Arithmetic modeling of k-ary n-cubes and toroidal k-ary grids

Abstract

Abstract The use of toroidal topologies offers many advantages in the computing field when it comes to both pattern spotting and path finding strategies. The former is covered by de Bruijn shapes, which permit to uniquely locate a single pattern throughout the shape. However, the latter is mainly carried out by k-ary n-cubes, which label node identifiers in a sequential order according to rows, columns, layers, and so on. This scheme facilitates the movement among nodes by just applying arithmetic operations, such as integer divisions and arithmetic modulo n. On the othe hand, toroidal k-ary grids are an alternative available in some specific cases, where determined patterns appear in all dimensions of each node, thus allowing to use those patterns to dictate paths to move among nodes. In this paper, the arithmetic bases of both path finding strategies have been presented and some pseudocode algorithms have been designed.