Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpi\'nski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system

Reflections on Tiles (in Self-Assembly)

We define the Reflexive Tile Assembly Model (RTAM), which is obtained from the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across their horizontal and/or vertical axes. We show that the class of directed temperature-1 RTAM systems is not computationally universal, which is conjectured but unproven for the aTAM, and like the aTAM, the RTAM is computationally universal at temperature 2. We then show that at temperature 1, when starting from a single tile seed, the RTAM is capable of assembling n x n squares for n odd using only n tile types, but incapable of assembling n x n squares for n even. Moreover, we show that n is a lower bound on the number of tile types needed to assemble n x n squares for n odd in the temperature-1 RTAM. The conjectured lower bound for temperature-1 aTAM systems is 2n-1. Finally, we give preliminary results toward the classification of which finite connected shapes in Z^2 can be assembled (strictly or weakly) by a singly seeded (i.e. seed of size 1) RTAM system, including a complete classification of which finite connected shapes be strictly assembled by a "mismatch-free" singly seeded RTAM system.Comment: New results which classify the types of shapes which can self-assemble in the RTAM have been adde

Simple Iterative Heuristics for Correlation Clustering

A straightforward natural iterative heuristic for correlation clustering in the general setting is to start from singleton clusters and whenever merging two clusters improves the current quality score merge them into a single cluster. We analyze the approximation and complexity aspects of this heuristic and its randomized variant where two clusters to merge are chosen uniformly at random among cluster pairs amenable to merge

Non-cooperatively Assembling Large Structures

International audienceAlgorithmic self-assembly is the study of the local, distributed, asynchronous algorithms ran by molecules to self-organise, in particular during crystal growth. The general cooperative model, also called temperature 2, uses synchronisation to simulate Turing machines, build shapes using the smallest possible amount of tile types, and other algorithmic tasks. However, in the non-cooperative (temperature 1) model, the growth process is entirely asynchronous, and mostly relies on geometry. Even though the model looks like a generalisation of finite automata to two dimensions, its 3D generalisation is capable of performing arbitrary (Turing) computation [SODA 2011], and of universal simulations [SODA 2014], whereby a single 3D non-cooperative tileset can simulate the dynamics of all possible 3D non-cooperative systems, up to a constant scaling factor. However, the original 2D non-cooperative model is not capable of universal simulations [STOC 2017], and the question of its computational power is still widely open and it is conjectured to be weaker than temperature or its 3D counterpart. Here, we show an unexpected result, namely that this model can reliably grow assemblies of diameter$$\varTheta (n \log n)$$ with only n tile types, which is the first asymptotically efficient positive construction