Parallel complexity of discrete problems

TLDR: By exhibiting a parallel algorithm, it is shown, for the first time, that the recognition of the consecutive 1's property for rows of a (0,1)-matrix is in NC, and the techniques and methods exhibited can be used to obtain efficient solutions to some other problems.

Abstract: Processor complexity and time complexity are two important measures of the efficiency of parallel algorithms. A problem is said to be in NC if it can be solved in polylogarithmic time using a polynomial number of processors. This dissertation centers on designing efficient parallel algorithms for discrete problems. By exhibiting a parallel algorithm, we show, for the first time, that the recognition of the consecutive 1's property for rows of a (0,1)-matrix is in NC. The algorithm gives a useful tool for solving many other problems. Based on the above procedure, we prove to be in NC the recognition of many classes of graphs, e.g., transformable convex bipartite (TCB) graphs, bipartite permutation (BP) graphs, $\Gamma$ circular arc graphs, $\Theta$ circular arc graphs (a.k.a. Helly circular arc graphs), proper circular arc graphs, proper interval graphs, etc. We also present the first NC algorithms for finding a maximum matching, a maximum independent set, and a minimum clique cover in a TCB graph, and for finding a Hamilton path/cycle in a BP graph. We also obtain more efficient parallel algorithms for some problems such as computing maximum cliques on comparability graphs, circle graphs, and circular arc graphs than the predecessors. Furthermore, we study a well known longstanding open problem, the graph isomorphism problem. We introduce the notion of identification (ID) matrices. We show to be in NC the isomorphism testing for graphs which have ID matrices with the circular 1's property. These graphs include, among others, $\Gamma$ circular arc graphs, $\Theta$ circular arc graphs, and circular convex bipartite graphs. This substantially broadens the class of graphs for which the graph isomorphism problem is known to be in NC, and therefore, in P. Included in the appendix are efficient sorting algorithms and efficient implementation of the sequential isomorphism testing algorithms, which are by no means less important. This research has produced many efficient algorithms. They will help us in at least the following two aspects: (0) The algorithms can be used as building blocks to develop some other efficient algorithms; (1) The techniques and methods exhibited can be used to obtain efficient solutions to some other problems.