An algebraic approach to communication complexity

Abstract

Let M be a finite monoid: define C^(k)(M) to be the maximum number of bits that need to be exchanged in the k-party communication game to decide membership in any language recognized by M. We prove the following:

• a) If M is a group then, for any k, C^(k)(M) = O(1) if M is nilpotent of class k − 1 and C^(k)(M) = θ(n) otherwise.

• b) If M is aperiodic, then C⁽²⁾(M) = O(1) if M is commutative, C⁽²⁾(M) = θ(log n) if M belongs to the variety DA but is not commutative and C⁽²⁾(M) = θ(n) otherwise.

We also show that when M is in DA, C^(k)(M) = O(1) for some k and conjecture that this algebraic condition is also necessary.

Research supported by FCAR and NSERC grants. We would like to thank Peter Bro Miltersen of the University of Aarhus, Denmark for his ideas and comments, that helped us improve some of our lower bounds.