Sensitive functions and approximate problems

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