Proceedings of the twelfth annual symposium on Computational geometry
Fully dynamic search trees for an extension of the BSP model
Proceedings of the eighth annual ACM symposium on Parallel algorithms and architectures
Randomized fully-scalable BSP techniques for multi-searching and convex hull construction
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Parallel Bridging Models and Their Impact on Algorithm Design
ICCS '01 Proceedings of the International Conference on Computational Science-Part II
Optimal, Output-Sensitive Algorithms for Constructing Upper Envelope of Line Segments in Parallel
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
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There are now a number of fundamental problems in computational geometry that have optimal algorithms on PRAM models. This paper presents randomized parallel algorithms that execute on an $n$-processor butterfly interconnection network in $O(\log n)$ time for the following problems of input size $n$: trapezoidal decomposition, visibility, triangulation, and two-dimensional convex hull. These algorithms involve tackling some of the very basic problems, like binary search and load balancing, that are taken for granted in PRAM models. Apart from a two-dimensional convex hull algorithm, these are the first nontrivial geometric algorithms that attain this performance on fixed connection networks. These techniques use a number of ideas from Flashsort that have to be modified to handle more difficult situations; it seems likely that they will have wider applications.