There are now a number of fundamental problems in computational geometry that have optimal algorithms on PRAM models. We present randomized parallel algorithms which execute on an n-processor buttery inter-connection network in O(log n) time for the following problems of input size n: trapezoidal decomposition, visibility, triangulation and 2-D convex hull. These algorithms involve tackling some of the very basic problems like binary search and load-balancing that we take for granted in PRAM models. Apart from a 2-D convex hull algorithm, these are the rst non-trivial geometric algorithms which attain this performance on xed connection networks. Our techniques use a number of ideas from Flashsort which have to be modied to handle more dicult situations; it seems likely that they will have wider applications. 1 Introduction 1.1 Motivation and overview In the past decade, we have witnessed a systematic growth in the state-of-art of parallelizing algorithms in the PRAM envi...

A survey of techniques for solving geometric problems in parallel is given, both for shared memory parallel machines and for networks of processors. Open problems are also discussed, as well as directions for future research. 'This work was supported by the office oi Naval Research under Contracts N00014-84-K-0502 and

) Omer Berkman 1,4 Dany Breslauer 1,2 Zvi Galil 1,2 Baruch Schieber 3 Uzi Vishkin 1,4,5 Summary of Results. We establish that several problems are highly parallelizable. For each of these problems, we design an optimal O (loglogn ) time parallel algorithm on the Common CRCW PRAM model which is the weakest among the CRCW PRAM models. These problems include: # all nearest smaller values, # preprocessing for answering range maxima queries, # several problems in Computational Geometry, # string matching. Until recently, such algorithms were known only for finding the maximum and merging. A new lower bound technique is presented showing that some of the new O (loglogn ) upper bounds cannot be improved even when non optimal algorithms are used. The technique extends Ramsey-like lower bound argumentation due to auf der Heide and Wigderson [MW-85]. Its most interesting applications are for Computational Geometry problems for which no previous lower bounds are known. ###############...