Abstract
In this paper we construct fixed finite tile systems that assemble into particular classes of shapes. Moreover, given an arbitrary n, we show how to calculate the tile concentrations in order to ensure that the expected size of the produced shape is n. For rectangles and squares our constructions are optimal (with respect to the size of the systems). We also introduce the notion of parallel time, which is a good approximation of the classical asynchronous time. We prove that our tile systems produce the rectangles and squares in linear parallel time (with respect to the diameter). Those results are optimal. Finally, we introduce the class of diamonds. For these shapes we construct a non trivial tile system having also a linear parallel time complexity.
This work was supported by programs Ecos-Conicyt, Conicyt “Anillo en Redes”, and Fondap on Applied Mathematics.
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© 2006 Springer-Verlag Berlin Heidelberg
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Becker, F., Rapaport, I., Rémila, É. (2006). Self-assemblying Classes of Shapes with a Minimum Number of Tiles, and in Optimal Time. In: Arun-Kumar, S., Garg, N. (eds) FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2006. Lecture Notes in Computer Science, vol 4337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11944836_7
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DOI: https://doi.org/10.1007/11944836_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-49994-7
Online ISBN: 978-3-540-49995-4
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