On Geometric Shape Construction via Growth Operations

Abstract

In this work, we investigate novel algorithmic growth processes. Our system runs on a 2-dimensional grid and operates in discrete time steps. The growth process begins with an initial shape of nodes $$S_I=S_0$$ and, in every time step $$t \ge 1$$ , by applying (in parallel) one or more growth operations of a specific type to the current shape-instance $$S_{t-1}$$ , generates the next instance $$S_t$$ , always satisfying $$|S_t| > |S_{t-1}|$$ . 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 $$S_F$$ , and determine whether a shape $$S_F$$ can be constructed from an initial shape $$S_I$$ using a finite sequence of growth operations of a given type, called a constructor of $$S_F$$ . In particular, we propose three growth operations, full doubling, row and column doubling, which we call RC doubling, and doubling, and explore the algorithmic and structural properties of their resulting processes under a geometric setting. For full doubling, in which, in every time step, every node generates a new node in a given direction, we completely characterize the structure of the class of shapes that can be constructed from a given initial shape. For RC doubling, in which complete columns or rows double, our main contribution is a linear-time centralized algorithm that for any pair of shapes $$S_I$$ , $$S_F$$ decides if $$S_F$$ can be constructed from $$S_I$$ and, if the answer is yes, returns an $$O(\log n)$$ -time step constructor of $$S_F$$ from $$S_I$$ . For the most general doubling operation, where a subset of individual nodes can double, we show that some shapes cannot be constructed in sublinear time steps and give two universal constructors of any $$S_F$$ from a singleton $$S_I$$ , which are efficient (i.e., up to polylogarithmic time steps) for large classes of shapes. Both constructors can be computed by polynomial-time centralized algorithms for any shape $$S_F$$ .