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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.
