) time on a mesh with muliiple broadcasting of site n x n . This is the first instance of time-optimal solutionsfor these problems on this architecture.