Compression methods for graph algorithms

Abstract

Two compression methods for representing graphs are presented, in conjunction with algorithms applying these methods. A decomposition technique for networks that can be generated in O(m) time is presented. The components of the decomposition and the shortest path matrix of the compressed network can be used to find the shortest path between any pair of vertices in the original network in linear time. A compression method for boolean matrices and a method for applying the compression to boolean matrix multiplication is developed. The algorithms have an expected running time of O(n²*log ₂n). From this compression method a simple heuristic that may be applied to any algorithm for boolean matrix multiplication has been developed. This heuristic will improve the average running time of boolean matrix multiplication algorithms. An order of magnitude analysis of the results published by Loukakis and Tsouris [1981], on the efficiency of algorithms for finding all maximal independent sets of a graph has been performed. This analysis showed that their conclusions, which are based on a direct comparison of the running times of the algorithms, do not take into account implementation factors. An average constant factor improvement is developed for the algorithm of Tsukiyama, Ide, Ariyoshi and Shirakawa [1977] for finding all maximal independent sets of a graph. Analysis of the running time results from the algorithm comparisons presented in this thesis show that the Bron-Kerbosch algorithm has the smallest rate of increase in running time as the size of the graphs increase