Communication complexity of permutation-invariant functions

Abstract

Motivated by the quest for a broader understanding of upper bounds in communication complexity, at least for simple functions, we introduce the class of functions. A partial function f: {0, 1}n × {0, 1}n → {0, 1, ?} is permutation-invariant if for every bijection π: {1, . . ., n} → {1, . . ., n} and every x, y ∈ {0, 1}n, it is the case that f(x, y) = f(xπ, yπ). Most of the commonly studied functions in communication complexity are permutation-invariant. For such functions, we present a simple complexity measure (computable in time polynomial in n given an implicit description of f) that describes their communication complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the communication complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as Set-Disjointness and Indexing, while complementing them with the relatively lesser-known upper bounds for Gap-Inner-Product (from the sketching literature) and Sparse-Gap-Inner-Product (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of communication complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in communication complexity after an additive O(log log n) overhead.