Efficient Synthesis for Two-Dimensional Strand Arrays with Row Constraints

In large-scale array-based DNA synthesis, optical and chemical coupling between nearby sites can limit simultaneous activations. Motivated by this constraint, we study strands synthesized according to a fixed global synthesis sequence, with at most one strand per row advancing in each cycle. We focus on the fundamental case of two strands in a single row and analyze the expected completion time of row-constrained synthesis. We introduce the laggard-first (LF) policy, a simple rule that always advances the strand with fewer synthesized symbols when a conflict arises, and establish that it is asymptotically optimal among online policies without look-ahead. In the binary case, one-symbol look-ahead strictly improves on the no-look-ahead bound. We further show that even complete advance knowledge does not eliminate the scheduling loss, as even a globally optimal schedule incurs an unavoidable expected overhead that grows linearly with the strand length. Finally, we complement these scheduling results with a dynamic programming algorithm for computing an optimal offline synthesis order and a constant-redundancy binary coding scheme that yields a deterministic worst-case synthesis time guarantee.