The abstract tile assembly model (aTam) is a model of DNA self-assembly. Most of the studies focus on cooperative aTam where a form of synchronization between the tiles is possible. Simulating Turing machines is achievable in this context. Few results and constructions are known for the non-cooperative case (a variant of Wang tilings where assemblies do not need to cover the whole plane and some mismatches may occur). Introduced by P.E. Meunier and D. Regnault, efficient paths are a non-trivial construction for non-cooperative aTam. These paths of width nlog(n) are designed with n different tile types. Assembling them relies heavily on a form of ``non-determinism''. Indeed, the set of tiles may produced different finite terminal assemblies but they all contain the same efficient path. Directed non-cooperative aTam does not allow this non-determinism as only one assembly may be produced by a tile assembly system. This variant of aTam is the only one who was shown to be decidable. In this paper, we show that if the terminal assembly of a directed non-cooperative tile assembly system is finite then its width and length are of linear size according to the size of the tile assembly system. This result implies that the construction of efficient paths cannot be generalized to the directed case and that some computation must rely on a competition between different paths. It also implies that the construction of a square of width n using 2n-1 tiles types is asymptotically optimal. Moreover, we hope that the techniques introduced here will lead to a better comprehension of the non-directed case.

翻译：抽象瓦片组装模型（aTam）是一种DNA自组装模型。大多数研究集中于合作型aTam，其中瓦片间可实现某种形式的同步。在此背景下可实现对图灵机的模拟。对于非合作情形（王氏铺砖的变体，其中组装无需覆盖整个平面且允许部分错配），已知的结果和构造较少。由P.E. Meunier与D. Regnault引入的高效路径是非合作aTam中一种非平凡构造。这些宽度为nlog(n)的路径使用n种不同瓦片类型设计而成。其组装过程严重依赖于某种形式的"非确定性"：虽然瓦片集合可能产生不同的有限终态组装，但它们都包含相同的高效路径。有向非合作aTam不允许这种非确定性，因为瓦片组装系统只能产生单一组装。该aTam变体是目前唯一被证明具有可判定性的模型。本文证明：若有向非合作瓦片组装系统的终态组装是有限的，则其宽度和长度相对于系统规模呈线性尺寸。该结果意味着高效路径的构造无法推广至有向情形，且某些计算必须依赖于不同路径间的竞争。同时表明使用2n-1种瓦片类型构造宽度为n的正方形在渐近意义下是最优的。此外，我们希望本文引入的技术将有助于更好地理解非定向情形。