Derandomization and Group Testing

The rapid development of derandomization theory, which is a fundamental area in theoretical computer science, has recently led to many surprising applications outside its initial intention. We will review some recent such developments related to combinatorial group testing. In its most basic setting, the aim of group testing is to identify a set of "positive" individuals in a population of items by taking groups of items and asking whether there is a positive in each group. In particular, we will discuss explicit constructions of optimal or nearly-optimal group testing schemes using "randomness-conducting" functions. Among such developments are constructions of error-correcting group testing schemes using randomness extractors and condensers, as well as threshold group testing schemes from lossless condensers.

Applications of Derandomization Theory in Coding

Mahdi Cheraghchi Bashi Astaneh

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. Towards this goal, numerous fundamental notions have been developed to provide a unified framework for approaching various derandomization problems and to improve our general understanding of the power of randomness in computation. Two important classes of such tools are pseudorandom generators and randomness extractors. Pseudorandom generators transform a short, purely random, sequence into a much longer sequence that looks random, while extractors transform a weak source of randomness into a perfectly random one (or one with much better qualities, in which case the transformation is called a randomness condenser). 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. We utilize randomness extractors to 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. Next, we use randomness condensers and extractors to 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. Finally, we consider the problem of explicit construction of error-correcting codes on the Gilbert-Varshamov bound and extend the original idea of Nisan and Wigderson to obtain a small ensemble of codes, mostly achieving the bound, under suitable computational hardness assumptions.

EPFL2010

Capacity Achieving Codes from Randomness Conductors

Mahdi Cheraghchi Bashi Astaneh

We give a general framework for construction of small ensembles of capacity achieving linear codes for a wide range of (not necessarily memoryless) discrete symmetric channels, and in particular, the binary erasure and symmetric channels. The main tool used in our constructions is the notion of randomness extractors and lossless condensers that are regarded as central tools in theoretical computer science. Same as random codes, the resulting ensembles preserve their capacity achieving properties under any change of basis. Our methods can potentially lead to polynomial-sized ensembles; however, using known explicit constructions of randomness conductors we obtain specific ensembles whose size is as small as quasipolynomial in the block length. By applying our construction to Justesen's concatenation scheme (Justesen, 1972) we obtain explicit capacity achieving codes for BEC (resp., BSC) with almost linear time encoding and almost linear time (resp., quadratic time) decoding and exponentially small error probability. The explicit code for BEC is defined and capacity achieving for every block length, a property lacked in previously known explicit constructions.

2009

Applications of Derandomization Theory in Coding

Mahdi Cheraghchi Bashi Astaneh

EPFL2010