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Register a new userAlgorithmic Number On the Forehead Protocols Yielding Dense Ruzsa-Szemer{e}di Graphs and Hypergraphs
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Added by
Adi Shraibman
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa-Szemer{e}di graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemer{e}di. The graphs induced by this protocol have $n$ vertices, $Omega(n^2/log n)$ edges, and are decomposable into $n^{1+O(1/log log n)}$ induced matchings. Another protocol is an explicit (and slightly simpler) version of the construction of Alon, Moitra and Sudakov, producing graphs with similar properties. We also generalize the above protocols to more than three players, in order to construct dense uniform hypergraphs in which every edge lies in a positive small number of simplices.
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Information-theoretic methods have proven to be a very powerful tool in communication complexity, in particular giving an elegant proof of the linear lower bound for the two-party disjointness function, and tight lower bounds on disjointness in the multi-party number-in-the-hand (NIH) model. In this paper, we study the applicability of information theoretic methods to the multi-party number-on-the-forehead model (NOF), where determining the complexity of disjointness remains an important open problem. There are two basic parts to the NIH disjointness lower bound: a direct sum theorem and a lower bound on the one-bit AND function using a beautiful connection between Hellinger distance and protocols revealed by Bar-Yossef, Jayram, Kumar and Sivakumar [BYJKS04]. Inspired by this connection, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party NOF protocols and provide a small toolbox that allows one to manipulate several Hellinger volume terms and lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. We then apply these new tools to obtain a lower bound on the informational complexity of the AND_k function in the NOF setting. Finally, we discuss the difficulties of proving a direct sum theorem for information cost in the NOF model.
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