Average and randomized communication complexity

The communication complexity of a two-variable function f (x , y) is the number of information bits two communicators need to exchange to compute f when, initially, each knows only one of the variables. There are several communication-complexity measures corresponding to whether 1) the worst case or average number of bits is considered, 2) computation errors are allowed or not, and 3) randomization is allowed or not. Tight bounds are provided for the typical behavior of all bounded-error communication-complexity measures of Boolean functions. every n I s I t i 2 / 2 , the communication-complexity measures fall into two classes: logpi cluss-the error-free worst case randomized complexity and, more importantly, the error-free worst case deterministic complexity of most functions in 3 are between log n-4 and log n + 1 bits (this holds even f o r s = n) ; /og(s/n) cluss-the c-error worst case randomized complexity and the c-error average randomized complexity of most functions in e are between (1-2c)(log (s/ n)-2 log log (s / n) and (1-2c)(log(s/n) + 5.3 loglog n) bits. More importantly, the error-free average deterministic complexity of all functions in is less than log(s/n)+8.3loglogn bits. For most of these functions it is also t log(s/n)-2loglog(s/n) bits. The difference between the complexities of the log n class and the l o g (s / n) class ranges from a constant (for s = n2/2) to exponential (for s-n log n). In particular, since most functions have about n2/2 ones, all their complexity measures are around log n bits.