From expanders to hitting distributions and simulation theorems

TL;DR

This paper constructs new gadgets with good hitting distributions from specific expanders, enabling improved simulation theorems in communication complexity and demonstrating their computational properties.

Contribution

It shows how to derive gadgets with strong hitting distributions from certain expanders, extending the applicability of simulation theorems in communication complexity.

Findings

Gadgets from affine plane expanders achieve optimal arity trade-offs.

Hitting distributions in these gadgets are polynomial-time listable.

Application to Ramanujan graphs extends the class of expanders usable for this purpose.

Abstract

Recently, Chattopadhyay et al. (\cite{chattopadhyay2017simulation}) proved that any gadget having so called \emph{hitting distributions} admits deterministic "query-to-communication" simulation theorem. They applied this result to Inner Product, Gap Hamming Distance and Indexing Function. They also demonstrated that previous works used hitting distributions implicitly (\cite{goos2015deterministic} for Indexing Function and \cite{wu2017raz} for Inner Product). In this paper we show that any expander in which any two distinct vertices have at most one common neighbor can be transformed into a gadget possessing good hitting distributions. We demonstrate that this result is applicable to affine plane expanders and to Lubotzky-Phillips-Sarnak construction of Ramanujan graphs . In particular, from affine plane expanders we extract a gadget achieving the best known trade-off between the arity of outer function and the size of gadget. More specifically, when this gadget has $k$ bits on input, it admits a simulation theorem for all outer function of arity roughly $2^{k/2}$ or less (the same was also known for $k$-bit Inner Product, (\cite{chattopadhyay2017simulation})). In addition we show that, unlike Inner Product, underlying hitting distributions in our new gadget are "polynomial-time listable" in the sense that their supports can be written down in time $2^{O(k)}$, i.e, in time polynomial in size of gadget's matrix.