The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Synchronizing Finite Automata on Eulerian Digraphs
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
A Note on Synchronized Automata and Road Coloring Problem
DLT '01 Revised Papers from the 5th International Conference on Developments in Language Theory
Decision version of the road coloring problem is NP-complete
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
An algorithm for road coloring
Journal of Discrete Algorithms
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A coloring of edges of a finite directed graph turns the graph into a finite-state automaton. The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph of uniform outdegree (constant outdegree of any vertex) is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph of uniform outdegree if the greatest common divisor of the lengths of all its cycles is one. The problem posed in 1970 has evoked noticeable interest among the specialists in the theory of graphs, automata, codes, symbolic dynamics as well as among the wide mathematical community. A polynomial time algorithm of O(n3) complexity in the worst case and quadratic in the majority of studied cases for the road coloring of the considered graph is presented below. The work is based on the recent positive solution of the road coloring problem. The algorithm was implemented in the freeware package TESTAS.