By Mittelmann H.D., Weber H. (eds.)
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Extra info for Bifurcation problems and their numerical solution. Proc. Dortmund
J. Atallah and M. T. Goodrich, Efficient plane sweeping in parallel (preliminary version), Proceedings of the Second Annual ACM Symposium on Computational Geometry, Yorktown Heights, New York, June 1986, 216-225. M. J. Atallah, R. Cole, and M. T. Goodrich, Cascading divide-and-conquer: a technique for designing parallel algorithms, SIAM Journal on Computing, Vol. 18, No. 3, 1989, 499-532. M. J. Atallah, F. Dehne, R. Miller, A. -J. Tsay, Multisearch techniques for implementing data structures on a mesh-connected computer, Proceedings of the Third ACM Symposium on ParallelAlgorithms and Architectures, Hilton Head, South Carolina, July 1991, 204-214.
It is interesting to point out that this algorithm is essentially a parallelization of the algorithm due to Jarvis [Jarv73], long believed to be inherently sequential because of the incremental (point-by-point) way it constructs the convex hull. Note further that no algorithm is known for constructing the convex hull in constant time in the worst case while using asymptotically fewer that n 2 processors. By contrast, the CRCW PRAM algorithm described in [Stou88], which requires 0(n) processors and the COLLISION rule for resolving write conflicts, assumes that the data are chosen from a uniform distribution and runs in constant expected time.
Suppose that the convex hull of a point set has been computed. We now remove the hull points and compute the convex hull of the remaining set. This process, which is repeated until no points are left, is referred to as peeling. Sequentially, peeling can be performed optimally in 0(n log n) time for a planar point set [Prep85]. Peeling can also be applied to the boundary points. No efficient parallel algorithm is known for peeling, in either of its forms, at the time of this writing [ElGi9O]. 7 summarizes the results in the preceding section.
Bifurcation problems and their numerical solution. Proc. Dortmund by Mittelmann H.D., Weber H. (eds.)