The equality problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worst-case input) for a protocol that uses limited interaction—i.e., a bounded number of rounds of communication—and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal rounds-versus-cost tradeoffs for equality: both expected communication cost and information cost scale as Θ(log log · · · logn), with r − 1 logs, where r is the number of rounds. These bounds hold even when the false negative rate approaches 1. For the case of zero-error communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications.

Abstract. We obtain a strong direct product theorem for two-party bounded round communication complexity. Let sucr(µ, f, C) denote the maximum success probability of an r-round communication protocol that uses at most C bits of communication in computing f(x, y) when (x, y) ∼ µ. Jain et al. [12] have recently showed that if sucr(µ, f, C) ≤ 23 and T (C − Ω(r2)) · n r, then sucr(µ

In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give match-ing lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is to determine whether the sets intersect. For this problem, we give a protocol that communicates a total of O(k log(r) k) bits over r rounds and errs with very small probability. Here we can take r = log ∗ k to obtain a O(k) total communication log ∗ k-round protocol with exponentially small error probability, improving on the O(k)-bits O(log k)-round constant error probability protocol of H̊astad and Wigderson from 1997. In the exist-equal problem, the players receive vectors x, y ∈ [t]n and the goal is to determine whether there exists a coordinate i such that xi = yi. Namely, the exists-equal problem is the OR of n equality problems. Observe that exists-equal is an instance of sparse set disjointness with k = n, hence the protocol above applies here as well, giving an O(n log(r) n) upper bound. Our main technical contribution