On Geometric Shape Construction via Growth Operations
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In this work, we investigate novel algorithmic growth processes. In particular, we propose three growth operations, full doubling, RC doubling and doubling, and explore the algorithmic and structural properties of their resulting processes under a geometric setting. In terms of modeling, our system runs on a 2-dimensional grid and operates in discrete time-steps. The process begins with an initial shape S_I=S_0 and, in every time-step t ≥ 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 and determine whether a final 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. 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 up to individual nodes can double, we show that some shapes cannot be constructed in sub-linear 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.
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