8,049 research outputs found
A Fast Algorithm Finding the Shortest Reset Words
In this paper we present a new fast algorithm finding minimal reset words for
finite synchronizing automata. The problem is know to be computationally hard,
and our algorithm is exponential. Yet, it is faster than the algorithms used so
far and it works well in practice. The main idea is to use a bidirectional BFS
and radix (Patricia) tries to store and compare resulted subsets. We give both
theoretical and practical arguments showing that the branching factor is
reduced efficiently. As a practical test we perform an experimental study of
the length of the shortest reset word for random automata with states and 2
input letters. We follow Skvorsov and Tipikin, who have performed such a study
using a SAT solver and considering automata up to states. With our
algorithm we are able to consider much larger sample of automata with up to
states. In particular, we obtain a new more precise estimation of the
expected length of the shortest reset word .Comment: COCOON 2013. The final publication is available at
http://link.springer.com/chapter/10.1007%2F978-3-642-38768-5_1
On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs
We tackle the problem of the randomized generation of slowly synchronizing
deterministic automata (DFAs) by generating random primitive sets of matrices.
We show that when the randomized procedure is too simple the exponent of the
generated sets is O(n log n) with high probability, thus the procedure fails to
return DFAs with large reset threshold. We extend this result to random
nondeterministic automata (NDFAs) by showing, in particular, that a uniformly
sampled NDFA has both a 2-directing word and a 3-directing word of length O(n
log n) with high probability. We then present a more involved randomized
algorithm that manages to generate DFAs with large reset threshold and we
finally leverage this finding for exhibiting new families of DFAs with reset
threshold of order .Comment: 31 pages, 9 figures. arXiv admin note: text overlap with
arXiv:1805.0672
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
Adaptive Probabilistic Flooding for Multipath Routing
In this work, we develop a distributed source routing algorithm for topology
discovery suitable for ISP transport networks, that is however inspired by
opportunistic algorithms used in ad hoc wireless networks. We propose a
plug-and-play control plane, able to find multiple paths toward the same
destination, and introduce a novel algorithm, called adaptive probabilistic
flooding, to achieve this goal. By keeping a small amount of state in routers
taking part in the discovery process, our technique significantly limits the
amount of control messages exchanged with flooding -- and, at the same time, it
only minimally affects the quality of the discovered multiple path with respect
to the optimal solution. Simple analytical bounds, confirmed by results
gathered with extensive simulation on four realistic topologies, show our
approach to be of high practical interest.Comment: 6 pages, 6 figure
DFAs and PFAs with Long Shortest Synchronizing Word Length
It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on
states always has a shortest synchronizing word of length at most ,
and he gave a sequence of DFAs for which this bound is reached. Until now a
full analysis of all DFAs reaching this bound was only given for ,
and with bounds on the number of symbols for . Here we give the full
analysis for , without bounds on the number of symbols.
For PFAs the bound is much higher. For we do a similar analysis as
for DFAs and find the maximal shortest synchronizing word lengths, exceeding
for . For arbitrary n we give a construction of a PFA on
three symbols with exponential shortest synchronizing word length, giving
significantly better bounds than earlier exponential constructions. We give a
transformation of this PFA to a PFA on two symbols keeping exponential shortest
synchronizing word length, yielding a better bound than applying a similar
known transformation.Comment: 16 pages, 2 figures source code adde
Labeling Schemes with Queries
We study the question of ``how robust are the known lower bounds of labeling
schemes when one increases the number of consulted labels''. Let be a
function on pairs of vertices. An -labeling scheme for a family of graphs
\cF labels the vertices of all graphs in \cF such that for every graph
G\in\cF and every two vertices , the value can be inferred
by merely inspecting the labels of and .
This paper introduces a natural generalization: the notion of -labeling
schemes with queries, in which the value can be inferred by inspecting
not only the labels of and but possibly the labels of some additional
vertices. We show that inspecting the label of a single additional vertex (one
{\em query}) enables us to reduce the label size of many labeling schemes
significantly
An Improved Algorithm for Finding the Shortest Synchronizing Words
A synchronizing word of a deterministic finite complete automaton is a word
whose action maps every state to a single one. Finding a shortest or a short
synchronizing word is a central computational problem in the theory of
synchronizing automata and is applied in other areas such as model-based
testing and the theory of codes. Because the problem of finding a shortest
synchronizing word is computationally hard, among \emph{exact} algorithms only
exponential ones are known. We redesign the previously fastest known exact
algorithm based on the bidirectional breadth-first search and improve it with
respect to time and space in a practical sense. We develop new algorithmic
enhancements and adapt the algorithm to multithreaded and GPU computing. Our
experiments show that the new algorithm is multiple times faster than the
previously fastest one and its advantage quickly grows with the hardness of the
problem instance. Given a modest time limit, we compute the lengths of the
shortest synchronizing words for random binary automata up to 570 states,
significantly beating the previous record. We refine the experimental
estimation of the average reset threshold of these automata. Finally, we
develop a general computational package devoted to the problem, where an
efficient and practical implementation of our algorithm is included, together
with several well-known heuristics.Comment: Full version of ESA 2022 pape
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