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On Computing the Upper Envelope of Segments in Parallel
January 2002 (vol. 13 no. 1)
pp. 5-13

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 (0,+\infty). 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 n line segments optimally in O(\log n) time using O(n) processors. Furthermore, if the segments are nonintersecting and their endpoints are sorted in x-coordinate, then we can reduce the number of processors to O(n/\log n). Our method implies that we can find the upper envelope sequentially in O(n\log \log n) time, which improves previous results. We also show that we can find the upper envelope of nk-intersecting segments (any pair of the segments intersects at most k times) with a slightly larger time and processor bound.

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Index Terms:
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 and Distributed Systems, vol. 13, no. 1, pp. 5-13, Jan. 2002, doi:10.1109/71.980023
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