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Almost random graphs with simple hash functions


Dietzfelbinger,  Martin
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Dietzfelbinger, M., & Woelfel, P.(2003). Almost random graphs with simple hash functions (MPI-I-2003-1-005). Saarbrücken: Max-Planck-Institut für Informatik.

We describe a simple randomized construction for generating pairs of hash functions h_1,h_2 from a universe U to ranges V=[m]={0,1,...,m-1} and W=[m] so that for every key set S\subseteq U with n=|S| <= m/(1+epsilon) the (random) bipartite (multi)graph with node set V + W and edge set {(h_1(x),h_2(x)) | x in S} exhibits a structure that is essentially random. The construction combines d-wise independent classes for d a relatively small constant with the well-known technique of random offsets. While keeping the space needed to store the description of h_1 and h_2 at O(n^zeta), for zeta<1 fixed arbitrarily, we obtain a much smaller (constant) evaluation time than previous constructions of this kind, which involved Siegel's high-performance hash classes. The main new technique is the combined analysis of the graph structure and the inner structure of the hash functions, as well as a new way of looking at the cycle structure of random (multi)graphs. The construction may be applied to improve on Pagh and Rodler's ``cuckoo hashing'' (2001), to obtain a simpler and faster alternative to a recent construction of "Ostlin and Pagh (2002/03) for simulating uniform hashing on a key set S, and to the simulation of shared memory on distributed memory machines. We also describe a novel way of implementing (approximate) d-wise independent hashing without using polynomials.