Robust Set Reconciliation via Locality Sensitive Hashing

Abstract

We consider variations of set reconciliation problems where two parties, Alice and Bob, each hold a set of points in a metric space, and the goal is for Bob to conclude with a set of points that is close to Alice's set of points in a well-defined way. This setting has been referred to as robust set reconciliation. More specifically, in one variation we examine the goal is for Bob to end with a set of points that is close to Alice's in earth mover's distance, and in another the goal is for Bob to have a point that is close to each of Alice's. The first problem has been studied before; our results scale better with the dimension of the space. The second problem appears new.Our primary novelty is utilizing Invertible Bloom Lookup Tables in combination with locality sensitive hashing. This combination allows us to cope with the geometric setting in a communication-efficient manner.Gap Guarantee model. In the second model, which we introduce, we aim for a stronger guarantee of closeness for every point, and consider the necessary communication. Here Bob's final point set S B will be of the form S B ∪ T A , where T A ⊂ S A includes every point in S A which is at least some chosen distance r 2 from every point in S B . Note that T A is allowed to contain additional points from S A beyond these. That is, Bob is guaranteed that every point in the union of Alice's and Bob's original sets is close to some point in his final set. In order to achieve nontrivial 1 We note that Definition 2 of [7] makes the additional stipulation that S B ⊂ SA ∪SB, however neither our protocol nor the protocol of [7] meet this requirement. Both include points in S B that approximate, without necessarily equaling, points from SA.2 TheÕ here hides log factors of n and log factors of parameters depending on the metric space, in particular |U |.