Optimal self-assembly of finite shapes at temperature 1 in 3D

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape $X \subset \mathbb{Z}^2$, there is a tile set that uniquely self-assembles into a 3D representation of a scaled-up version of $X$ at temperature 1 in 3D with optimal program-size complexity (the "program-size complexity", also known as "tile complexity", of a shape is the minimum number of tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it only places tiles in the $z = 0$ and $z = 1$ planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SICOMP 2007)

Limited discrepancy search revisited

Harvey and Ginsberg's limited discrepancy search (LDS) is based on the assumption that costly heuristic mistakes are made early in the search process. Consequently, LDS repeatedly probes the state space, going against the heuristic (i.e., taking discrepancies) a specified number of times in all possible ways and attempts to take those discrepancies as early as possible. LDS was improved by Richard Korf, to become improved LDS (ILDS), but in doing so, discrepancies were taken as late as possible, going against the original assumption. Many subsequent algorithms have faithfully inherited Korf's interpretation of LDS, and take discrepancies late. This then raises the question: Should we take our discrepancies late or early? We repeat the original experiments performed by Harvey and Ginsberg and those by Korf in an attempt to answer this question. We also investigate the early stopping condition of the YIELDS algorithm, demonstrating that it is simple, elegant and efficient