Régnier M., Rostami L. (1993) A unifying look at d-dimensional periodicities and space coverings. In: Apostolico A., Crochemore M., Galil Z., Manber U. (eds) Combinatorial Pattern Matching. CPM 1993. Lecture Notes in Computer Science, vol 684. Springer, Berlin, Heidelberg
We propose a formal characterization of d-dimensional periodicities. We show first that any periodic pattern has a canonical decomposition and a minimal generator, generalizing the 1D property, This allows to classify the d-dimensional patterns in 2d−1 + 1 classes, according to their periodicities, each class having subclasses. A full classification of the coverings of a 2-dimensional space by a pattern follows. These results have important algorithmic issues in pattern matching. First, the covering classification allows an efficient use of the now classical “duel” paradigm. Second, d-dimensional pattern matching complexity is intrinsically different for each class.