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<p>Given a collection of segments in the plane, if we regard the segments as opaque barriers, their upper envelope consists of the portions of the segments visible from point <tmath>(0,+\infty)</tmath>. In this paper, we present deterministic parallel methods for constructing the upper envelope of segments on the weakest shared-memory model, the EREW PRAM. We show that we can find the upper envelope of <it>n</it> line segments optimally in <tmath>O(\log n)</tmath> time using <tmath>O(n)</tmath> processors. Furthermore, if the segments are nonintersecting and their endpoints are sorted in <it>x</it>-coordinate, then we can reduce the number of processors to <tmath>O(n/\log n)</tmath>. Our method implies that we can find the upper envelope sequentially in <tmath>O(n\log \log n)</tmath> time, which improves previous results. We also show that we can find the upper envelope of <it>n</it><it>k</it>-intersecting segments (any pair of the segments intersects at most <it>k</it> times) with a slightly larger time and processor bound.</p>
Computational geometry, upper envelope, Davenport-Schinzel sequence, visibility, convex hull, EREW PRAM model.
Wei Chen, Koichi Wada, "On Computing the Upper Envelope of Segments in Parallel", IEEE Transactions on Parallel & Distributed Systems, vol. 13, no. , pp. 5-13, January 2002, doi:10.1109/71.980023
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