On the Exponential Growth of Geometric Shapes

TL;DR

This work investigates how geometric structures on a square grid can be grown from a single node with exponential speed under different growth rules. It introduces connectivity and adjacency graph models, along with cycle-preserving and cycle-breaking strategies, to analyze the complexity of constructing final shapes from initial ones. The authors establish polylogarithmic upper bounds for broad shape classes (e.g., $O(k \log n)$ for trees with $k$ turns per root-to-leaf path and $O(\log n)$ for spirals) and matching lower bounds (e.g., $\Omega(\log n)$ and $\Omega(k \log k)$) that illuminate inherent limitations. In the strongest cycle-breaking model with neighbor handover, they present a universal algorithm achieving $O(\log n)$ time for any final shape, highlighting the potential for rapid, centralized shape growth and informing programmable-matter design and self-assembly theory.

Abstract

In this paper, we explore how geometric structures can be grown exponentially fast. The studied processes start from an initial shape and apply a sequence of centralized growth operations to grow other shapes. We focus on the case where the initial shape is just a single node. A technical challenge in growing shapes that fast is the need to avoid collisions caused when the shape breaks, stretches, or self-intersects. We identify a parameter $k$, representing the number of turning points within specific parts of a shape. We prove that, if edges can only be formed when generating new nodes and cannot be deleted, trees having $O(k)$ turning points on every root-to-leaf path can be grown in $O(k\log n)$ time steps and spirals with $O(\log n)$ turning points can be grown in $O(\log n)$ time steps, $n$ being the size of the final shape. For this case, we also show that the maximum number of turning points in a root-to-leaf path of a tree is a lower bound on the number of time steps to grow the tree and that there exists a class of paths such that any path in the class with $Ω(k)$ turning points requires $Ω(k\log k)$ time steps to be grown. If nodes can additionally be connected as soon as they become adjacent, we prove that if a shape $S$ has a spanning tree with $O(k)$ turning points on every root-to-leaf path, then the adjacency closure of $S$ can be grown in $O(k \log n)$ time steps. In the strongest model that we study, where edges can be deleted and neighbors can be handed over to newly generated nodes, we obtain a universal algorithm: for any shape $S$ it gives a process that grows $S$ from a single node exponentially fast.

Figures (7)

• Figure 1: (a) Initial tree $T$. (b) The tree $T'$ after a growth operation on node $u$ moves north to generate a new node $u'$ without any collision. (c) Illustration of a node collision scenario: the $T'$ here is a result of a growth operation applied on node $u$ toward the east, where a node $v$ already exists and $uv\notin E$. The newly generated node $u'$ occupies the same position as $v$, leading to a node collision. (d) Another scenario of a node collision: two nodes $u_1$ and $u_2$ simultaneously grow in the east direction and generate $u_1'$ and $u_2'$, though $u_1'$ and $u_2'$ do not collide directly, their growth pushes their branch into an adjacent branch, leading to a collision.
• Figure 2: An example of a cycle collision within the shape $S$ due to unequal displacement vectors along the two paths $p_1$ and $p_2$, thus, $\vec{v}_{p_1}\neq \vec{v}_{p_2}$. In particular, the number of generated nodes (gray nodes) along $p_2$ is greater than that along $p_1$. This difference in the number of generated nodes leads to a collision within the cycle, indicating an irregularity in the shape $S'$.
• Figure 3: (a) An illustration of turning points (drawn by cross points) of an incompressible spiral path $P$. (b) An illustration of the path $\widehat{P}$ constructed by $\sigma$ in the Base Case in the proof of Lemma \ref{['lem:lowerBound2']}. The black (blue) part of $\widehat{P}$ shows the subpath, which is the same as (different from) $P$. The dotted green line shows the direction in which $tp_5$ will be generated. (c) An illustration of the path $\widehat{P}$ constructed by $\sigma$ in the Inductive step in the proof of Lemma \ref{['lem:lowerBound2']} for $t > 6$. The black (blue) part of $\widehat{P}$ shows the subpath, which is the same as (different from) $P$. The dotted green line shows the direction in which $tp_t$ will be generated.
• Figure 4: An illustration of an incompressible path consisting of a red and a blue spiral used in the proof of Theorem \ref{['the:lowerBound']}. The green square vertex $u$ denotes the internal endpoint of both spirals.
• Figure 5: An illustration of the modified BFS approach for growing adjacent line segments in the shape without collisions. In this example, the growth of sub-segment $s_{2,1}$ of $s_2$ during phase $i=2$ involves growing its length from the adjacent previously grown line segment $s_1$. Furthermore, when two adjacent line segments are growing in parallel, such as $s_3$, they are separated based on whether they are positioned in an even or odd row of $T$.

Theorems & Definitions (25)

• Definition 1
• Proposition 2
• Proposition 3
• Definition 4: Staircase
• Definition 5: Spiral
• Proposition 6
• Lemma 7
• Corollary 8
• Theorem 9
• Theorem 10