2 Search Results for "Vorel, Vojtěch"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.44
Tilt Automata: Gathering Particles with Uniform External Control

Authors: Sándor P. Fekete, Jonas Friemel, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, and Christian Scheffer

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Motivated by targeted drug delivery, we investigate the gathering of particles in the full tilt model of externally controlled motion planning: A set of particles is located at the tiles of a polyomino with all particles reacting uniformly to an external force by moving as far as possible in one of the four axis-parallel directions until they hit the boundary. The goal is to choose a sequence of directions that moves all particles to a common position. Our results include a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences and for determining whether the particles in a partially filled polyomino can be gathered. We pay special attention to the impact of restricted geometry, particularly polyominoes without holes. As a corollary, we make progress on an open question from [Balanza-Martinez et al., SODA 2020] by showing that deciding whether a given position can be occupied remains NP-hard in polyominoes without holes. Our results build on a connection we establish between tilt models and the theory of synchronizing automata.
Cite as

Sándor P. Fekete, Jonas Friemel, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, and Christian Scheffer. Tilt Automata: Gathering Particles with Uniform External Control. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.44,
  author =	{Fekete, S\'{a}ndor P. and Friemel, Jonas and Kramer, Peter and Reinhardt, Jan-Marc and Rieck, Christian and Scheffer, Christian},
  title =	{{Tilt Automata: Gathering Particles with Uniform External Control}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{44:1--44:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.44},
  URN =		{urn:nbn:de:0030-drops-258508},
  doi =		{10.4230/LIPIcs.SoCG.2026.44},
  annote =	{Keywords: Uniform control, gathering, full tilt, polyominoes, synchronizing automata}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.46
Lower Bounds on Avoiding Thresholds

Authors: Robert Ferens, Marek Szykuła, and Vojtěch Vorel

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
For a DFA, a word avoids a subset of states, if after reading that word the automaton cannot be in any state from the subset regardless of its initial state. A subset that admits an avoiding word is avoidable. The k-avoiding threshold of a DFA is the smallest number such that every avoidable subset of size k can be avoided with a word no longer than that number. We study the problem of determining the maximum possible k-avoiding thresholds. For every fixed k ≥ 1, we show a general construction of strongly connected DFAs with n states and the k-avoiding threshold in Θ(n^k). This meets the known upper bound for k ≥ 3. For k = 1 and k = 2, the known upper bounds are respectively in 𝒪(n²) and in 𝒪(n³). For k = 1, we show that 2n-3 is attainable for every number of states n in the class of strongly connected synchronizing binary DFAs, which is supposed to be the best possible in the class of all DFAs for n ≥ 8. For k = 2, we show that the conjectured solution for k = 1 (an upper bound in 𝒪(n)) also implies a tight upper bound in 𝒪(n²) on 2-avoiding threshold. Finally, we discuss the possibility of using k-avoiding thresholds of synchronizing automata to improve upper bounds on the length of the shortest reset words.
Cite as

Robert Ferens, Marek Szykuła, and Vojtěch Vorel. Lower Bounds on Avoiding Thresholds. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ferens_et_al:LIPIcs.MFCS.2021.46,
  author =	{Ferens, Robert and Szyku{\l}a, Marek and Vorel, Vojt\v{e}ch},
  title =	{{Lower Bounds on Avoiding Thresholds}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.46},
  URN =		{urn:nbn:de:0030-drops-144869},
  doi =		{10.4230/LIPIcs.MFCS.2021.46},
  annote =	{Keywords: avoiding word, \v{C}ern\'{y} conjecture, rank conjecture, reset threshold, reset word, synchronizing automaton, synchronizing word}
}
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