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Tight bounds for some problems in computational geometry: the complete sub-logarithmic parallel time range

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons45456

Sen,  Sandeep
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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MPI-I-93-129.pdf
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Citation

Sen, S.(1993). Tight bounds for some problems in computational geometry: the complete sub-logarithmic parallel time range (MPI-I-93-129). Saarbrücken: Max-Planck-Institut für Informatik.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-B750-7
Abstract
There are a number of fundamental problems in computational geometry for which work-optimal algorithms exist which have a parallel running time of $O(\log n)$ in the PRAM model. These include problems like two and three dimensional convex-hulls, trapezoidal decomposition, arrangement construction, dominance among others. Further improvements in running time to sub-logarithmic range were not considered likely because of their close relationship to sorting for which an $\Omega (\log n/\log\log n )$ is known to hold even with a polynomial number of processors. However, with recent progress in padded-sort algorithms, which circumvents the conventional lower-bounds, there arises a natural question about speeding up algorithms for the above-mentioned geometric problems (with appropriate modifications in the output specification). We present randomized parallel algorithms for some fundamental problems like convex-hulls and trapezoidal decomposition which execute in time $O( \log n/\log k)$ in an $nk$ ($k > 1$) processor CRCW PRAM. Our algorithms do not make any assumptions about the input distribution. Our work relies heavily on results on padded-sorting and some earlier results of Reif and Sen [28, 27]. We further prove a matching lower-bound for these problems in the bounded degree decision tree.