Testing Equality Under the Local Broadcast Model

Abstract

In the multiparty equality problem, each of the n nodes starts with a k-bit input. If there is a mismatch between the inputs, then at least one node must be able to detect it. The cost of a multiparty equality protocol is the total number of bits sent in the protocol. We consider the problem of minimizing this communication cost under the local broadcast model for the case where the underlying communication graph is undirected. In the local broadcast model of communication, a message sent by a node is received identically by all of its neighbors. This is in contrast to the classical point-to-point communication model, where a message sent by a node to one of its neighbors is received only by its intended recipient.

Under point-to-point communication, there exists a simple protocol which is competitive within a factor 2 of the lower bound [1]. In this protocol, a rooted spanning tree is fixed and each node sends its entire input to its parent in the tree. On receiving a value from its child, a node compares it against its own input to check if the two values match. Ignoring lower order additive terms, a more complicated protocol comes within a factor ${4/3}$ of the lower bound and is tight for certain classes of graphs [1]. Tight results, ignoring lower order terms, are also known for complete graphs [2, 9].

We study the multiparty equality problem under the local broadcast model. Recently, our work has shown that the connectivity requirements for Byzantine consensus are lower in the local broadcast model as compared to the classical model [7, 8]. In this work,

• 1. we identify a lower bound for the multiparty equality problem in this model.

• 2. we first identify simple protocols, wherein nodes are restricted to either transmit their entire input or not transmit anything at all, and find that these can cost $\varOmega (\log n)$ times the lower bound using existing example for the set cover problem [12].

• 3. we then design a protocol to solve the problem within a constant factor of the lower bound.

This research is supported in part by the National Science Foundation awards 1409416 and 1733872, and Toyota InfoTechnology Center. Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the funding agencies or the U.S. government.