Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any t...

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Bibliographic Details
Published in:Algorithmica Vol. 83; no. 5; pp. 1316 - 1351
Main Authors: Akitaya, Hugo A., Arkin, Esther M., Damian, Mirela, Demaine, Erik D., Dujmović, Vida, Flatland, Robin, Korman, Matias, Palop, Belen, Parada, Irene, Renssen, André van, Sacristán, Vera
Format: Journal Article
Language:English
Published: New York Springer US 01.05.2021
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Configurations
Data Structures and Information Theory
Mathematics of Computing
Modules
Reconfiguration
Robots
Theory of Computation
Pivoting squares
Geometric algorithms
Modular robots
Reconfiguration
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses O ( n 2 ) pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-020-00784-6