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Abstract:
We investigate properties of functions that are good measures of the
CRCW PRAM complexity of computing them. While the {\em block
sensitivity} is known to be a good measure of the CREW PRAM
complexity, no such measure is known for CRCW PRAMs. We show that the
complexity of computing a function is related to its {\em everywhere
sensitivity}, introduced by Vishkin and Wigderson. Specifically we
show that the time required to compute a function $f:D^n \rightarrow R$ of everywhere
sensitivity
$ \es
(f)$ with $P \geq n$ processors and unbounded memory
is $
\Omega
(\log [\log \es(f)/(\log 4P|D| - \log \es(f))])$.
This
improves previous results of Azar, and Vishkin and Wigderson. We use
this lower bound to derive new lower bounds for some {\em approximate
problems}. These problems can often be solved faster than their exact
counterparts and for many applications, it is sufficient to solve the
approximate problem. We show
that {\em approximate selection} requires time
$\Omega(\log [\log n/\log k])$ with $kn$ processors and {\em
approximate counting} with accuracy $\lambda \geq 2$ requires time
$\Omega(\log [\log n/(\log k + \log \lambda)])$ with $kn$ processors.
In particular, for constant accuracy, no lower bounds were known for
these problems.
