Research Academic Computer Computer Technology Institute CTI

Pátra, Greece

Research Academic Computer Computer Technology Institute CTI

Pátra, Greece
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Michail O.,University of Liverpool | Skretas G.,University of Patras | Spirakis P.G.,University of Liverpool | Spirakis P.G.,University of Patras | Spirakis P.G.,Research Academic Computer Computer Technology Institute CTI
Leibniz International Proceedings in Informatics, LIPIcs | Year: 2017

In this work, we study theoretical models of programmable matter systems. The systems under consideration consist of spherical modules, kept together by magnetic forces and able to perform two minimal mechanical operations (or movements): rotate around a neighbor and slide over a line. In terms of modeling, there are n nodes arranged in a 2-dimensional grid and forming some initial shape. The goal is for the initial shape A to transform to some target shape B by a sequence of movements. Most of the paper focuses on transformability questions, meaning whether it is in principle feasible to transform a given shape to another. We first consider the case in which only rotation is available to the nodes. Our main result is that deciding whether two given shapes A and B can be transformed to each other is in P. We then insist on rotation only and impose the restriction that the nodes must maintain global connectivity throughout the transformation. We prove that the corresponding transformability question is in PSPACE and study the problem of determining the minimum seeds that can make feasible otherwise infeasible transformations. Next we allow both rotations and slidings and prove universality: any two connected shapes A,B of the same number of nodes, can be transformed to each other without breaking connectivity. The worst-case number of movements of the generic strategy is (n2). We improve this to O(n) parallel time, by a pipelining strategy, and prove optimality of both by matching lower bounds. We next turn our attention to distributed transformations. The nodes are now distributed processes able to perform communicate-compute-move rounds. We provide distributed algorithms for a general type of transformation. © Othon Michail, George Skretas, and Paul G. Spirakis;.

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