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DOI: 10.4230/LIPIcs.SAND.2024.14
Forming Large Patterns with Local Robots in the OBLOT Model

Authors: Christopher Hahn, Jonas Harbig, and Peter Kling

Published in: LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

Abstract
In the arbitrary pattern formation problem, n autonomous, mobile robots must form an arbitrary pattern P ⊆ R². The (deterministic) robots are typically assumed to be indistinguishable, disoriented, and unable to communicate. An important distinction is whether robots have memory and/or a limited viewing range. Previous work managed to form P under a natural symmetry condition if robots have no memory but an unlimited viewing range [Masafumi Yamashita and Ichiro Suzuki, 2010] or if robots have a limited viewing range but memory [Yukiko Yamauchi and Masafumi Yamashita, 2013]. In the latter case, P is only formed in a shrunk version that has constant diameter. Without memory and with limited viewing range, forming arbitrary patterns remains an open problem. We provide a partial solution by showing that P can be formed under the same symmetry condition if the robots' initial diameter is ≤ 1. Our protocol partitions P into rotation-symmetric components and exploits the initial mutual visibility to form one cluster per component. Using a careful placement of the clusters and their robots, we show that a cluster can move in a coordinated way through its component while "drawing" P by dropping one robot per pattern coordinate.
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Christopher Hahn, Jonas Harbig, and Peter Kling. Forming Large Patterns with Local Robots in the OBLOT Model. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hahn_et_al:LIPIcs.SAND.2024.14,
  author =	{Hahn, Christopher and Harbig, Jonas and Kling, Peter},
  title =	{{Forming Large Patterns with Local Robots in the OBLOT Model}},
  booktitle =	{3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)},
  pages =	{14:1--14:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-315-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{292},
  editor =	{Casteigts, Arnaud and Kuhn, Fabian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.14},
  URN =		{urn:nbn:de:0030-drops-198924},
  doi =		{10.4230/LIPIcs.SAND.2024.14},
  annote =	{Keywords: Swarm Algorithm, Swarm Robots, Distributed Algorithm, Pattern Formation, Limited Visibility, Oblivious}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2022.15
A Unifying Approach to Efficient (Near)-Gathering of Disoriented Robots with Limited Visibility

Authors: Jannik Castenow, Jonas Harbig, Daniel Jung, Peter Kling, Till Knollmann, and Friedhelm Meyer auf der Heide

Published in: LIPIcs, Volume 253, 26th International Conference on Principles of Distributed Systems (OPODIS 2022)

Abstract
We consider a swarm of n robots in a d-dimensional Euclidean space. The robots are oblivious (no persistent memory), disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task Gathering requires that all robots reach the same, not predefined position. In the related NearGathering task, they must reach distinct positions in close proximity such that every robot sees the entire swarm. In the considered setting, Gathering can be solved in 𝒪(n + Δ²) synchronous rounds both in two and three dimensions, where Δ denotes the initial maximal distance of two robots [Hideki Ando et al., 1999; Michael Braun et al., 2020; Bastian Degener et al., 2011]. In this work, we formalize a key property of efficient Gathering protocols and use it to define λ-contracting protocols. Any such protocol gathers n robots in the d-dimensional space in 𝒪(Δ²) synchronous rounds, for d ≥ 2. For d = 1, any λ-contracting protocol gathers in optimal time 𝒪(Δ). Moreover, we prove a corresponding lower bound stating that any protocol in which robots move to target points inside the local convex hulls of their neighborhoods - λ-contracting protocols have this property - requires Ω(Δ²) rounds to gather all robots (d > 1). Among others, we prove that the d-dimensional generalization of the GTC-protocol [Hideki Ando et al., 1999] is λ-contracting. Remarkably, our improved and generalized runtime bound is independent of n and d. We also introduce an approach to make any λ-contracting protocol collision-free (robots never occupy the same position) to solve NearGathering. The resulting protocols maintain the runtime of Θ (Δ²) and work even in the semi-synchronous model. This yields the first NearGathering protocols for disoriented robots and the first proven runtime bound. In particular, combined with results from [Paola Flocchini et al., 2017] for robots with global visibility, we obtain the first protocol to solve Uniform Circle Formation (arrange the robots on the vertices of a regular n-gon) for oblivious, disoriented robots with limited visibility.
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Jannik Castenow, Jonas Harbig, Daniel Jung, Peter Kling, Till Knollmann, and Friedhelm Meyer auf der Heide. A Unifying Approach to Efficient (Near)-Gathering of Disoriented Robots with Limited Visibility. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 15:1-15:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{castenow_et_al:LIPIcs.OPODIS.2022.15,
  author =	{Castenow, Jannik and Harbig, Jonas and Jung, Daniel and Kling, Peter and Knollmann, Till and Meyer auf der Heide, Friedhelm},
  title =	{{A Unifying Approach to Efficient (Near)-Gathering of Disoriented Robots with Limited Visibility}},
  booktitle =	{26th International Conference on Principles of Distributed Systems (OPODIS 2022)},
  pages =	{15:1--15:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-265-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{253},
  editor =	{Hillel, Eshcar and Palmieri, Roberto and Rivi\`{e}re, Etienne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2022.15},
  URN =		{urn:nbn:de:0030-drops-176350},
  doi =		{10.4230/LIPIcs.OPODIS.2022.15},
  annote =	{Keywords: mobile robots, gathering, limited visibility, runtime, collision avoidance, near-gathering}
}
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