MPI-I-93-129
Tight bounds for some problems in computational geometry: the complete sub-logarithmic parallel time range
Sen, Sandeep
July 1993, 12 pages.
.
Status: available - back from printing
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.
URL to this document: https://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1993-129
BibTeX
@TECHREPORT{Sandeep93,
AUTHOR = {Sen, Sandeep},
TITLE = {Tight bounds for some problems in computational geometry: the complete sub-logarithmic parallel time range},
TYPE = {Research Report},
INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
NUMBER = {MPI-I-93-129},
MONTH = {July},
YEAR = {1993},
ISSN = {0946-011X},
}