This dissertation is an algorithmic study of sparse random graphs which are parametrized by the distribution of vertex degrees. Our contributions include: a formula for the diameter of various sparse random graphs, including the Erdös-Rényi random graphs G[subscript n,m] and G[subscript n,p] and certain power-law graphs; a heuristic for the k-orientability problem, which performs optimally for certain classes of random graphs, again including the Erdös-Rényi models G[subscript n,m] and G[subscript n,p]; an improved lower bound for the independence ratio of random 3-regular graphs. In addition to these structural results, we also develop a technique for reasoning abstractly about random graphs by representing discrete structures topologically.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of California, Berkeley, 1981. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (p. 46-48).

Randomised algorithms are an effective method of attacking computationally intractable problems. A simple and fast randomised algorithm may produce results to an accuracy sufficient for many purposes, especially in the average case. In this thesis we consider average case analyses of heuristics for certain NP-hard graph optimisation problems. In particular, we consider algorithms that find dominating sets of random regular directed graphs. As well as providing an average case analysis, our results also determine new upper bounds on domination numbers of random regular directed graphs. The algorithms for random regular directed graphs considered in this thesis are known as prioritised algorithms. Each prioritised algorithm determines a discrete random process. This discrete process may be continuously approximated using differential equations. Under certain conditions, the solutions to these differential equations describe the behaviour of the prioritised algorithm. Applying such an analysis to prioritised algorithms directly is difficult. However, we are able to use prioritised algorithms to define new algorithms, called deprioritised algorithms, that can be analysed in this fashion. Defining a deprioritised algorithm based on a given prioritised algorithm, and then analysing the deprioritised algorithm, is called the deprioritised approach. The initial theory describing the deprioritised approach was developed by Wormald and has been successfully applied in many cases. However not all algorithms are covered by Wormald??s theory: for example, algorithms for random regular directed graphs. The main contribution of this thesis is the extension of the deprioritised approach to a larger class of prioritised algorithms. We demonstrate the new theory by applying it to two algorithms which find dominating sets of random regular directed graphs.

The probabilistic method in combinatorics is a nonconstructive tool popularized through the work of Paul Erd˝os. Many difficult problems can be solved through a relatively simple application of probability theory that can lead to solutions which are better than known constructive methods. This thesis presents some of the basic tools used throughout the probabilistic method along with some of the applications of the probabilistic method throughout the fields of Ramsey theory, graph theory and other areas of combinatorial analysis. Then the topic of random graphs is covered. The theory of random graphs was founded during the late fifties and early sixties to study questions involving the effect of probability distributions upon graphical properties. This thesis presents some of the basic results involving graph models and graph properties.

The probabilistic method in combinatorics is a nonconstructive tool popularized through the work of Paul Erd˝os. Many difficult problems can be solved through a relatively simple application of probability theory that can lead to solutions which are better than known constructive methods. This thesis presents some of the basic tools used throughout the probabilistic method along with some of the applications of the probabilistic method throughout the fields of Ramsey theory, graph theory and other areas of combinatorial analysis. Then the topic of random graphs is covered. The theory of random graphs was founded during the late fifties and early sixties to study questions involving the effect of probability distributions upon graphical properties. This thesis presents some of the basic results involving graph models and graph properties.

Graduation date: 1992

Randomised algorithms are an effective method of attacking computationally intractable problems. A simple and fast randomised algorithm may produce results to an accuracy sufficient for many purposes, especially in the average case. In this thesis we consider average case analyses of heuristics for certain NP-hard graph optimisation problems. In particular, we consider algorithms that find dominating sets of random regular directed graphs. As well as providing an average case analysis, our results also determine new upper bounds on domination numbers of random regular directed graphs. The algorithms for random regular directed graphs considered in this thesis are known as prioritised algorithms. Each prioritised algorithm determines a discrete random process. This discrete process may be continuously approximated using differential equations. Under certain conditions, the solutions to these differential equations describe the behaviour of the prioritised algorithm. Applying such an analysis to prioritised algorithms directly is difficult. However, we are able to use prioritised algorithms to define new algorithms, called deprioritised algorithms, that can be analysed in this fashion. Defining a deprioritised algorithm based on a given prioritised algorithm, and then analysing the deprioritised algorithm, is called the deprioritised approach. The initial theory describing the deprioritised approach was developed by Wormald and has been successfully applied in many cases. However not all algorithms are covered by Wormald??s theory: for example, algorithms for random regular directed graphs. The main contribution of this thesis is the extension of the deprioritised approach to a larger class of prioritised algorithms. We demonstrate the new theory by applying it to two algorithms which find dominating sets of random regular directed graphs.

Thesis (Ph. D.)--Oregon State University, 1992. / Typescript (photocopy). Includes bibliographical references (leaves 49-50). Also available on the World Wide Web.

Thesis (M.S.)--Ohio University, March, 1990. / Title from PDF t.p.