×

The rectangle complexity of functions on two-dimensional lattices. (English) Zbl 0989.68062

Summary: Let \(X\) be a non-empty set. Let \(f: \mathbb{Z}^2\to X\). All vectors which occur have integer coefficients, and for \(\vec a= (a_1, a_2)\), \(\vec b= (b_1, b_2)\) we write \(\vec a\leq\vec b\) or \(\vec a< \vec b\) if \(a_j\leq b_j\) or \(a_j< b_j\) for \(j= 1,2\), respectively. Let \(\vec b>0\). A \(\vec b\)-block is a set of the form \(B_{\vec b}(\vec c):= \{\vec x\in \mathbb{Z}^2|\vec c\leq\vec x< \vec c+ \vec b\}\). A \(\vec b\)-pattern is the restriction of \(f\) to some \(\vec b\)-block. The total number of distinct \(\vec b\)-patterns is called the \(\vec b\)-complexity of \(f\). A conjecture of the authors implies that \(f\) is periodic if there is a \(\vec b> 0\) such that the \(\vec b\)-complexity of \(f\) does not exceed \(b_1b_2\). In this paper, we prove the statement for \(b= (n,2)\) where \(n\) is any positive integer.

MSC:

68Q25 Analysis of algorithms and problem complexity
68R10 Graph theory (including graph drawing) in computer science
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berthé, V.; Vuillon, L., Tilings and rotations, on the torusa two-dimensional generalization of Sturmian sequences. Discrete Math., 223, 27-53 (2000) · Zbl 0970.68124
[2] H. Boender, Factoring large integers with the quadratic sieve, Ph.D. Thesis, Leiden University, 1997.; H. Boender, Factoring large integers with the quadratic sieve, Ph.D. Thesis, Leiden University, 1997.
[3] Morse, M.; Hedlund, G. A., Symbolic dynamics, Amer. J. Math., 60, 815-866 (1938) · JFM 64.0798.04
[4] Sander, J. W.; Tijdeman, R., The complexity of functions on lattices, Theor. Comput. Sci., 246, 195-225 (2000) · Zbl 1005.68118
[5] Sander, J. W.; Tijdeman, R., Low complexity functions and convex sets in \(Z^k\), Math. Z., 233, 205-218 (2000) · Zbl 1022.37011
[6] Vuillon, L., Combinatoire des motifs d’une suite sturmienne bidimensionelle, Theor. Comput. Sci., 209, 261-285 (1998) · Zbl 0913.68206
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.