Growing Fast without Colliding: Polylogarithmic Time Step Construction of Geometric Shapes
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Building on two recent models of [Almalki and Michail, 2022] and [Gupta et al., 2023], we explore the constructive power of a set of geometric growth processes. The studied processes, by applying a sequence of centralized, parallel, and linear-strength growth operations, can construct shapes from smaller shapes or from a singleton exponentially fast. A technical challenge in growing shapes that fast is the need to avoid collisions caused, for example, when the shape breaks, stretches, or self-intersects. We distinguish two types of growth operations – one that avoids collisions by preserving cycles and one that achieves the same by breaking them – and two types of graph models. We study the following types of shape reachability questions in these models. Given a class of initial shapes ℐ and a class of final shapes ℱ, our objective is to determine whether any (some) shape S ∈ℱ can be reached from any shape S_0 ∈ℐ in a number of time steps which is (poly)logarithmic in the size of S. For the reachable classes, we additionally present the respective growth processes. In cycle-preserving growth, we study these problems in basic classes of shapes such as paths, spirals, and trees and reveal the importance of the number of turning points as a parameter. We give both positive and negative results. For cycle-breaking growth, we obtain a strong positive result – a general growth process that can grow any connected shape from a singleton fast.
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