Separating Deterministic from Randomized Multiparty Communication Complexity

Abstract We solve some fundamental,problems in the number-on-forehead (NOF) k-player communication model. We show that there exists a function which has at most logarithmic communication,complexity for randomized,protocols with a one-sided error probability of 1/3 but which has linear communication complexity for deterministic protocols. The result is true for k = n,(1) players, where n is the number of bits on each players’ forehead. This separates the analogues of RP and P in the NOF communication model. We also show that there exists a function which has constant randomized,complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential and we do not know of anyexplicitfunctionwhichallowssuchseparations. However,forthe 3-playercase we exhibitan explicit function which has (loglog n) randomized,complexity for private coins but only constant complexity for public coins. It follows from our existential result that any function that is complete for the class of functions with polylogarithmic nondeterministic k-player communication,complexity,does not have polylogarithmic deterministic complexity. We show that the set intersection function, which is complete in the number- in-hand model, is not complete in the NOF model under cylindrical reductions. 2