Building on two recent models of [Almalki and Michail, 2022] and [Gupta et al., 2023], we explore the constructive power of a set of geometric growth processes. The studied processes, by applying a sequence of centralized, parallel, and linear-strength growth operations, can construct shapes from smaller shapes or from a singleton exponentially fast. A technical challenge in growing shapes that fast is the need to avoid collisions caused, for example, when the shape breaks, stretches, or self-intersects. We distinguish two types of growth operations -- one that avoids collisions by preserving cycles and one that achieves the same by breaking them -- and two types of graph models. We study the following types of shape reachability questions in these models. Given a class of initial shapes $\mathcal{I}$ and a class of final shapes $\mathcal{F}$, our objective is to determine whether any (some) shape $S \in \mathcal{F}$ can be reached from any shape $S_0 \in \mathcal{I}$ in a number of time steps which is (poly)logarithmic in the size of $S$. For the reachable classes, we additionally present the respective growth processes. In cycle-preserving growth, we study these problems in basic classes of shapes such as paths, spirals, and trees and reveal the importance of the number of turning points as a parameter. We give both positive and negative results. For cycle-breaking growth, we obtain a strong positive result -- a general growth process that can grow any connected shape from a singleton fast.

翻译：基于[Almalki与Michail, 2022]及[Gupta等, 2023]近期提出的两个模型，我们探索了一类几何生长过程的构造能力。通过应用一系列集中式、并行且具有线性强度的生长操作，所研究的这些过程能够从较小形状或单元素出发，以指数级速度构造出形状。在如此快速的形状生长过程中，一个技术挑战是避免碰撞——例如因形状断裂、拉伸或自交而引发的问题。我们区分了两类生长操作：一类通过保留环来避免碰撞，另一类通过打破环实现相同目标；并区分了两类图模型。我们在这些模型中研究了以下类型的形状可达性问题：给定初始形状类$\mathcal{I}$和最终形状类$\mathcal{F}$，目标是确定是否存在某个（或某些）形状$S \in \mathcal{F}$能够从任意形状$S_0 \in \mathcal{I}$出发，在$S$大小的（多）对数时间步内到达。对于可达类，我们额外给出了相应的生长过程。在保环生长中，我们针对路径、螺旋和树等基本形状类研究了这些问题，并揭示了转折点数量作为关键参数的重要性。我们给出了肯定与否定两类结果。对于破环生长，我们获得了一个强肯定结果——一个通用的生长过程，能够从单元素快速生长出任意连通形状。