Linear Visualization of a Road Coloring Algorithm

The visualization has become essential in many application areas. The finite graphs and automata undoubtedly belong to such areas. A problem of a visual image of a directed finite graph has appeared in the study of the road coloring conjecture. Given a finite directed graph, a coloring of its edges turns the graph into finitestate automaton. The visual perception of the structure properties of automata is an important goal. A synchronizing word of a deterministic automaton is a word in the alphabet of colors of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring conjecture [1], [2] was stated about forty yeas ago for a complete strongly connected directed finite graph with constant outdegree of all its vertices where the greatest common divisor (gcd) of lengths of all its cycles is one. The edges of the graph being unlabelled, the task is to find a labelling that turns the graph into a deterministic finite automaton possessing a synchronizing word. Such graph has according to the conjecture a synchronizing coloring. The problem belonged to the most fascinating problems in the theory of finite automata [9], [4] and was mentioned in the popular Internet Encyclopedia ”Wikipedia” on the list of the most interesting unsolved problems in mathematics. The positive solution of the road coloring problem [13] is a basis of a polynomial-time implemented algorithm of O(n) complexity in the worst case. The realization of the considered algorithm is demonstrated by a high-speed visualization program. The visibility of inner structure of a digraph without doubt is a matter of interest not only for road coloring, the range of the application may be significantly wider. Crucial role in the visualization plays for us the correspondence of the layout to the human intuition, the perception of the structure properties of the graph and the rapidity of the appearance of the image. We use for this aim some known approaches [11], [14] together with some new productive ideas. Our algorithm for the visualization is linear in the size of the automaton. This algorithm not complicated at first sight successfully solves a whole series of tasks of the disposal of the objects. The visualization of the transition graph of the automaton is a help tool of the Preprint submitted to CTW 2008 study of the automata. Thus the linearity of the algorithm is comfortably and important. Both the road coloring algorithm and the visualization algorithm are implemented in the package TESTAS (www.cs.biu.ac.il/∼trakht/syn.html). As usual, we regard a directed graph with colors assigned to its edges as a finite automaton, whose input alphabet consists of these colors. The graph is called transition graph of the automaton. An automaton is deterministic if no state has two outgoing edges of the same color. In complete automaton each state has outgoing edges of any color. Let |P | denote the size of the subset P of states from an automaton (of vertices from a graph). Let Ps be the set of states ps for p ∈ P s ∈ Σ. For the transition graph Γ of an automaton let Γs denote the map of the set of states of the automaton. A word s ∈ Σ is called a k-synchronizing word of the automaton with transition graph Γ if both |Γs| = k and for all words t ∈ Σ∗ holds |Γt| ≥ k. A pair of distinct states p,q of an automaton (of vertices of the transition graph) will be called synchronizing if ps = qs for some s ∈ Σ. A synchronizing pair of states p, q of an automaton is called stable if for every word u the pair pu,qu is also synchronizing [4], [?]. We call the set of all outgoing edges of a vertex a bunch if all these edges are incoming edges of only one vertex. Imagine a map with roads which are colored in such a way that a fixed sequence of colors, called a synchronizing sequence, leads to a fixed place whatever is the starting point. Finding such a coloring is called em road coloring problem. The roads of the map are considered as edges of a directed graph. The visual presentation of a road coloring algorithm is essentially based on the paths of the graph. The paths must be visible as well as cycles, bunches and other structure components of the graph. In particular, the notion of the bunch according to the following lemma plays some role in the road coloring algorithm. Lemma 1 [13] If some vertex of graph Γ has two incoming bunches then there exists a stable pair by any coloring. The role of the length of a path is also important. Lemma 2 [13] Let any vertex of the graph Γ have no two incoming bunches. Then a subgraph of Γ of some color has maximal subtree. A crucial role in the visualization plays in our opinion the correspondence of the layout to the human intuition, the perception of the structure properties of the graph and the rapidity of the appearance of the image. The automatically drawn graphical image must resemble the last one of a human being. The considered visualization is a help tool for any program dealing with transition graph of DFA and in particular for the road coloring algorithm. Our main objective is the visual representation of the transition graph of a deterministic finite automaton based on the structure properties of the graph. Any deterministic finite automaton is accepted by the algorithm.