A Clique Tree Algorithm For Partitioning A Chordal Graph Into Transitive Subgraphs
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A Clique Tree Algorithm for Partitioning a Chordal Graph Into Transitive Subgraphs
Author: University of Waterloo. Dept. of Computer Science
language: en
Publisher: Faculty of Mathematics, University of Waterloo
Release Date: 1993
Abstract: "A partitioning problem on chordal graphs that arises in the solution of sparse triangular systems of equations on parallel computers is considered. Roughly the problem is to partition a chordal graph G into the fewest transitively orientable subgraphs over all perfect elimination orderings of G, subject to a certain precedence relationship on its vertices. In earlier work, a greedy scheme that solved the problem by eliminating a largest subset of vertices at each step was described, and an algorithm implementing the scheme in time and space linear in the number of edges of the graph was provided. Here a more efficient greedy scheme, obtained by representing the chordal graph in terms of its maximal cliques, which eliminates a subset of the leaf cliques at each step is described. Several new results about minimal vertex separators in chordal graphs, and in particular the concept of a critical separator of a leaf clique, are employed to prove that the new scheme solves the partitioning problem. We provide an algorithm implementing the scheme in time and space linear in the size of the clique tree."
Graph Theory and Sparse Matrix Computation
Author: Alan George
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
When reality is modeled by computation, matrices are often the connection between the continuous physical world and the finite algorithmic one. Usually, the more detailed the model, the bigger the matrix, the better the answer, however, efficiency demands that every possible advantage be exploited. The articles in this volume are based on recent research on sparse matrix computations. This volume looks at graph theory as it connects to linear algebra, parallel computing, data structures, geometry, and both numerical and discrete algorithms. The articles are grouped into three general categories: graph models of symmetric matrices and factorizations, graph models of algorithms on nonsymmetric matrices, and parallel sparse matrix algorithms. This book will be a resource for the researcher or advanced student of either graphs or sparse matrices; it will be useful to mathematicians, numerical analysts and theoretical computer scientists alike.
Topics in Intersection Graph Theory
Finally there is a book that presents real applications of graph theory in a unified format. This book is the only source for an extended, concentrated focus on the theory and techniques common to various types of intersection graphs. It is a concise treatment of the aspects of intersection graphs that interconnect many standard concepts and form the foundation of a surprising array of applications to biology, computing, psychology, matrices, and statistics.