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http://hdl.handle.net/11375/12491
Title: | On the Parallelization of a Search for Counterexamples to a Conjecture of Erd\H{o}s |
Authors: | Shen, ShengWei |
Advisor: | Deza, Antoine Frantisek Franek, Ned Nedialkov |
Department: | Computer Science |
Keywords: | clique;coclique;Cayley graph;complete graph;subgraph;parallelization;Computer Sciences;Discrete Mathematics and Combinatorics;Theory and Algorithms;Computer Sciences |
Publication Date: | Oct-2012 |
Abstract: | <p>Denote by $k_t(G)$ the number of cliques of order $t$ in a graph $G$ having $n$ vertices. Let $k_t(n) = \min\{k_t(G)+k_t(\overline{G}) \}$ where $\overline{G}$ denotes the complement of $G$. Let $c_t(n) = {k_t(n)}/{\tbinom{n}{t}}$ and $c_t$ be the limit of $c_t(n)$ for $n$ going to infinity. A 1962 conjecture of Erd\H{o}s stating that $c_t = 2^{1-\tbinom{t}{2}}$ was disproved by Thomason in 1989 for all $t\geq 4$. Tighter counterexamples have been constructed by Jagger, {\v S}{\v t}ov{\' \i}{\v c}ek and Thomason in 1996, by Thomason for $t\leq 6$ in 1997, and by Franek for $t=6$ in 2002. Further tightenings $t=6,7$ and $8$ was recently obtained by Deza, Franek, and Liu.</p> <p>We investigate the computational framework used by Deza, Franek, and Liu. In particular, we present the benefits and limitations of different parallel computer memory architectures and parallel programming models. We propose a functional decomposition approach which is implemented in C++ with POSIX thread (Pthread) libraries for multi-threading. Computational benchmarking on the parallelized framework and a performance analysis including a comparison with the original computational framework are presented.</p> |
URI: | http://hdl.handle.net/11375/12491 |
Identifier: | opendissertations/7374 8429 3326384 |
Appears in Collections: | Open Access Dissertations and Theses |
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