981 research outputs found
Black Box White Arrow
The present paper proposes a new and systematic approach to the so-called
black box group methods in computational group theory. Instead of a single
black box, we consider categories of black boxes and their morphisms. This
makes new classes of black box problems accessible. For example, we can enrich
black box groups by actions of outer automorphisms.
As an example of application of this technique, we construct Frobenius maps
on black box groups of untwisted Lie type in odd characteristic (Section 6) and
inverse-transpose automorphisms on black box groups encrypting .
One of the advantages of our approach is that it allows us to work in black
box groups over finite fields of big characteristic. Another advantage is
explanatory power of our methods; as an example, we explain Kantor's and
Kassabov's construction of an involution in black box groups encrypting .
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
The paper is further development of our text "Fifty shades of black"
[arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black
box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248
Trivalent Graph isomorphism in polynomial time
It's important to design polynomial time algorithms to test if two graphs are
isomorphic at least for some special classes of graphs.
An approach to this was presented by Eugene M. Luks(1981) in the work
\textit{Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial
Time}. Unfortunately, it was a theoretical algorithm and was very difficult to
put into practice. On the other hand, there is no known implementation of the
algorithm, although Galil, Hoffman and Luks(1983) shows an improvement of this
algorithm running in .
The two main goals of this master thesis are to explain more carefully the
algorithm of Luks(1981), including a detailed study of the complexity and, then
to provide an efficient implementation in SAGE system. It is divided into four
chapters plus an appendix.Comment: 48 pages. It is a Master Thesi
Isomorphism test for digraphs with weighted edges
Colour refinement is at the heart of all the most efficient graph isomorphism software packages. In this paper we present a method for extending the applicability of refinement algorithms to directed graphs with weighted edges. We use Traces as a reference software, but the proposed solution is easily transferrable to any other refinement-based graph isomorphism tool in the literature. We substantiate the claim that the performances of the original algorithm remain substantially unchanged by showing experiments for some classes of benchmark graphs
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