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Documents authored by Knollmann, Till

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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.12
Mincut Sensitivity Data Structures for the Insertion of an Edge

Authors: Surender Baswana, Shiv Gupta, and Till Knollmann

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Let G = (V,E) be an undirected graph on n vertices with non-negative capacities on its edges. The mincut sensitivity problem for the insertion of an edge is defined as follows. Build a compact data structure for G and a given set S ⊆ V of vertices that, on receiving any edge (x,y) ∈ S×S of positive capacity as query input, can efficiently report the set of all pairs from S× S whose mincut value increases upon insertion of the edge (x,y) to G. The only result that exists for this problem is for a single pair of vertices (Picard and Queyranne, Mathematical Programming Study, 13 (1980), 8-16). We present the following results for the single source and the all-pairs versions of this problem. 1) Single source: Given any designated source vertex s, there exists a data structure of size 𝒪(|S|) that can output all those vertices from S whose mincut value to s increases upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(|S|). 2) All-pairs: There exists an 𝒪(|S|²) size data structure that can output all those pairs of vertices from S× S whose mincut value gets increased upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(k), where k is the number of pairs of vertices whose mincut increases. For both these versions, we also address the problem of reporting the values of the mincuts upon insertion of any given edge. To derive our results, we use interesting insights into the nearest and the farthest mincuts for a pair of vertices. In addition, a crucial result, that we establish and use in our data structures, is that there exists a directed acyclic graph of 𝒪(n) size that compactly stores the farthest mincuts from all vertices of V to a designated vertex s in the graph. We believe that this result is of independent interest, especially, because it also complements a previously existing result by Hariharan et al. (STOC 2007) that the nearest mincuts from all vertices of V to s is a laminar family, and hence, can be stored compactly in a tree of 𝒪(n) size.
Cite as

Surender Baswana, Shiv Gupta, and Till Knollmann. Mincut Sensitivity Data Structures for the Insertion of an Edge. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{baswana_et_al:LIPIcs.ESA.2020.12,
  author =	{Baswana, Surender and Gupta, Shiv and Knollmann, Till},
  title =	{{Mincut Sensitivity Data Structures for the Insertion of an Edge}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{12:1--12:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.12},
  URN =		{urn:nbn:de:0030-drops-128781},
  doi =		{10.4230/LIPIcs.ESA.2020.12},
  annote =	{Keywords: Mincut, Sensitivity, Data Structure}
}
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