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

Abstract

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 Håstad 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 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 exists-equal show 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.[16]). We note that these lower bounds do not apply to the exists-equal problem, as the input distribution they use generates instances inherently specific to the disjointness problem; furthermore this distribution admits a O(log k) protocol in two rounds. The exists-equal problemIn the equality problem Alice and Bob receive elements x and y of a universe [t] and they have to decide whether x = y. We define the two player communication game existsequal with parameters t and n as follows. Each player is given an n-dimensional vector from [t] n , namely x and y. The value of the game is one if there exists a coordinate i ∈ [n] such that x i = y i , zero otherwise. Clearly, this problem is the OR of n independent instances of the equality problem.The direct sum problem in communication complexity is the study of whether n instances ...

The Communication Complexity of Set Intersection and Multiple Equality Testing

“…By applying the reduction of Theorem 1 to Theorem 9, we conclude that SetDisjointness can be solved in r + 1 rounds using O(Ek 1/r ) bits of communication. In this particular case we actually do not need Theorem 1; it is possible to solve SetDisjointness directly in r rounds with O(Ek 1/r ) communication by an algorithm along the lines of Theorem 9 or [ST13]. Theorem 1 can also be applied to Theorem 8 to yield a SetIntersection protocol using r + 1 rounds and O(rEk 1/r ) communication, but here we do not see how to solve the problem directly in r rounds.…”

Near Log-Convexity of Measured Heat in (Discrete) Time and Consequences

“…To summarize the above results, the 1-round communication complexity of both Disj n k and Ham n k is Θ(k log(k/δ)) by [Buh+12; Sag11; JW11] and [Hua+06]. We know that Disj n k can be solved much more efficiently if one is allowed multiple rounds: firstly the log k factor can be removed [HW07] and secondly the error probability can be brought down to exp(−k) [ST13], by using no more than log * k rounds. It is an interesting question whether similar efficiency improvements can be obtained for Ham n k also, by using multiple rounds.…”

“…Our tight Ω(k log(k/δ)) lower bound for the δ-error communication complexity of the k-Hamming distance problem (that applies whenever k 2 < δn) answers affirmatively a conjecture stated in [BBG14] (Conjecture 1.4). Prior to our work, the best impossibility results for this problem were an Ω(k log (r) k) bits lower bound (log (r) being the iterated logarithm) that applies to any randomized r-round communication protocol [ST13], and an Ω(k log(1/δ)) lower bound that applies to any δ-error randomized protocol for k < δn [BBG14].…”

A Lower Bound for Sampling Disjoint Sets

“…Because of its central role, Set-Disjointness has become the de facto testbed for proving new types of communication bounds. This function has been studied in the contexts of randomized [9,49,62,10,17] and quantum [25,43,63,2,66,70] protocols; multi-party number-in-hand [6,10,27,41,48,18,22] and number-on-forehead [40,71,12,66,28,57,11,69,68,61,60] models; Merlin-Arthur and related models [50,3,35,39,38,4,64,29]; with a bounded number of rounds of interaction [52,46,80,19,23]; with bounds on the sizes of the sets [42,56,59,31,26,65]; very precise relationships between communication and error...…”