Tight Bounds on the Directed Tile Complexity of a Just-Barely 3D $$2 \times N$$ Rectangle at Temperature 1

We study the problem of determining the size of the smallest tile set in which a given target shape uniquely self-assembles in Winfree’s abstract Tile assembly Model (aTAM), an elegant combinatorial model of DNA tile self-assembly. This problem is also known as the directed tile complexity problem. We work in a variant of the aTAM, restricted to having the minimum binding strength threshold (temperature) set to 1 but mildly generalized to allow self-assembly to take place in a just-barely 3D setting, where unit cubes are allowed to be placed in the $$z=0$$ and $$z=1$$ planes. Furcy, Summers and Withers recently proved lower and upper bounds on the directed tile complexity of a just-barely 3D $$k \times N$$ rectangle at temperature-1 of $$\varOmega \left( N^{\frac{1}{k}}\right) $$ and $$O\left( N^{\frac{1}{k-1}}+\log N\right) $$ , respectively. However, their upper bound does not hold for $$k=2$$ . We close this gap for $$k=2$$ by proving an asymptotically tight bound of $$\varTheta (N)$$ on the directed tile complexity of a just-barely 3D $$2 \times N$$ rectangle at temperature-1. The proof of our lower bound is based on an algorithm that uses a novel projection of a given just-barely 3D assembly onto an equivalent, 2D assembly.