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# On the Exponential Growth of Geometric Shapes

Feb 2024
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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 \$\Omega(k)\$ turning points requires \$\Omega(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.

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