3 research outputs found
Highest Trees of Random Mappings
We prove the exact asymptotic
for the
probability that the underlying graph of a random mapping of elements
possesses a unique highest tree. The property of having a unique highest tree
turned out to be crucial in the solution of the famous Road Coloring Problem as
well as the generalization of this property in the proof of the author's result
about the probability of being synchronizable for a random automaton
Implementation of the algorithm for testing an automaton for synchronization in linear expected time
Berlinkov has suggested an algorithm that, given a deterministic finite
automaton , verifies whether or not is synchronizing
in linear (of the number of states and letters) expected time. We present a
modification of Berlinkov's algorithm which we have implemented and tested. Our
experiments show that the implementation outperforms the standard quadratic
algorithm even for automata of modest size and allow us to give a statistically
accurate approximation of the ratio of non-synchronizing automata amongst all
automata with a given number of states
On the probability of being synchronizable
We prove that a random automaton with states and any fixed non-singleton
alphabet is synchronizing with high probability. Moreover, we also prove that
the convergence rate is exactly as conjectured by
Cameron \cite{CamConj} for the most interesting binary alphabet case. Finally,
we describe a deterministic algorithm which decides whether a given random
automaton is synchronizing in linear expected time.Comment: Numerous fixes in the proo