24 research outputs found
Strong inapproximability of the shortest reset word
The \v{C}ern\'y conjecture states that every -state synchronizing
automaton has a reset word of length at most . We study the hardness
of finding short reset words. It is known that the exact version of the
problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and
complete for the DP class, and that approximating the length of the shortest
reset word within a factor of is NP-hard [Gerbush and Heeringa,
CIAA'10], even for the binary alphabet [Berlinkov, DLT'13]. We significantly
improve on these results by showing that, for every , it is NP-hard
to approximate the length of the shortest reset word within a factor of
. This is essentially tight since a simple -approximation
algorithm exists.Comment: extended abstract to appear in MFCS 201
Algebraic synchronization criterion and computing reset words
We refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain new upper bounds for automata with a short word of small rank. The results are applied to make several improvements in the area. In particular, we improve the upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most O(nlog3n). In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary n-state decoder is at most O(nlog n). We prove the Černý conjecture for n-state automata with a letter of rank ≤6n−63. In another corollary, we show that the probability that the Černý conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and that the expected value of the reset threshold is at most n3/2+o(1). Moreover, all of our bounds are constructible. We present suitable polynomial algorithms for the task of finding a reset word of length within our bounds. © 201
Algebraic synchronization criterion and computing reset words
We refine results about relations between Markov chains and synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain upper bounds for the reset thresholds of automata with a short word of a small rank. The results are applied to make several improvements in the area. We improve the best general upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most O(n log3 n). Also, we prove the Černý conjecture for n-state automata with a letter of rank at most 3√6n-6. In another corollary, based on the recent results of Nicaud, we show that the probability that the Čern conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states. It follows that the expected value of the reset threshold of an n-state random synchronizing binary automaton is at most n7/4+o(1). Moreover, reset words of the lengths within our bounds are computable in polynomial time. We present suitable algorithms for this task for various classes of automata for which our results can be applied. These include (quasi-)one-cluster and (quasi-)Eulerian automata. © Springer-Verlag Berlin Heidelberg 2015
Approximating the Minimum Length of Synchronizing Words is Hard
We prove that, unless P = NP, no polynomial-time algorithm can approximate the minimum length of synchronizing words for a given synchronizing automaton within a constant factor. © 2010 Springer-Verlag.The author acknowledges support from the Federal Education Agency of Russia, grant 2.1.1/3537, and from the Russian Foundation for Basic Research, grant 09-01-12142
Approximating the Minimum Length of Synchronizing Words Is Hard
We prove that, unless P=NP, no polynomial-time algorithm can approximate the minimum length of reset words for a given synchronizing automaton within a constant factor. © 2013 Springer Science+Business Media New York
A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata
Černý's conjecture asserts the existence of a synchronizing word of length at most (n - 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p·ar = q·as for some integers r, s (for a state p and a word w, we denote by p·w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n2). This applies in particular to Huffman codes. © 2011 World Scientific Publishing Company
Synchronizing Automata on Quasi Eulerian Digraph
In 1964 \v{C}ern\'{y} conjectured that each -state synchronizing automaton
posesses a reset word of length at most . From the other side the best
known upper bound on the reset length (minimum length of reset words) is cubic
in . Thus the main problem here is to prove quadratic (in ) upper bounds.
Since 1964, this problem has been solved for few special classes of \sa. One of
this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In
this paper we introduce a new approach to prove quadratic upper bounds and
explain it in terms of Markov chains and Perron-Frobenius theories. Using this
approach we obtain a quadratic upper bound for a generalization of Eulerian
automata.Comment: 8 pages, 1 figur
Algebraic synchronization criterion and computing reset words
We refine a uniform algebraic approach for deriving upper bounds on reset
thresholds of synchronizing automata. We express the condition that an
automaton is synchronizing in terms of linear algebra, and obtain upper bounds
for the reset thresholds of automata with a short word of a small rank. The
results are applied to make several improvements in the area.
We improve the best general upper bound for reset thresholds of finite prefix
codes (Huffman codes): we show that an -state synchronizing decoder has a
reset word of length at most . In addition to that, we prove
that the expected reset threshold of a uniformly random synchronizing binary
-state decoder is at most . We also show that for any non-unary
alphabet there exist decoders whose reset threshold is in .
We prove the \v{C}ern\'{y} conjecture for -state automata with a letter of
rank at most . In another corollary, based on the recent
results of Nicaud, we show that the probability that the \v{C}ern\'y conjecture
does not hold for a random synchronizing binary automaton is exponentially
small in terms of the number of states, and also that the expected value of the
reset threshold of an -state random synchronizing binary automaton is at
most .
Moreover, reset words of lengths within all of our bounds are computable in
polynomial time. We present suitable algorithms for this task for various
classes of automata, such as (quasi-)one-cluster and (quasi-)Eulerian automata,
for which our results can be applied.Comment: 18 pages, 2 figure
Checking Whether an Automaton Is Monotonic Is NP-complete
An automaton is monotonic if its states can be arranged in a linear order
that is preserved by the action of every letter. We prove that the problem of
deciding whether a given automaton is monotonic is NP-complete. The same result
is obtained for oriented automata, whose states can be arranged in a cyclic
order. Moreover, both problems remain hard under the restriction to binary
input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at
http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2
Packing and Hausdorff measures of stable trees
In this paper we discuss Hausdorff and packing measures of random continuous
trees called stable trees. Stable trees form a specific class of L\'evy trees
(introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum
random tree (1991) which corresponds to the Brownian case. We provide results
for the whole stable trees and for their level sets that are the sets of points
situated at a given distance from the root. We first show that there is no
exact packing measure for levels sets. We also prove that non-Brownian stable
trees and their level sets have no exact Hausdorff measure with regularly
varying gauge function, which continues previous results from a joint work with
J-F Le Gall (2006).Comment: 40 page