Date: 17 Jun 2005

A unifying look at d-dimensional periodicities and space coverings

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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 2 d−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.

This work was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract No. 3075 (project ALCOM).