Document Type

Presentation Abstract

Presentation Date

3-9-2015

Abstract

We will consider the bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form "Is x in S?" by adaptively probing the bit vector at t places. Let s(m,n,t) be the minimum number of bits of storage needed for such a scheme. Alon and Feige showed that for t=2 (two bit probes), such schemes can be obtained from dense graphs with large girth. In particular, they showed that for n < \log m,

s(m,n,2) = O(m n \log((\log m) / n) / \log m).

We improve their analysis and obtain a better upper bound and a corresponding lower bound.

Upper bound: There is a constant C>0, such that for all large m,

s(m,n,2) \leq C \cdot m^{1-\frac{1}{(4n+1)}}.

Lower bound: There is a constant D>0, such that for n\geq 4 and all large m, we have

s(m,n,2) \geq D \cdot m^{1-\frac{1}{\lfloor n/4 \rfloor}}.

(This is joint work with Mohit Garg.)

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Additional Details

Monday, March 9, 2015 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

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