Self-assemblying Classes of Shapes with a Minimum Number of Tiles, and in Optimal Time

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.

• Type: Conference
• Year: 2006
• Where: FSTTCS
• Authors: Florent Becker, Ivan Rapaport, Eric Rémila
• Keywords: FSTTCS 2006, Linear Parallel Time, Parallel Time, Software Engineering, Tile Systems