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ItemMarkov chains and emergent behavior for problems from discrete geometry(Georgia Institute of Technology, 2018-07-02) Cannon, SarahThe problem of generating random samples from large, complex sets is widespread across the sciences, where such samples provide one way to begin to learn about the sets' typical properties. However, when the samples generated are unexpectedly correlated or drawn from the wrong distribution, this can produce misleading conclusions. One way to generate random samples is with Markov chains, which are widely used but often applied without careful analysis of their mixing time, how long they must run for until they are guaranteed to produce good samples. We present new mixing time bounds for two sampling problems from discrete geometry: dyadic tilings, combinatorial structures with applications in machine learning and harmonic analysis, and 3-colorings on a grid, an instance of the celebrated antiferromagnetic Potts model from statistical physics. Both of these results required the development of new techniques. In addition, we use Markov chains in a novel way to address research questions in programmable matter. Here, a main goal is to understand how simple computational elements can collectively accomplish complicated system-level goals. In an abstracted setting, we show that groups of particles executing our simple processes, based on Markov chains, can accomplish various tasks. This includes compression, a behavior exhibited by natural distributed systems such as fire ants and honey bees, and shortcut bridging, where the particles build bridges that optimize the same global trade-off as certain bridge-building ant colonies. Throughout, a key ingredient is the interplay between global properties of Markov chains, including but not limited to mixing time, and their dependence on local moves, or Markov chain transitions that change only a small part of the configuration. We call the global behavior that arises out of these local moves and their probabilities emergent behavior. In addition to understanding the relationship between local moves and mixing times in order to give sampling guarantees, our work on programmable matter harnesses this interaction between local and emergent behavior in a novel way, to develop distributed algorithms.
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ItemEfficient high-dimensional sampling and integration(Georgia Institute of Technology, 2017-05-31) Cousins, BenjaminVolume computation is an algorithmic version of the fundamental geometric problem to figure out how much space an object occupies. Related problems of sampling and integration have numerous applications to other fields, thus it is key to develop efficient algorithmic solutions to these problems. This thesis pushes the computational frontier of volume computation, randomized sampling, and integration, both in theory and practice. The search for efficient algorithms for volume computation has been an active area of research over the last few decades. Many geometric problems suffer from computational inefficiency in high-dimensions: the so-called \emph{curse of dimensionality}, where the problem efficiency grows exponentially with the dimension. For volume computation, Dyer, Frieze, and Kannan gave a polynomial time randomized algorithm to approximate the volume of any convex body. While their algorithm complexity was prohibitively high, the fundamental ideas inspired further improvements. Building upon the tools and ideas developed in this line of work, we obtain an $O*(n^3) algorithm to approximate the volume of well-rounded convex bodies. As a crucial tool to our faster volume algorithm, we obtain a faster algorithm for sampling a Gaussian distribution restricted by a convex set. We also generalize the Gaussian sampling restricted by a logconcave function. The Gaussian sampling algorithm additionally yields a faster algorithm for generating a uniform random sample from a convex body. The ideas for the O*(n^3) volume algorithm also lead to an efficient algorithm in practice. While the theoretical developments were inspiring, there was still no satisfying real-world implementation of the algorithm. We implement a variant of our algorithm in MATLAB, which can estimate volume in hundreds of dimensions with relative ease. The implementation allows for volume estimations which were previously far out of the realm of computational feasibility. In collaboration with systems biologists, we additionally explore a direct application of the sampling and volume implementations to the analysis of metabolic networks.
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ItemTuring machine algorithms and studies in quasi-randomness(Georgia Institute of Technology, 2011-11-09) Kalyanasundaram, SubrahmanyamRandomness 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.