Raz-McKenzie simulation: new gadget and unimprovability of Thickness Lemma
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Assume that we have an outer function f:{0, 1}^n →{0, 1} and a gadget function g:A× B →{0, 1} and we want to lower bound deterministic communication complexity of a composed function f∘ g^n, denoted here by D^cc(f∘ g^n), in terms of deterministic query complexity of f, denoted here by D^dt(f). The study of such questions, now usually called Simulation Theorems, was initiated in the work of Raz and McKenzie (raz1997separation). Recently, Chattopadhyaay et al. (chattopadhyay2017simulation) introduced a very convenient tool to study such questions, they called this tool hitting distributions. Using this tool, they proved that D^cc(f ∘ g^n) = Ω(D^dt(f) · D^cc(g)), where g is either Inner Product or Gap Hamming Distance gadget on k bits, and f is any function depending on at most 2^ck variables for some absolute constant c. In this work we use hitting distributions to show Simulation Theorem for a new gadget. Our gadget is the following one: Alice holds a∈F_p^2, Bob holds b∈F_p^2 and they want to know whether a - b is a square in F_p^2. We show that D^cc(f ∘ g^n) = Ω(D^dt(f) · D^cc(g)), where g is our gadget and f is any function depending on at most p^c variables for some constant c (note that this bound is exponential in the size of gadget, which is O( p)). We also investigate hitting distributions for disjointness predicate on k-element subsets of {1, 2, ..., n}. Finally, we study Thickness Lemma, a combinatorial statement from the proof of Simulation Theorems of Göös et al and Chattopadhyaay et al., and show that this Lemma can not be improved.
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