De Bruijn-Based and k-Ary n-Cube-Based Algebraic Models in Fog Environments

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

“…On the other hand, de Bruijn sequences may be extended to higher dimensions in order to expand the capabilities of pattern spotting [15]. In this sense, if two dimensions are considered, then a de Bruijn torus is defined as a toroidal bidimensional matrix of dimensions r 1 × r 2 , where all available bidimensional submatrices of dimensions m 1 × m 2 , being constructed with a k-ary alphabet, appears exactly once.…”

“…In that case, one of the corners is selected as the origin, hence all its d i digits will be set to zero. From that point on, taking a random node d, whose value related to a given dimension i is d i , then the node attached to the left in that dimension i is assigned to the predecessor modulo k of the value d i , whilst the node attached to the right is assigned to the successor modulo k of the value d i , as expressed in (15). However, if the binary alphabet is employed, then such an expression gets simplified because there are only two neighbors in each dimension, as in (16).…”