From expanders to hitting distributions and simulation theorems
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Recently, Chattopadhyay et al. [chattopadhyay 2017 simulation] proved that any gadget having so called 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 [goos 2015 deterministic] for Indexing Function and [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, [chattopadhyay 2017 simulation]). We also obtain several results showing that with current technique no better trade-off can be achieved. Namely, we observe that no gadget can have hitting distributions with significantly better parameters than Inner Product or our new affine plane gadget. We also show that Thickness Lemma, which causes restrictions on the arity of outer functions in proofs of simulation theorems, is unimprovable. Finally, we prove a very weak simulation theorem for Disjointness predicate, by exploring hitting distributions of its sparse version.
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