Brief Announcement: On the Exponential Growth of Geometric Shapes

Abstract

We explore how geometric structures (or shapes) can be grown exponentially fast from a single node, through a sequence of centralized growth operations, and if collisions during growth are to be avoided. 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(klog 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(klog k) time steps to be grown. In the stronger model, 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.