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Connected Coordinated Motion Planning with Bounded Stretch

Authors Sándor P. Fekete , Phillip Keldenich , Ramin Kosfeld , Christian Rieck , Christian Scheffer

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Computing methodologies → Motion path planning
Keywords
  • Motion planning
  • parallel motion
  • bounded stretch
  • scaled shape
  • makespan
  • connectivity
  • swarm robotics

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Abstract

We consider the problem of coordinated motion planning for a swarm of simple, identical robots: From a given start grid configuration of robots, we need to reach a desired target configuration via a sequence of parallel, continuous, collision-free robot motions, such that the set of robots induces a connected grid graph at all integer times. The objective is to minimize the makespan of the motion schedule, i.e., to reach the new configuration in a minimum amount of time. We show that this problem is NP-hard, even for deciding whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved.
On the algorithmic side, we establish simultaneous constant-factor approximation for two fundamental parameters, by achieving constant stretch for constant scale. Scaled shapes (which arise by increasing all dimensions of a given object by the same multiplicative factor) have been considered in previous seminal work on self-assembly, often with unbounded or logarithmic scale factors; we provide methods for a generalized scale factor, bounded by a constant. Moreover, our algorithm achieves a constant stretch factor: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is 𝒪(d), which is optimal up to constant factors.

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Sándor P. Fekete, Phillip Keldenich, Ramin Kosfeld, Christian Rieck, and Christian Scheffer. Connected Coordinated Motion Planning with Bounded Stretch. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.9

Author Details

Sándor P. Fekete
  • Department of Computer Science, TU Braunschweig, Germany
Phillip Keldenich
  • Department of Computer Science, TU Braunschweig, Germany
Ramin Kosfeld
  • Department of Computer Science, TU Braunschweig, Germany
Christian Rieck
  • Department of Computer Science, TU Braunschweig, Germany
Christian Scheffer
  • Faculty of Electrical Engineering and Computer Science, Bochum University of Applied Sciences, Bochum, Germany

Acknowledgements

We thank Linda Kleist and Arne Schmidt for helpful algorithmic discourse and suggestions that improved the presentation of this paper.

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