1,968 research outputs found
A quadratic algorithm for road coloring
The Road Coloring Theorem states that every aperiodic directed graph with
constant out-degree has a synchronized coloring. This theorem had been
conjectured during many years as the Road Coloring Problem before being settled
by A. Trahtman. Trahtman's proof leads to an algorithm that finds a
synchronized labeling with a cubic worst-case time complexity. We show a
variant of his construction with a worst-case complexity which is quadratic in
time and linear in space. We also extend the Road Coloring Theorem to the
periodic case
On the Number of Synchronizing Colorings of Digraphs
We deal with -out-regular directed multigraphs with loops (called simply
\emph{digraphs}). The edges of such a digraph can be colored by elements of
some fixed -element set in such a way that outgoing edges of every vertex
have different colors. Such a coloring corresponds naturally to an automaton.
The road coloring theorem states that every primitive digraph has a
synchronizing coloring.
In the present paper we study how many synchronizing colorings can exist for
a digraph with vertices. We performed an extensive experimental
investigation of digraphs with small number of vertices. This was done by using
our dedicated algorithm exhaustively enumerating all small digraphs. We also
present a series of digraphs whose fraction of synchronizing colorings is equal
to , for every and the number of vertices large enough.
On the basis of our results we state several conjectures and open problems.
In particular, we conjecture that is the smallest possible fraction of
synchronizing colorings, except for a single exceptional example on 6 vertices
for .Comment: CIAA 2015. The final publication is available at
http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
Parametric Multi-Step Scheme for GPU-Accelerated Graph Decomposition into Strongly Connected Components
The problem of decomposing a directed graph into strongly connected components (SCCs) is a fundamental graph problem that is inherently present in many scientific and commercial applications. Clearly, there is a strong need for good high-performance, e.g., GPU-accelerated, algorithms to solve it. Unfortunately, among existing GPU-enabled algorithms to solve the problem, there is none that can be considered the best on every graph, disregarding the graph characteristics. Indeed, the choice of the right and most appropriate algorithm to be used is often left to inexperienced users. In this paper, we introduce a novel parametric multi-step scheme to evaluate existing GPU-accelerated algorithms for SCC decomposition in order to alleviate the burden of the choice and to help the user to identify which combination of existing techniques for SCC decomposition would fit an expected use case the most. We support our scheme with an extensive experimental evaluation that dissects correlations between the internal structure of GPU-based algorithms and their performance on various classes of graphs. The measurements confirm that there is no algorithm that would beat all other algorithms in the decomposition on all of the classes of graphs. Our contribution thus represents an important step towards an ultimate solution of automatically adjusted scheme for the GPU-accelerated SCC decomposition
All finite transitive graphs admit self-adjoint free semigroupoid algebras
In this paper we show that every non-cycle finite transitive directed graph
has a Cuntz-Krieger family whose WOT-closed algebra is . This
is accomplished through a new construction that reduces this problem to
in-degree -regular graphs, which is then treated by applying the periodic
Road Coloring Theorem of B\'eal and Perrin. As a consequence we show that
finite disjoint unions of finite transitive directed graphs are exactly those
finite graphs which admit self-adjoint free semigroupoid algebras.Comment: Added missing reference. 16 pages 2 figure
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