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The Communication Complexity of Set Intersection and Multiple Equality Testing
In this paper we explore fundamental problems in randomized communication
complexity such as computing Set Intersection on sets of size and Equality
Testing between vectors of length . Sa\u{g}lam and Tardos and Brody et al.
showed that for these types of problems, one can achieve optimal communication
volume of bits, with a randomized protocol that takes
rounds. Aside from rounds and communication volume, there is a \emph{third}
parameter of interest, namely the \emph{error probability} .
It is straightforward to show that protocols for Set Intersection or Equality
Testing need to send bits. Is it
possible to simultaneously achieve optimality in all three parameters, namely
communication and rounds? In
this paper we prove that there is no universally optimal algorithm, and
complement the existing round-communication tradeoffs with a new tradeoff
between rounds, communication, and probability of error. In particular:
1. Any protocol for solving Multiple Equality Testing in rounds with
failure probability has communication volume .
2. There exists a protocol for solving Multiple Equality Testing in rounds with communication, thereby essentially
matching our lower bound and that of Sa\u{g}lam and Tardos.
Our original motivation for considering as an independent
parameter came from the problem of enumerating triangles in distributed
() networks having maximum degree . We prove that
this problem can be solved in time with
high probability .Comment: 44 page