On the exponential growth of geometric shapes

In this paper, we explore the exponential growth of geometric structures starting from a single node, focusing on centralized growth operations. 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 between a newly generated node and the node that created it and cannot be deleted, trees having at most k turning points on every root-to-leaf path can be grown in $O(k\log k+\log n)$ 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 model, 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 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 at most k turning points on every root-to-leaf path, then the adjacency closure of S can be grown in $O(k\log k+\log n)$ time steps. In the strongest version of the model, where, additionally, edges can be deleted and neighbors handed over to new nodes, we present a universal algorithm for growing any shape S exponentially fast.