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Documents authored by Dhar, Anubhav

Document
DOI: 10.4230/LIPIcs.OPODIS.2024.27
Local Problems in Trees Across a Wide Range of Distributed Models

Authors: Anubhav Dhar, Eli Kujawa, Henrik Lievonen, Augusto Modanese, Mikail Muftuoglu, Jan Studený, and Jukka Suomela

Published in: LIPIcs, Volume 324, 28th International Conference on Principles of Distributed Systems (OPODIS 2024)

Abstract
The randomized online-LOCAL model captures a number of models of computing; it is at least as strong as all of these models: - the classical LOCAL model of distributed graph algorithms, - the quantum version of the LOCAL model, - finitely dependent distributions [e.g. Holroyd 2016], - any model that does not violate physical causality [Gavoille, Kosowski, Markiewicz, DISC 2009], - the SLOCAL model [Ghaffari, Kuhn, Maus, STOC 2017], and - the dynamic-LOCAL and online-LOCAL models [Akbari et al., ICALP 2023]. In general, the online-LOCAL model can be much stronger than the LOCAL model. For example, there are locally checkable labeling problems (LCLs) that can be solved with logarithmic locality in the online-LOCAL model but that require polynomial locality in the LOCAL model. However, in this work we show that in trees, many classes of LCL problems have the same locality in deterministic LOCAL and randomized online-LOCAL (and as a corollary across all the above-mentioned models). In particular, these classes of problems do not admit any distributed quantum advantage. We present a near-complete classification for the case of rooted regular trees. We also fully classify the super-logarithmic region in unrooted regular trees. Finally, we show that in general trees (rooted or unrooted, possibly irregular, possibly with input labels) problems that are global in deterministic LOCAL remain global also in the randomized online-LOCAL model.
Cite as

Anubhav Dhar, Eli Kujawa, Henrik Lievonen, Augusto Modanese, Mikail Muftuoglu, Jan Studený, and Jukka Suomela. Local Problems in Trees Across a Wide Range of Distributed Models. In 28th International Conference on Principles of Distributed Systems (OPODIS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 324, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dhar_et_al:LIPIcs.OPODIS.2024.27,
  author =	{Dhar, Anubhav and Kujawa, Eli and Lievonen, Henrik and Modanese, Augusto and Muftuoglu, Mikail and Studen\'{y}, Jan and Suomela, Jukka},
  title =	{{Local Problems in Trees Across a Wide Range of Distributed Models}},
  booktitle =	{28th International Conference on Principles of Distributed Systems (OPODIS 2024)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-360-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{324},
  editor =	{Bonomi, Silvia and Galletta, Letterio and Rivi\`{e}re, Etienne and Schiavoni, Valerio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2024.27},
  URN =		{urn:nbn:de:0030-drops-225633},
  doi =		{10.4230/LIPIcs.OPODIS.2024.27},
  annote =	{Keywords: Distributed algorithms, quantum-LOCAL model, randomized online-LOCAL model, locally checkable labeling problems, trees}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.25
Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets

Authors: Anubhav Dhar, Soumita Hait, and Sudeshna Kolay

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
The Euclidean Steiner Minimal Tree problem takes as input a set P of points in the Euclidean plane and finds the minimum length network interconnecting all the points of P. In this paper, in continuation to the works of [Du et al., 1987] and [Weng and Booth, 1995], we study Euclidean Steiner Minimal Tree when P is formed by the vertices of a pair of regular, concentric and parallel n-gons. We restrict our attention to the cases where the two polygons are not very close to each other. In such cases, we show that Euclidean Steiner Minimal Tree is polynomial-time solvable, and we describe an explicit structure of a Euclidean Steiner minimal tree for P. We also consider point sets P of size n where the number of input points not on the convex hull of P is f(n) ≤ n. We give an exact algorithm with running time 2^𝒪(f(n) log n) for such input point sets P. Note that when f(n) = 𝒪(n/(log n)), our algorithm runs in single-exponential time, and when f(n) = o(n) the running time is 2^o(n log n) which is better than the known algorithm in [Hwang et al., 1992]. We know that no FPTAS exists for Euclidean Steiner Minimal Tree unless P = NP [Garey et al., 1977]. On the other hand FPTASes exist for Euclidean Steiner Minimal Tree on convex point sets [Scott Provan, 1988]. In this paper, we show that if the number of input points in P not belonging to the convex hull of P is 𝒪(log n), then an FPTAS exists for Euclidean Steiner Minimal Tree. In contrast, we show that for any ε ∈ (0,1], when there are Ω(n^ε) points not belonging to the convex hull of the input set, then no FPTAS can exist for Euclidean Steiner Minimal Tree unless P = NP.
Cite as

Anubhav Dhar, Soumita Hait, and Sudeshna Kolay. Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dhar_et_al:LIPIcs.ISAAC.2023.25,
  author =	{Dhar, Anubhav and Hait, Soumita and Kolay, Sudeshna},
  title =	{{Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.25},
  URN =		{urn:nbn:de:0030-drops-193273},
  doi =		{10.4230/LIPIcs.ISAAC.2023.25},
  annote =	{Keywords: Steiner minimal tree, Euclidean Geometry, Almost Convex point sets, FPTAS, strong NP-completeness}
}
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