arXiv subjects

Marc Tedder

Publications and source records attributed to Marc Tedder.

4 recordsLinked to original sources

arXiv2015

Simpler, Linear-Time Transitive Orientation via Lexicographic Breadth-First Search

Comparability graphs are the undirected graphs whose edges can be directed so that the resulting directed graph is transitive. They are related to posets and have applications in scheduling theory. This paper considers the problem of finding a transitive orientation of a comparability graph, a requirement for many of its applications. A linear-time algorithm is presented based on an elegant partition refinement scheme developed elsewhere for the problem. The algorithm is intended as a simpler and more practical alternative to the existing lineartime solution, which is commonly understood to be difficult and mainly of theoretical value. It accomplishes this by using Lexicographic Breadth-First Search to achieve the same effect as produced by modular decomposition in the earlier linear-time algorithm.

cs.DS↗

arXiv2011

Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time $O(n+m)\alpha(n+m)$, where $\alpha$ is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.] uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm.

cs.DS↗

arXiv2011

Practical and Efficient Circle Graph Recognition

Circle graphs are the intersection graphs of chords in a circle. This paper presents the first sub-quadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n + m) times the inverse Ackermann function, {\alpha}(n + m), whose value is smaller than 4 for any practical graph. The algorithm is based on a new incremental Lexicographic Breadth-First Search characterization of circle graphs, and a new efficient data-structure for circle graphs, both developed in the paper. The algorithm is an extension of a Split Decomposition algorithm with the same running time developed by the authors in a companion paper.

cs.DS↗

arXiv2007

A recursive linear time modular decomposition algorithm via LexBFS

A module of a graph G is a set of vertices that have the same set of neighbours outside. Modules of a graphs form a so-called partitive family and thereby can be represented by a unique tree MD(G), called the modular decomposition tree. Motivated by the central role of modules in numerous algorithmic graph theory questions, the problem of efficiently computing MD(G) has been investigated since the early 70's. To date the best algorithms run in linear time but are all rather complicated. By combining previous algorithmic paradigms developed for the problem, we are able to present a simpler linear-time that relies on very simple data-structures, namely slice decomposition and sequences of rooted ordered trees.

cs.DM↗