Scaleable Parallel Algorithms for Lower Envelopes with Applications

This paper considers a variety of geometric problems based in description of thelower envelopefunction, on input sets of sizenusing acoarse grained multicomputermodel consisting ofpprocessors withΩ(n/p) local memory each (i.e.,Ω(n/p) memory cells ofΘ(logn) bits apiece), where the processors are connected to an arbitrary interconnection network. We give an efficient scaleable parallel algorithm for computation of the lower envelope and use this algorithm to obtain efficient solutions for a variety of geometric problems, including •the minimization of the Hausdorff distance between two finite sets on the real line when one is subject to translation; •theCommon Intersection Problemfor vertically convex planar polygons; and •several problems inDynamic Computational Geometry, in which we consider geometric questions for systems of moving objects. All of the algorithms presented arescaleablein that they are applicable and efficient over a very wide range of ratios of problem size to number of processors. In addition to the practicality imparted by scaleability, these algorithms are easy to implement in that all required communications can be achieved by a small number of calls to standard global routing operations.