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Randomised Techniques in Combinatorial Algorithmics
, 1999
"... ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ..."
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ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...
Computation in Permutation Groups: Counting and Randomly Sampling Orbits
"... Let be a finite set and let G be a permutation group acting on The permutation group G partitions into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input ( ; G) 2 I, are (1) count the or ..."
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Let be a finite set and let G be a permutation group acting on The permutation group G partitions into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input ( ; G) 2 I, are (1) count the orbits (exactly), (2) approximately count the orbits, and (3) choose an orbit uniformly at random. The goal is to quantify the computational diculty of the problems. In particular, we would like to know for which input sets I the problems are tractable.
Randomly Sampling Unlabelled Structures
, 1999
"... Informally, an \unlabelled combinatorial structure" is an object such as an unlabelled graph (in which the vertices are indistinguishable) or a structural isomer in chemistry (in which dierent atoms of the same type are indistinguishable). Computational experiments such as those described in this ..."
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Informally, an \unlabelled combinatorial structure" is an object such as an unlabelled graph (in which the vertices are indistinguishable) or a structural isomer in chemistry (in which dierent atoms of the same type are indistinguishable). Computational experiments such as those described in this volume often rely on random sampling to generate inputs for the experiments. This paper surveys work on the problem of eciently sampling unlabelled combinatorial structures from a uniform distribution. 1 Introduction Most of the experimental work described in this volume involves rst randomly sampling combinatorial structures and second using the randomlychosen structures as inputs to computational experiments. In order for the experiments to be valid, the distribution from which the combinatorial structures are drawn must be precisely specied. In order for the experiments to be computationally feasible, the randomsampling algorithms must be ecient. This survey is devoted to the ...
COMPUTATIONAL IMPROVEMENTS IN THE SUBSTITUTION METHOD FOR BOUNDING PERCOLATION THRESHOLDS
, 2005
"... The substitution method has proven to be an effective tool for bounding the critical probability of a variety of percolation models. Nevertheless, until recently substitution method calculations have been done by hand. This has severely restricted the size of computationally feasible substitution re ..."
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The substitution method has proven to be an effective tool for bounding the critical probability of a variety of percolation models. Nevertheless, until recently substitution method calculations have been done by hand. This has severely restricted the size of computationally feasible substitution regions. We examine the computational problems posed by the substitution method, beginning with an analysis of the component calculations. We seek to better understand the nature of the computational problem, hoping that better understanding will lead to improvements. Our goal is a little different from that of most algorithmic investigations. Since the substitution method constitutes a proof, there is little reason to perform a particular computation more than once. We use each speed improvement to attempt a new, larger computation that will lead to tighter bounds on the critical probability. Our major results can be grouped into two categories: recognition of links between substitution method calculations and wellknown results in other areas of mathematics,
Time Hierarchies for Sampling Distributions
, 2012
"... We prove that for every constant k ≥ 2, every polynomial time bound t, and every polynomially small ǫ, there exists a family of distributions on k elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1−1/k−ǫ in time t. Our proof involves reducing ..."
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We prove that for every constant k ≥ 2, every polynomial time bound t, and every polynomially small ǫ, there exists a family of distributions on k elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1−1/k−ǫ in time t. Our proof involves reducing the problem to a communication problem over a certain type of noisy channel. We solve the latter problem by giving a construction of a new type of listdecodable code, for a setting where there is no bound on the number of errors but each error gives more information than an erasure. 1
The Computational Complexity of Randomness
, 2013
"... This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to wh ..."
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This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem’s output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling throughthelens ofcomputational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that “a little more time gives a lot more power to sampling algorithms.” Part II concerns settings where the algorithm’s output is random. Even when the specificationofacomputational problem involves no randomness, onecanstill consider randomized