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 exist-equal problem, the players receive vectors x,y\in [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=\Omega(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size \Omega(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

Exponential Separation of Quantum Communication and Classical Information

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha. As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.Comment: v1, 36 pages, 3 figure