Multiparty communication complexity and very hard functions

• Authors: Pavol Ďuriš
• Venue: Information and Computation
• Year: 2004

A boolean function f(x1,...,xn) with xi ∈ {0,1}m for each i is hard if its nondeterministic multiparty communication complexity (introduced in [in: Proceedings of the 30th IEEE FOCS, 1989, p. 428-433]), C(f), is at least nm. Note that C(f) ≤ nm for each f(x1,...,xn) with xi ∈ {0, 1}m for each i. A boolean function is very hard if it is hard and its complementary function is also hard. In this paper, we show that randomly chosen boolean function f(x1,...,xn) with xi ∈ {0,1}m for each i is very hard with very high probability (for n ≥ 3 and m large enough). In [in: Proceedings of the 12th Symposium on Theoretical Aspects of Computer Science, LNCS 900, 1995, p. 350-360], it has been shown that if f(x1,...,xk,...,xn) = f1(x1,..., xk) ċ f2(xk+1,...,xn), where C(f1) 0 and C(f2) 0, then C(f) = C(f1) + C(f2). We prove here an analogical result: If f(x1,..., xk,..., xn) = f1(x1,...,xk) ⊕ f2(xk+1,..., xn) then DC(f) = DC(f1) + DC(f2), where DC(g) denotes the deterministic multiparty communication complexity of the function g and "⊕" denotes the parity function.