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Pseudo-random graphs and bit probe schemes with one-sided error
We study probabilistic bit-probe schemes for the membership problem. Given a
set A of at most n elements from the universe of size m we organize such a
structure that queries of type "Is x in A?" can be answered very quickly.
H.Buhrman, P.B.Miltersen, J.Radhakrishnan, and S.Venkatesh proposed a bit-probe
scheme based on expanders. Their scheme needs space of bits, and
requires to read only one randomly chosen bit from the memory to answer a
query. The answer is correct with high probability with two-sided errors. In
this paper we show that for the same problem there exists a bit-probe scheme
with one-sided error that needs space of O(n\log^2 m+\poly(\log m)) bits. The
difference with the model of Buhrman, Miltersen, Radhakrishnan, and Venkatesh
is that we consider a bit-probe scheme with an auxiliary word. This means that
in our scheme the memory is split into two parts of different size: the main
storage of bits and a short word of bits that is
pre-computed once for the stored set A and `cached'. To answer a query "Is x in
A?" we allow to read the whole cached word and only one bit from the main
storage. For some reasonable values of parameters our space bound is better
than what can be achieved by any scheme without cached data.Comment: 19 page
Applications of Derandomization Theory in Coding
Randomized techniques play a fundamental role in theoretical computer science
and discrete mathematics, in particular for the design of efficient algorithms
and construction of combinatorial objects. The basic goal in derandomization
theory is to eliminate or reduce the need for randomness in such randomized
constructions. In this thesis, we explore some applications of the fundamental
notions in derandomization theory to problems outside the core of theoretical
computer science, and in particular, certain problems related to coding theory.
First, we consider the wiretap channel problem which involves a communication
system in which an intruder can eavesdrop a limited portion of the
transmissions, and construct efficient and information-theoretically optimal
communication protocols for this model. Then we consider the combinatorial
group testing problem. In this classical problem, one aims to determine a set
of defective items within a large population by asking a number of queries,
where each query reveals whether a defective item is present within a specified
group of items. We use randomness condensers to explicitly construct optimal,
or nearly optimal, group testing schemes for a setting where the query outcomes
can be highly unreliable, as well as the threshold model where a query returns
positive if the number of defectives pass a certain threshold. Finally, we
design ensembles of error-correcting codes that achieve the
information-theoretic capacity of a large class of communication channels, and
then use the obtained ensembles for construction of explicit capacity achieving
codes.
[This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi
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