@MISC{_averageand,

author = {},

title = {Average and Randomized Communication Complexity},

year = {}

}

Abstract-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. Let 3 bethesetoffunctionsf: {0;~~,n-l)X{0;~~,n-l}-t{O,l) that contain s ones (i.e., I{(x, v): f(x, y) =1}1=s). It is shown that for every n I s I ti2/2, 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 fors = 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 log(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. I.

randomized communication complexity two-variable function bounded-error communication-complexity measure log log typical behavior tight bound logpi cluss-the error-free worst case abstract-the communication complexity complexity measure log class log bit average number c-error average randomized complexity several communication-complexity measure error-free worst case deterministic complexity cluss-the c-error boolean function information bit error-free average deterministic complexity computation error

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