This paper studies the problem of enumerating all maximal collinear subsets of size at least three in a given set of @math points. An algorithm for this problem, besides solving degeneracy testing and the exact fitting problem, can also help with other problems, such as point line cover and general position subset selection. The classic algorithm of Edelsbrunner and Guibas can find these subsets in @math time in the dual plane. We present an alternative algorithm that, although asymptotically slower than their algorithm in the worst case, is simpler to implement and more amenable to parallelization. If the input points are decomposed into @math convex polygons, our algorithm has time complexity @math and space complexity @math . Our algorithm can be parallelized on the CREW PRAM with time complexity @math using @math processors.
We propose the jirst optimal parallel algorithm computing arrangements of hyperplanes in Ed (d 2 2). The algorithm is randomized and computes the arrangement of n hyperplanes within expected logarithmic time on a CRCW-PRAM with O(nd log n) processors. Parallel algorithms used in Very Large Scale Integration physical design bring significant challenges for their efficient and effective design and implementation. The rectangle intersection problem is a subset of the plane sweep problem, a topic of computational geometry and a component in design rule checking, parasitic resistance-capacitance extraction, and mask processing flows. A variant of a plane sweep algorithm that is embarrassingly parallel and therefore easily scalable on multicore machines and clusters, while exceeding the best-known parallel plane sweep algorithms on real-world tests, is presented in this letter. In this paper we consider the following problem: Given a set ? ofn lines in the plane, partition the plane intoO(r2) triangles so that no triangle meets more thanO(n r) lines of ?. We present a deterministic algorithm for this problem withO(nr logn r) running time, where? is a constant <3.33. The plane sweep algorithm is a foundational algorithm for many geometric and spatial computations; thus, improvements in the algorithm have far reaching effects in many applications. In this paper, we examine the performance of the serial plane sweep algorithm, and introduce a parallelization technique for the algorithm that is suitable to multi-core computers. The parallelization technique is described in detail and shown to be correct. Finally, experiments are performed using multiple data sets on computers with varying numbers of processing cores. We show that our algorithm achieves significant speedups over the serial plane sweep algorithm using a wide range of input parameters; thus, our algorithm achieves good performance without the need to tune the input parameters for specific input cases. In this paper we give efficient parallel algorithms for a number of problems from computational geometry by using versions of parallel plane sweeping. We illustrate our approach with a number of applications, which include:General hidden-surface elimination (even if the overlap relation contains cycles).CSG boundary evaluation.Computing the contour of a collection of rectangles.Hidden-surface elimination for rectangles. There are interesting subproblems that we solve as a part of each parallelization. For example, we give an optimal parallel method for building a data structure for line-stabbing queries (which, incidentally, improves the sequential complexity of this problem). Our algorithms are for the CREW PRAM, unless otherwise noted.