We study the theoretical problem of synthesizing multiple DNA strands under spatial constraints, motivated by large-scale DNA synthesis technologies. In this setting, strands are arranged in an array and synthesized according to a fixed global synthesis sequence, with the restriction that at most one strand per row may be synthesized in any synthesis cycle. We focus on the basic case of two strands in a single row and analyze the expected completion time under this row-constrained model. By decomposing the process into a Markov chain, we derive analytical upper and lower bounds on the expected synthesis time. We show that a simple laggard-first policy achieves an asymptotic expected completion time of (q+3)L/2 for any alphabet of size q, and that no online policy without look-ahead can asymptotically outperform this bound. For the binary case, we show that allowing a single-symbol look-ahead strictly improves performance, yielding an asymptotic expected completion time of 7L/3. Finally, we present a dynamic programming algorithm that computes the optimal offline schedule for any fixed pair of sequences. Together, these results provide the first analytical bounds for synthesis under spatial constraints and lay the groundwork for future studies of optimal synthesis policies in such settings.

翻译：本研究探讨了在大规模DNA合成技术背景下，受空间约束的多条DNA链合成的理论问题。在此设定中，链被排列成阵列，并按照固定的全局合成序列进行合成，其限制条件是每个合成周期中每行最多只能合成一条链。我们聚焦于单行中两条链的基本情况，分析了在此行约束模型下的期望完成时间。通过将过程分解为马尔可夫链，我们推导出了期望合成时间的解析上界与下界。我们证明，对于任何大小为q的字母表，一种简单的滞后优先策略能达到(q+3)L/2的渐近期望完成时间，并且任何无前瞻能力的在线策略都无法渐近地超越此界限。对于二进制情况，我们证明允许单符号前瞻能严格提升性能，产生7L/3的渐近期望完成时间。最后，我们提出了一种动态规划算法，可为任意固定的序列对计算最优离线调度方案。这些结果共同为空间约束下的合成问题提供了首个解析界限，并为未来在此类设定中研究最优合成策略奠定了基础。