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#### Almost random graphs with simple hash functions

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##### Fulltext (public)

2003-1-005

(Any fulltext), 11KB

##### Supplementary Material (public)

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##### Citation

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

Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-6BB3-C

##### Abstract

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.