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TR15-091 | 3rd June 2015 00:37

Dependent Random Graphs and Multiparty Pointer Jumping


Authors: Joshua Brody, Mario Sanchez
Publication: 3rd June 2015 09:25
Downloads: 542


We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs dependent random graphs. Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size $\frac{(1-o(1))\log(n)}{\log(1/p)}$, and the chromatic number will be at most $\frac{n\log(1/1-p)}{\log n}$. We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the $k$-player Multiparty Pointer Jumping ($MPJ_k$) problem in the number-on-the-forehead (NOF) model. Multiparty Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for $MPJ_3$ costs $O(n(\log \log n)/\log n)$ communication, improving on a bound from Brody and Chakrabarti [BC08]. We extend our protocol to the non-Boolean pointer jumping problem $\hat{MPJ}_k$, achieving an upper bound which is $o(n)$ for any $k \geq 4$ players. This is teh first $o(n)$ protocol for $\hat{MPJ}_k$ and improves on a bound of Damm, Jukna, and Sgall [DJS98], which has stood for almost twenty years.

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