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 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 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.

We study algorithmic growth processes under a geometric setting. Each process begins with an initial shape of nodes SI=S0 and, in every time step t≥1, by applying (in parallel) one or more growth operations of a specific type to the current shape, St−1, generates the next, St, always satisfying |St|>|St−1|. We define three types of growth operations and explore the algorithmic and structural properties of their resulting processes. Our goal is to characterize the classes of shapes that can be constructed in O(log⁡n) or polylog n time steps, n being the size of the final shape SF. Moreover, we want to determine whether a given shape SF can be constructed from a given initial shape SI using a finite sequence of growth operations of a given type, called a constructor of SF. We give exact and partial characterizations of classes of shapes that can be constructed in polylog n time steps, polynomial-time centralized algorithms for deciding reachability between pairs of input shapes (SI,SF) and for generating constructors when SF can be constructed from SI, as well as some negative results.