On the Communication Complexity of Sparse Set Disjointness and Exists-Equal Problems

In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching 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 Hastad and Wigderson from 1997. In the exists-equal problem, the players receive vectors x, y ∈ [t]ⁿ and the goal is to determine whether there exists a coordinate i such that x_i = y_i. 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 in this paper is a matching lower bound: we show that when t = Ω(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size Ω(n log^(r) n). Our lower bound holds even for super-constant r ≤ log* n, showing that any O(n) bits exists-equal protocol should have log* n - O(1) rounds. Note that the protocol we give errs only with less than polynomially small probability and provides guarantees on the total communication for the harder set disjointness problem, whereas our lower bound holds even for constant error probability protocols and for the easier exists-equal problem with guarantees on the max-communication. Hence our upper and lower bounds match in a strong sense. Our lower bound on the constant round protocols for exist-sequal shows that solving the OR of n instances of the equality problems requires strictly more than n times the cost of a single instance. To our knowledge this is the first example of such a super-linear increase in complexity.