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Documents authored by Schmidt, Arne

Document
DOI: 10.4230/LIPIcs.ISAAC.2022.17
Efficiently Reconfiguring a Connected Swarm of Labeled Robots

Authors: Sándor P. Fekete, Peter Kramer, Christian Rieck, Christian Scheffer, and Arne Schmidt

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
When considering motion planning for a swarm of n labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, continuous, collision-free robot motions. The objective is to reach the new configuration in a minimum amount of time; an important constraint is to keep the swarm connected at all times. Problems of this type have been considered before, with recent notable results achieving constant stretch for not necessarily connected reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, the total duration of an overall schedule can be bounded to 𝒪(d), which is optimal up to constant factors. However, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations (which arise by increasing all dimensions of a given object by the same multiplicative factor) of unlabeled robots. We resolve these major open problems by (1) establishing a lower bound of Ω(√n) for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected reconfiguration can be achieved. In addition, we show that (3) it is NP-hard to decide 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.
Cite as

Sándor P. Fekete, Peter Kramer, Christian Rieck, Christian Scheffer, and Arne Schmidt. Efficiently Reconfiguring a Connected Swarm of Labeled Robots. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fekete_et_al:LIPIcs.ISAAC.2022.17,
  author =	{Fekete, S\'{a}ndor P. and Kramer, Peter and Rieck, Christian and Scheffer, Christian and Schmidt, Arne},
  title =	{{Efficiently Reconfiguring a Connected Swarm of Labeled Robots}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.17},
  URN =		{urn:nbn:de:0030-drops-173028},
  doi =		{10.4230/LIPIcs.ISAAC.2022.17},
  annote =	{Keywords: Motion planning, parallel motion, bounded stretch, makespan, connectivity, swarm robotics}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2021.62
Can You Walk This? Eulerian Tours and IDEA Instructions (Media Exposition)

Authors: Aaron T. Becker, Sándor P. Fekete, Matthias Konitzny, Sebastian Morr, and Arne Schmidt

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
We illustrate and animate the classic problem of deciding whether a given graph has an Eulerian path. Starting with a collection of instances of increasing difficulty, we present a set of pictorial instructions, and show how they can be used to solve all instances. These IDEA instructions ("A series of nonverbal algorithm assembly instructions") have proven to be both entertaining for experts and enlightening for novices. We (w)rap up with a song and dance to Euler’s original instance.
Cite as

Aaron T. Becker, Sándor P. Fekete, Matthias Konitzny, Sebastian Morr, and Arne Schmidt. Can You Walk This? Eulerian Tours and IDEA Instructions (Media Exposition). In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 62:1-62:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{becker_et_al:LIPIcs.SoCG.2021.62,
  author =	{Becker, Aaron T. and Fekete, S\'{a}ndor P. and Konitzny, Matthias and Morr, Sebastian and Schmidt, Arne},
  title =	{{Can You Walk This? Eulerian Tours and IDEA Instructions}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{62:1--62:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.62},
  URN =		{urn:nbn:de:0030-drops-138616},
  doi =		{10.4230/LIPIcs.SoCG.2021.62},
  annote =	{Keywords: Eulerian tours, algorithms, education, IDEA instructions}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2020.72
Coordinated Particle Relocation with Global Signals and Local Friction (Media Exposition)

Authors: Victor M. Baez, Aaron T. Becker, Sándor P. Fekete, and Arne Schmidt

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
In this video, we present theoretical and practical methods for achieving arbitrary reconfiguration of a set of objects, based on the use of external forces, such as a magnetic field or gravity: Upon actuation, each object is pushed in the same direction. This concept can be used for a wide range of applications in which particles do not have their own energy supply or in which they are subject to the same global control commands. A crucial challenge for achieving any desired target configuration is breaking global symmetry in a controlled fashion. Previous work (some of which was presented during SoCG 2015) made use of specifically placed barriers; however, introducing precisely located obstacles into the workspace is impractical for many scenarios. In this paper, we present a different, less intrusive method: making use of the interplay between static friction with a boundary and the external force to achieve arbitrary reconfiguration. Our key contributions are theoretical characterizations of the critical coefficient of friction that is sufficient for rearranging two particles in triangles, convex polygons, and regular polygons; a method for reconfiguring multiple particles in rectangular workspaces, and deriving practical algorithms for these rearrangements. Hardware experiments show the efficacy of these procedures, demonstrating the usefulness of this novel approach.
Cite as

Victor M. Baez, Aaron T. Becker, Sándor P. Fekete, and Arne Schmidt. Coordinated Particle Relocation with Global Signals and Local Friction (Media Exposition). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 72:1-72:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{baez_et_al:LIPIcs.SoCG.2020.72,
  author =	{Baez, Victor M. and Becker, Aaron T. and Fekete, S\'{a}ndor P. and Schmidt, Arne},
  title =	{{Coordinated Particle Relocation with Global Signals and Local Friction}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{72:1--72:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.72},
  URN =		{urn:nbn:de:0030-drops-122305},
  doi =		{10.4230/LIPIcs.SoCG.2020.72},
  annote =	{Keywords: Global control, reconfiguration, geometric algorithms, friction}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2020.73
Space Ants: Constructing and Reconfiguring Large-Scale Structures with Finite Automata (Media Exposition)

Authors: Amira Abdel-Rahman, Aaron T. Becker, Daniel E. Biediger, Kenneth C. Cheung, Sándor P. Fekete, Neil A. Gershenfeld, Sabrina Hugo, Benjamin Jenett, Phillip Keldenich, Eike Niehs, Christian Rieck, Arne Schmidt, Christian Scheffer, and Michael Yannuzzi

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
In this video, we consider recognition and reconfiguration of lattice-based cellular structures by very simple robots with only basic functionality. The underlying motivation is the construction and modification of space facilities of enormous dimensions, where the combination of new materials with extremely simple robots promises structures of previously unthinkable size and flexibility. We present algorithmic methods that are able to detect and reconfigure arbitrary polyominoes, based on finite-state robots, while also preserving connectivity of a structure during reconfiguration. Specific results include methods for determining a bounding box, scaling a given arrangement, and adapting more general algorithms for transforming polyominoes.
Cite as

Amira Abdel-Rahman, Aaron T. Becker, Daniel E. Biediger, Kenneth C. Cheung, Sándor P. Fekete, Neil A. Gershenfeld, Sabrina Hugo, Benjamin Jenett, Phillip Keldenich, Eike Niehs, Christian Rieck, Arne Schmidt, Christian Scheffer, and Michael Yannuzzi. Space Ants: Constructing and Reconfiguring Large-Scale Structures with Finite Automata (Media Exposition). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 73:1-73:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{abdelrahman_et_al:LIPIcs.SoCG.2020.73,
  author =	{Abdel-Rahman, Amira and Becker, Aaron T. and Biediger, Daniel E. and Cheung, Kenneth C. and Fekete, S\'{a}ndor P. and Gershenfeld, Neil A. and Hugo, Sabrina and Jenett, Benjamin and Keldenich, Phillip and Niehs, Eike and Rieck, Christian and Schmidt, Arne and Scheffer, Christian and Yannuzzi, Michael},
  title =	{{Space Ants: Constructing and Reconfiguring Large-Scale Structures with Finite Automata}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{73:1--73:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.73},
  URN =		{urn:nbn:de:0030-drops-122310},
  doi =		{10.4230/LIPIcs.SoCG.2020.73},
  annote =	{Keywords: Finite automata, reconfiguration, construction, scaling}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2017.11
Tilt Assembly: Algorithms for Micro-Factories that Build Objects with Uniform External Forces

Authors: Aaron T. Becker, Sándor P. Fekete, Phillip Keldenich, Dominik Krupke, Christian Rieck, Christian Scheffer, and Arne Schmidt

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract
We present algorithmic results for the parallel assembly of many micro-scale objects in two and three dimensions from tiny particles, which has been proposed in the context of programmable matter and self-assembly for building high-yield micro-factories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle; particles can bond when being forced together with another appropriate particle. Due to the physical and geometric constraints, not all shapes can be built in this manner; this gives rise to the Tilt Assembly Problem (TAP) of deciding constructibility. For simply-connected polyominoes P in 2D consisting of N unit-squares ("tiles"), we prove that TAP can be decided in O(N log N) time. For the optimization variant MaxTAP (in which the objective is to construct a subshape of maximum possible size), we show polyAPX-hardness: unless P=NP, MaxTAP cannot be approximated within a factor of N^(1/3); for tree-shaped structures, we give an N^(1/2)-approximation algorithm. For the efficiency of the assembly process itself, we show that any constructible shape allows pipelined assembly, which produces copies of P in O(1) amortized time, i.e., N copies of P in O(N) time steps. These considerations can be extended to three-dimensional objects: For the class of polycubes P we prove that it is NP-hard to decide whether it is possible to construct a path between two points of P; it is also NP-hard to decide constructibility of a polycube P. Moreover, it is expAPX-hard to maximize a path from a given start point.
Cite as

Aaron T. Becker, Sándor P. Fekete, Phillip Keldenich, Dominik Krupke, Christian Rieck, Christian Scheffer, and Arne Schmidt. Tilt Assembly: Algorithms for Micro-Factories that Build Objects with Uniform External Forces. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{becker_et_al:LIPIcs.ISAAC.2017.11,
  author =	{Becker, Aaron T. and Fekete, S\'{a}ndor P. and Keldenich, Phillip and Krupke, Dominik and Rieck, Christian and Scheffer, Christian and Schmidt, Arne},
  title =	{{Tilt Assembly: Algorithms for Micro-Factories that Build Objects with Uniform External Forces}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{11:1--11:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.11},
  URN =		{urn:nbn:de:0030-drops-82214},
  doi =		{10.4230/LIPIcs.ISAAC.2017.11},
  annote =	{Keywords: Programmable matter, micro-factories, tile assembly, tilt, approximation, hardness}
}
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