The Unbounded-Error Communication Complexity of Symmetric Functions

We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e., f(x, y) = D(vertical bar x Lambda y vertical bar), where D : (0, 1,..., n) -> {0, 1} is a given predicate and x, y range over {0, 1}(n). Specifically, we show that the communication complexity of f is between Theta(k/ log(5) n) and Theta(k log n), where k is the number of value changes of D in (0, 1,..., n). The unbounded-error model is the most powerful of the basic models of communication (both classical and quantum), and proving lower bounds in it is a considerable challenge. The only previous nontrivial lower bounds for explicit functions in this model appear in the ground breaking work of Forster (2001) and its extensions. Our proof is built around two novel ideas. First, we show that a given predicate D gives rise to a rapidly mixing random walk on Z(2)(n), which allows us to reduce the problem to communication lower bounds for "typical" predicates. Second, we use Paturi's approximation lower bounds (1992), suitably generalized here to clusters of real nodes in [0, n] and interpreted in their dual form, to prove that a typical predicate behaves analogous to PARITY with respect to a smooth distribution on the inputs.