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 φ of the reconciled data S
_A
∪ 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.