Sensitive Functions and Approximate Problems

We investigate properties offunctions that are good measures of the CRCW PRAM complexity of computing them. While the 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 everywhere ·sepsitivity, introduced by 'Vishkin and Wigderson. Specijically we show thatthe time required to compute a function f. : Dn R of everywhere sensitivity es(f) with P ~ n processors and unbounded memory is O(logpoges(f)j(log 4PIDI loges(f))]). This improves previous results of Azar, and Vishkin and Wigderson. We US'e this lower bound to derive new lower bounds for some approximate problems. These problems can often besolved faster than their exacf counterparts and for many applications, it is sufficient to solve the approximate problem. We ,show that approximate selection requires timeO(logpognjlogk]) with kn processors and approximate counting with accuracy >. ~ 2 requires time O(logpognj(logk+log>.)1) with kn processors. In particular, for constant accuracy, no lower bounds were known for these problems.