About an Algorithmic Approach to Tilings {p,q} of the Hyperbolic Plane

In this paper, we remind previous results about the tilings {p,q} of the hyperbolic plane. As proved in [Margenstern and Skordev 2003a], these tilings are combinatoric, a notion which we recall in the introduction. It turned out that in this case, most of these tilings also have the interesting property that the language of the splitting associated to the tiling is regular. In this paper, we investigate the consequence of the regularity of the language by providing algorithms to compute the path from a tile to the root of the spanning tree as well as to compute the coordinates of the neighbouring tiles. These algorithms are linear in the coordinate of the given node