An Algorithm for Road Coloring

TL;DR

This paper presents a polynomial-time algorithm for the road coloring problem, which involves finding a synchronizing coloring for a directed graph with uniform outdegree, based on a recent solution to the problem.

Contribution

It introduces an efficient $O(n^3)$ algorithm for the road coloring problem, advancing the computational methods for synchronizing automata.

Findings

Algorithm runs in $O(n^3)$ worst-case time

Implementation available in TESTAS package

Works for strongly connected graphs with uniform outdegree

Abstract

A coloring of edges of a finite directed graph turns the graph into 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 had evoked a 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 $O(n^3)$ complexity in the most worst case and quadratic in majority of studied cases for the road coloring of the considered graph is presented below. The work is based on recent positive solution of the road coloring problem. The algorithm was implemented in the package TESTAS