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On the kindependence required by linear probing and minwise independence
 In Proc. 37th International Colloquium on Automata, Languages and Programming (ICALP
, 2010
"... )independent hash functions are required, matching an upper bound of [Indyk, SODA’99]. We also show that the multiplyshift scheme of Dietzfelbinger, most commonly used in practice, fails badly in both applications. Abstract. We show that linear probing requires 5independent hash functions for exp ..."
Abstract

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)independent hash functions are required, matching an upper bound of [Indyk, SODA’99]. We also show that the multiplyshift scheme of Dietzfelbinger, most commonly used in practice, fails badly in both applications. Abstract. We show that linear probing requires 5independent hash functions for expected constanttime performance, matching an upper bound of [Pagh et al. STOC’07]. For (1 + ε)approximate minwise independence, we show that Ω(lg 1 ε 1
Dependent Random Graphs and MultiParty Pointer Jumping ˚
"... We initiate a study of a relaxed version of the standard ErdősRényi 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 als ..."
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We initiate a study of a relaxed version of the standard ErdősRényi 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 p1´op1qq logpnqlogp1{pq, and the chromatic number will be at most n logp1{p1´pqqlogn. We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the kplayer MultiParty Pointer Jumping (mpjk) problem in the numberontheforehead (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 mpj3 costs Opnplog lognq { lognq communication, improving on a bound from [9]. We extend our protocol to the nonBoolean pointer jumping problem ympjk, achieving an upper bound which is opnq for any k ě 4 players. This is the first opnq protocol for ympjk and improves on a bound of Damm, Jukna, and Sgall [12], which has stood for almost twenty years.