Two-Party Function Computation on the Reconciled Data

In this paper, we initiate a study of a new problem termed function computation on the reconciled data, which generalizes a set reconciliation problem in the literature. Assume a distributed data storage system with two users \(A\) and \(B\). The users possess a collection of binary vectors \(S_{A}\) and \(S_{B}\), respectively. They are interested in computing a function \(\phi\) of the reconciled data \(S_{A} \cup S_{B}\). It is shown that any deterministic protocol, which computes a sum and a product of reconciled sets of binary vectors represented as nonnegative integers, has to communicate at least \(2^n + n - 1\) and \(2^n + n - 2\) bits in the worst-case scenario, respectively, where \(n\) is the length of the binary vectors. Connections to other problems in computer science, such as set disjointness and finding the intersection, are established, yielding a variety of additional upper and lower bounds on the communication complexity. A protocol for computation of a sum function, which is based on use of a family of hash functions, is presented, and its characteristics are analyzed.