We analyze the complexity of building linear assemblies, sets of linear assemblies, and \({\mathcal{O}}(1)\)-scale general shapes in the staged tile assembly model. For systems with at most b bins and t tile types, we prove that the minimum number of stages to uniquely assemble a \(1 \times n\)line is \(\varTheta (\log _t{n} + \log _b{\frac{n}{t}} + 1)\). Generalizing to \({\mathcal{O}}(1) \times n\) lines, we prove the minimum number of stages is \({\mathcal{O}}(\frac{\log {n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})\) and \(\varOmega (\frac{\log {n} - tb - t\log t}{b^2})\). We also obtain similar upper and lower bounds in a model permitting flexible glues using non-diagonal glue functions. Next, we consider assembling sets of lines and general shapes using \(t = {\mathcal{O}}(1)\) tile types. We prove that the minimum number of stages needed to assemble a set of k lines of size at most \({\mathcal{O}}(1) \times n\) is \({\mathcal{O}}(\frac{k\log n}{b^2}+\frac{k\sqrt{\log n}}{b}+\log \log n)\) and \(\varOmega (\frac{k\log n}{b^2})\). In the case that \(b = \mathcal {O}(\sqrt{k})\), the minimum number of stages is \(\varTheta (\log {n})\). The upper bound in this special case is then used to assemble “hefty” shapes of at least logarithmic edge-length-to-edge-count ratio at \(\mathcal {O}(1)\)-scale using \(\mathcal {O}(\sqrt{k})\) bins and optimal \(\mathcal {O}(\log {n})\) stages.

中文翻译：

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线性装配的最佳分段自装配

我们分析了在分阶段的瓷砖装配模型中构建线性装配，线性装配集以及\（{\ mathcal {O}}（1）\）-比例通用形状的复杂性。对于具有最多b个bin和t个图块类型的系统，我们证明了唯一组装\（1 \ times n \）行的最小阶段数是\（\ varTheta（\ log _t {n} + \ log bb { \ frac {n} {t}} + 1）\）。推广到\（{\ mathcal {O}}（1）\ times n \）行，我们证明最小阶段数为\（{\ mathcal {O}}（\ frac {\ log {n}-tb- t \ log t} {b ^ 2} + \ frac {\ log \ log b} {\ log t}）\）和\（\ varOmega（\ frac {\ log {n}-tb-t \ log t} {b ^ 2}）\）。我们还可以在使用非对角胶功能的允许柔性胶的模型中获得相似的上下边界。接下来，我们考虑使用\（t = {\\ mathcal {O}}（1）\）拼贴类型来组装线和一般形状的集合。我们证明，最多可以组装一组大小为\（{\ mathcal {O}}（1）\ times n \）的k行的最小阶段数是\（{\ mathcal {O}}（\ frac {k \ log n} {b ^ 2} + \ frac {k \ sqrt {\ log n}} {b} + \ log \ log n）\）和\（\ varOmega（\ frac {k \ log n} {b ^ 2}）\）。在\（b = \ mathcal {O}（\ sqrt {k}）\）的情况下，最小阶段数为\（\ varTheta（\ log {n}）\）。然后使用这种特殊情况的上限，以\（\ mathcal {O}（1）\）- scale的\（\ mathcal ）组合至少具有对数边长与边数比的“重”形状{O}（\ sqrt {k}）\）分档和最佳\（\ mathcal {O}（\ log {n}）\）级。