Title:
Turing machine algorithms and studies in quasi-randomness

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dc.contributor.advisor Lipton, Richard J.
dc.contributor.advisor Shapira, Asaf
dc.contributor.author Kalyanasundaram, Subrahmanyam en_US
dc.contributor.committeeMember Randall, Dana
dc.contributor.committeeMember Tetali, Prasad
dc.contributor.committeeMember Venkateswaran, H.
dc.contributor.department Computing en_US
dc.date.accessioned 2012-02-17T19:20:29Z
dc.date.available 2012-02-17T19:20:29Z
dc.date.issued 2011-11-09 en_US
dc.description.abstract Randomness is an invaluable resource in theoretical computer science. However, pure random bits are hard to obtain. Quasi-randomness is a tool that has been widely used in eliminating/reducing the randomness from randomized algorithms. In this thesis, we study some aspects of quasi-randomness in graphs. Specifically, we provide an algorithm and a lower bound for two different kinds of regularity lemmas. Our algorithm for FK-regularity is derived using a spectral characterization of quasi-randomness. We also use a similar spectral connection to also answer an open question about quasi-random tournaments. We then provide a "Wowzer" type lower bound (for the number of parts required) for the strong regularity lemma. Finally, we study the derandomization of complexity classes using Turing machine simulations. 1. Connections between quasi-randomness and graph spectra. Quasi-random (or pseudo-random) objects are deterministic objects that behave almost like truly random objects. These objects have been widely studied in various settings (graphs, hypergraphs, directed graphs, set systems, etc.). In many cases, quasi-randomness is very closely related to the spectral properties of the combinatorial object that is under study. In this thesis, we discover the spectral characterizations of quasi-randomness in two different cases to solve open problems. A Deterministic Algorithm for Frieze-Kannan Regularity: The Frieze-Kannan regularity lemma asserts that any given graph of large enough size can be partitioned into a number of parts such that, across parts, the graph is quasi-random. . It was unknown if there was a deterministic algorithm that could produce a parition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. In this thesis, we answer this question by designing an O(n[superscript]w) time algorithm for constructing such a partition, where w is the exponent of fast matrix multiplication. Even Cycles and Quasi-Random Tournaments: Chung and Graham in had provided several equivalent characterizations of quasi-randomness in tournaments. One of them is about the number of "even" cycles where even is defined in the following sense. A cycle is said to be even, if when walking along it, an even number of edges point in the wrong direction. Chung and Graham showed that if close to half of the 4-cycles in a tournament T are even, then T is quasi-random. They asked if the same statement is true if instead of 4-cycles, we consider k-cycles, for an even integer k. We resolve this open question by showing that for every fixed even integer k geq 4, if close to half of the k-cycles in a tournament T are even, then T must be quasi-random. 2. A Wowzer type lower bound for the strong regularity lemma. The regularity lemma of Szemeredi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. Alon, Fischer, Krivelevich and Szegedy obtained a variant of the regularity lemma that allows one to have an arbitrary control on this measure of quasi-randomness. However, their proof only guaranteed to produce a partition where the number of parts is given by the Wowzer function, which is the iterated version of the Tower function. We show here that a bound of this type is unavoidable by constructing a graph H, with the property that even if one wants a very mild control on the quasi-randomness of a regular partition, then any such partition of H must have a number of parts given by a Wowzer-type function. 3. How fast can we deterministically simulate nondeterminism? We study an approach towards derandomizing complexity classes using Turing machine simulations. We look at the problem of deterministically counting the exact number of accepting computation paths of a given nondeterministic Turing machine. We provide a deterministic algorithm, which runs in time roughly O(sqrt(S)), where S is the size of the configuration graph. The best of the previously known methods required time linear in S. Our result implies a simulation of probabilistic time classes like PP, BPP and BQP in the same running time. This is an improvement over the currently best known simulation by van Melkebeek and Santhanam. en_US
dc.description.degree PhD en_US
dc.identifier.uri http://hdl.handle.net/1853/42808
dc.publisher Georgia Institute of Technology en_US
dc.subject Quasi-randomness en_US
dc.subject Algorithm en_US
dc.subject Lower bound en_US
dc.subject Turing machines en_US
dc.subject Regularity lemma en_US
dc.subject Combinatorics en_US
dc.subject Complexity theory en_US
dc.subject.lcsh Machine theory
dc.subject.lcsh Algorithms
dc.subject.lcsh Computer science
dc.subject.lcsh Computer science Mathematics
dc.title Turing machine algorithms and studies in quasi-randomness en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.corporatename College of Computing
local.contributor.corporatename School of Computer Science
local.relation.ispartofseries Doctor of Philosophy with a Major in Algorithms, Combinatorics, and Optimization
relation.isOrgUnitOfPublication c8892b3c-8db6-4b7b-a33a-1b67f7db2021
relation.isOrgUnitOfPublication 6b42174a-e0e1-40e3-a581-47bed0470a1e
relation.isSeriesOfPublication 186126ed-fc79-4186-8523-2cb526aa622e
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