We analyze the number of tile types t, bins b, and stages necessary to assemble $$n \times n$$ squares and scaled shapes in the staged tile assembly model. For $$n \times n$$ squares, we prove $$\mathcal {O}\left( \frac{\log {n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}\right) $$ stages suffice and $$\varOmega \left( \frac{\log {n} - tb - t\log t}{b^2}\right) $$ are necessary for almost all n. For shapes S with Kolmogorov complexity K(S), we prove $$\mathcal {O}\left( \frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}\right) $$ stages suffice and $$\varOmega \left( \frac{K(S) - tb - t\log t}{b^2}\right) $$ are necessary to assemble a scaled version of S, for almost all S. We obtain similarly tight bounds when the more powerful flexible glues are permitted.

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a $O(n\log^2{n})$-time algorithm for deciding if a polyomino with $n$ edges can tile the plane isohedrally. This improves on the $O(n^{18})$-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.

An example of a nonspecific molecular bond is the affinity of any positive charge for any negative charge (like-unlike), or of nonpolar material for itself when in aqueous solution (like-like). This contrasts specific bonds such as the affinity of the DNA base A for T, but not for C, G, or another A. Recent experimental breakthroughs in DNA nanotechnology [4], [11] demonstrate that a particular nonspecific like-like bond (“blunt-end DNA stacking” that occurs between the ends of any pair of DNA double-helices) can be used to create specific “macrobonds” by careful geometric arrangement of many nonspecific blunt ends, motivating the need for sets of macrobonds that are orthogonal: two macrobonds not intended to bind should have relatively low binding strength, even when misaligned. To address this need, we introduce geometric orthogonal codes that abstractly model the engineered DNA macrobonds as two-dimensional binary codewords. While motivated by completely different applications, geometric orthogonal codes share similar features to the optical orthogonal codes studied by Chung, Salehi, and Wei [3]. The main technical difference is the importance of 2D geometry in defining codeword orthogonality.

We prove a negative result on the power of a model of algorithmic self-assembly for which finding general techniques and results has been notoriously difficult. Specifically, we prove that Winfree's abstract Tile Assembly Model is not intrinsically universal when restricted to use noncooperative tile binding. This stands in stark contrast to the recent result that the abstract Tile Assembly Model is indeed intrinsically universal when cooperative binding is used (FOCS 2012). Noncooperative self-assembly, also known as “temperature 1”, is where all tiles bind to each other if they match on at least one side. On the other hand, cooperative self-assembly requires that some tiles bind on at least two sides. Our result shows that the change from non-cooperative to cooperative binding qualitatively improves the range of dynamics and behaviors found in these models of nanoscale self-assembly. The result holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. This shows that Turing universal behavior in self-assembly does not imply the ability to simulate all algorithmic self-assembly processes. In addition to the negative result, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. This tile set implies that, in a restricted sense, non-cooperative self-assembly is intrinsically universal for itself.

Agent-based simulation has become a key technique for modeling and simulating dynamic, complicated behaviors in social and behavioral sciences. As these simulations become more complex, they generate an increasingly large amount of data. Lacking the appropriate tools and support, it has become difficult for social scientists to interpret and analyze the results of these simulations. In this paper, we introduce the Aggregate Temporal Graph (ATG), a graph formulation that can be used to capture complex relationships between discrete simulation states in time. Using this formulation, we can assist social scientists in identifying critical simulation states by examining graph substructures. In particular, we define the concept of a Gateway and its inverse, a Terminal, which capture the relationships between pivotal states in the simulation and their inevitable outcomes. We propose two real-time computable algorithms to identify these relationships and provide a proof of correctness, complexity analysis, and empirical run-time analysis. We demonstrate the use of these algorithms on a large-scale social science simulation of political power and violence in present-day Thailand, and discuss broader applications of the ATG and associated algorithms in other domains such as analytic provenance.

We prove that every simple polygon can be made as a (2D) pop-up card/book that opens to any desired angle between 0 and 360°. More precisely, given a simple polygon attached to the two walls of the open pop-up, our polynomial-time algorithm subdivides the polygon into a single-degree-of-freedom linkage structure, such that closing the pop-up flattens the linkage without collision. This result solves an open problem of Hara and Sugihara from 2009. We also show how to obtain a more efficient construction for the special case of orthogonal polygons, and how to make 3D orthogonal polyhedra, from pop-ups that open to 90°, 180°, 270°, or 360°.

We study the difference between the standard seeded model (aTAM) of tile self-assembly, and the "seedless" two-handed model of tile self-assembly (2HAM). Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model.