7 Search Results for "Rozhoň, Václav"

Document
DOI: 10.4230/LIPIcs.SEA.2024.7
Local Search k-means++ with Foresight

Authors: Theo Conrads, Lukas Drexler, Joshua Könen, Daniel R. Schmidt, and Melanie Schmidt

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
Since its introduction in 1957, Lloyd’s algorithm for k-means clustering has been extensively studied and has undergone several improvements. While in its original form it does not guarantee any approximation factor at all, Arthur and Vassilvitskii (SODA 2007) proposed k-means++ which enhances Lloyd’s algorithm by a seeding method which guarantees a 𝒪(log k)-approximation in expectation. More recently, Lattanzi and Sohler (ICML 2019) proposed LS++ which further improves the solution quality of k-means++ by local search techniques to obtain a 𝒪(1)-approximation. On the practical side, the greedy variant of k-means++ is often used although its worst-case behaviour is provably worse than for the standard k-means++ variant. We investigate how to improve LS++ further in practice. We study two options for improving the practical performance: (a) Combining LS++ with greedy k-means++ instead of k-means++, and (b) Improving LS++ by better entangling it with Lloyd’s algorithm. Option (a) worsens the theoretical guarantees of k-means++ but improves the practical quality also in combination with LS++ as we confirm in our experiments. Option (b) is our new algorithm, Foresight LS++. We experimentally show that FLS++ improves upon the solution quality of LS++. It retains its asymptotic runtime and its worst-case approximation bounds.
Cite as

Theo Conrads, Lukas Drexler, Joshua Könen, Daniel R. Schmidt, and Melanie Schmidt. Local Search k-means++ with Foresight. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{conrads_et_al:LIPIcs.SEA.2024.7,
  author =	{Conrads, Theo and Drexler, Lukas and K\"{o}nen, Joshua and Schmidt, Daniel R. and Schmidt, Melanie},
  title =	{{Local Search k-means++ with Foresight}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.7},
  URN =		{urn:nbn:de:0030-drops-203727},
  doi =		{10.4230/LIPIcs.SEA.2024.7},
  annote =	{Keywords: k-means clustering, kmeans++, greedy, local search}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.42
Path-Reporting Distance Oracles with Logarithmic Stretch and Linear Size

Authors: Shiri Chechik and Tianyi Zhang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Given an undirected graph G = (V, E, 𝐰) on n vertices with positive edge weights, a distance oracle is a space-efficient data structure that answers pairwise distance queries in fast runtime. The quality of a distance oracle is measured by three parameters: space, query time, and stretch. In a landmark paper by [Thorup and Zwick, 2001], they showed that for any integer parameter k ≥ 1, there exists a distance oracle with size O(kn^{1+1/k}), O(k) query time, and (2k-1)-stretch error on the approximate distances. After that, there has been a line of subsequent improvements which culminated in the optimal trade-off of O(n^{1+1/k}) space, O(1) query time, and (2k-1)-stretch [Chechik, 2015]. However, these line of constructions did not require that the distance oracle is capable of printing an actual path besides an approximate distance estimate, and there has been a performance gap between path-reporting distance oracles and ones that are not path-reporting. It is known that the earliest construction by [Thorup and Zwick, 2001] is path-reporting, but the parameters are worse by a factor of k. In a later construction by [Wulff-Nilsen, 2013], the query time was improved from O(k) to O(log k). Better trade-offs were discovered in [Elkin and Pettie, 2015] where the authors broke the O(kn^{1+1/k}) space barrier and achieved O(n^{1+1/k}log k) space with O(log k) query time, but their stretch was blown up to a polynomial O(k^{log_{4/3}7}); they also gave an alternative choice of O(n^{1+1/k}) space which is optimal, and O(k)-stretch which is also optimal up to a constant factor, but their query time rose exponentially to O(n^ε). In a recent work [Elkin and Shabat, 2023], the authors obtained significant improvements of O(n^{1+1/k}log k) space, O(k)-stretch, and O(log log k) query time, or O(n^{1+1/k}) space, O(klog k)-stretch, and O(log log k) query time. All the above constructions of path-reporting distance oracles share a common barrier; that is, they could not achieve optimal space O(n^{1+1/k}) and stretch O(k) simultaneously within logarithmic query time; for example, in the natural regime where k = ⌈log n⌉, previous distance oracles had to pay an extra factor of log log n either in the space or stretch. As our result, we bypass this barrier by a new construction of path-reporting distance oracles with O(n^{1+1/k}) space and O(k)-stretch and O(log log k) query time.
Cite as

Shiri Chechik and Tianyi Zhang. Path-Reporting Distance Oracles with Logarithmic Stretch and Linear Size. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chechik_et_al:LIPIcs.ICALP.2024.42,
  author =	{Chechik, Shiri and Zhang, Tianyi},
  title =	{{Path-Reporting Distance Oracles with Logarithmic Stretch and Linear Size}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{42:1--42:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.42},
  URN =		{urn:nbn:de:0030-drops-201859},
  doi =		{10.4230/LIPIcs.ICALP.2024.42},
  annote =	{Keywords: graph algorithms, shortest paths, distance oracles}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.65
Optimal Electrical Oblivious Routing on Expanders

Authors: Cella Florescu, Rasmus Kyng, Maximilian Probst Gutenberg, and Sushant Sachdeva

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
In this paper, we investigate the question of whether the electrical flow routing is a good oblivious routing scheme on an m-edge graph G = (V, E) that is a Φ-expander, i.e. where |∂ S| ≥ Φ ⋅ vol(S) for every S ⊆ V, vol(S) ≤ vol(V)/2. Beyond its simplicity and structural importance, this question is well-motivated by the current state-of-the-art of fast algorithms for 𝓁_∞ oblivious routings that reduce to the expander-case which is in turn solved by electrical flow routing. Our main result proves that the electrical routing is an O(Φ^{-1} log m)-competitive oblivious routing in the 𝓁₁- and 𝓁_∞-norms. We further observe that the oblivious routing is O(log² m)-competitive in the 𝓁₂-norm and, in fact, O(log m)-competitive if 𝓁₂-localization is O(log m) which is widely believed. Using these three upper bounds, we can smoothly interpolate to obtain upper bounds for every p ∈ [2, ∞] and q given by 1/p + 1/q = 1. Assuming 𝓁₂-localization in O(log m), we obtain that in 𝓁_p and 𝓁_q, the electrical oblivious routing is O(Φ^{-(1-2/p)}log m) competitive. Using the currently known result for 𝓁₂-localization, this ratio deteriorates by at most a sublogarithmic factor for every p, q ≠ 2. We complement our upper bounds with lower bounds that show that the electrical routing for any such p and q is Ω(Φ^{-(1-2/p)} log m)-competitive. This renders our results in 𝓁₁ and 𝓁_∞ unconditionally tight up to constants, and the result in any 𝓁_p- and 𝓁_q-norm to be tight in case of 𝓁₂-localization in O(log m).
Cite as

Cella Florescu, Rasmus Kyng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Optimal Electrical Oblivious Routing on Expanders. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 65:1-65:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{florescu_et_al:LIPIcs.ICALP.2024.65,
  author =	{Florescu, Cella and Kyng, Rasmus and Gutenberg, Maximilian Probst and Sachdeva, Sushant},
  title =	{{Optimal Electrical Oblivious Routing on Expanders}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{65:1--65:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.65},
  URN =		{urn:nbn:de:0030-drops-202083},
  doi =		{10.4230/LIPIcs.ICALP.2024.65},
  annote =	{Keywords: Expanders, Oblivious routing for 𝓁\underlinep, Electrical flow routing}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.55
Noisy k-Means++ Revisited

Authors: Christoph Grunau, Ahmet Alper Özüdoğru, and Václav Rozhoň

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
The k-means++ algorithm by Arthur and Vassilvitskii [SODA 2007] is a classical and time-tested algorithm for the k-means problem. While being very practical, the algorithm also has good theoretical guarantees: its solution is O(log k)-approximate, in expectation. In a recent work, Bhattacharya, Eube, Roglin, and Schmidt [ESA 2020] considered the following question: does the algorithm retain its guarantees if we allow for a slight adversarial noise in the sampling probability distributions used by the algorithm? This is motivated e.g. by the fact that computations with real numbers in k-means++ implementations are inexact. Surprisingly, the analysis under this scenario gets substantially more difficult and the authors were able to prove only a weaker approximation guarantee of O(log² k). In this paper, we close the gap by providing a tight, O(log k)-approximate guarantee for the k-means++ algorithm with noise.
Cite as

Christoph Grunau, Ahmet Alper Özüdoğru, and Václav Rozhoň. Noisy k-Means++ Revisited. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 55:1-55:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{grunau_et_al:LIPIcs.ESA.2023.55,
  author =	{Grunau, Christoph and \"{O}z\"{u}do\u{g}ru, Ahmet Alper and Rozho\v{n}, V\'{a}clav},
  title =	{{Noisy k-Means++ Revisited}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{55:1--55:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.55},
  URN =		{urn:nbn:de:0030-drops-187080},
  doi =		{10.4230/LIPIcs.ESA.2023.55},
  annote =	{Keywords: clustering, k-means, k-means++, adversarial noise}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.107
Approximate Triangle Counting via Sampling and Fast Matrix Multiplication

Authors: Jakub Tětek

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
There is a simple O(n³/{ε²T}) time algorithm for 1±ε-approximate triangle counting where T is the number of triangles in the graph and n the number of vertices. At the same time, one may count triangles exactly using fast matrix multiplication in time Õ(n^ω). Is it possible to get a negative dependency on the number of triangles T while retaining the state-of-the-art n^ω dependency on n? We answer this question positively by providing an algorithm which runs in time O({n^ω}/T^{ω-2})⋅poly(n^o(1)/ε). This is optimal in the sense that as long as the exponent of T is independent of n, T, it cannot be improved while retaining the dependency on n. Our algorithm improves upon the state of the art when T ≫ 1 and T ≪ n. We also consider the problem of approximate triangle counting in sparse graphs, parameterized by the number of edges m. The best known algorithm runs in time Õ_ε(m^{3/2}/T) [Eden et al., SIAM Journal on Computing, 2017]. An algorithm by Alon et al. [JACM, 1995] for exact triangle counting that runs in time Õ(m^{2ω/(ω + 1)}). We again get an algorithm whose complexity has a state-of-the-art dependency on m while having negative dependency on T. Specifically, our algorithm runs in time O({m^{2ω/(ω+1)}}/{T^{2(ω-1)/(ω+1)}}) ⋅ poly(n^o(1)/ε). This is again optimal in the sense that no better constant exponent of T is possible without worsening the dependency on m. This algorithm improves upon the state of the art when T ≫ 1 and T ≪ √m. In both cases, algorithms with time complexity matching query complexity lower bounds were known on some range of parameters. While those algorithms have optimal query complexity for the whole range of T, the time complexity departs from the query complexity and is no longer optimal (as we show) for T ≪ n and T ≪ √m, respectively. We focus on the time complexity in this range of T. To the best of our knowledge, this is the first paper considering the discrepancy between query and time complexity in graph parameter estimation.
Cite as

Jakub Tětek. Approximate Triangle Counting via Sampling and Fast Matrix Multiplication. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 107:1-107:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{tetek:LIPIcs.ICALP.2022.107,
  author =	{T\v{e}tek, Jakub},
  title =	{{Approximate Triangle Counting via Sampling and Fast Matrix Multiplication}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{107:1--107:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.107},
  URN =		{urn:nbn:de:0030-drops-164485},
  doi =		{10.4230/LIPIcs.ICALP.2022.107},
  annote =	{Keywords: Approximate triangle counting, Fast matrix multiplication, Sampling}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2021.19
Distributed Recoloring of Interval and Chordal Graphs

Authors: Nicolas Bousquet, Laurent Feuilloley, Marc Heinrich, and Mikaël Rabie

Published in: LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

Abstract
One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions. First, the problem of recoloring, that is computing an efficient transformation between two given colorings (instead of computing a new coloring), has been considered, both to model radio network updates, and as a useful subroutine for coloring. Second, as it appears that general graphs and bounded-degree graphs do not model real networks very well (with, respectively, pathological worst-case topologies and too strong assumptions), coloring has been studied in more specific graph classes. In this paper, we study the intersection of these two directions: distributed recoloring in two relevant graph classes, interval and chordal graphs. More formally, the question of recoloring a graph is as follows: we are given a network, an input coloring α and a target coloring β, and we want to find a schedule of colorings to reach β starting from α. In a distributed setting, the schedule needs to be found within the LOCAL model, where nodes communicate with their direct neighbors synchronously. The question we want to answer is: how many rounds of communication {are} needed to produce a schedule, and what is the length of this schedule? In the case of interval and chordal graphs, we prove that, if we have less than 2ω colors, ω being the size of the largest clique, extra colors will be needed in the intermediate colorings. For interval graphs, we produce a schedule after O(poly(Δ)log*n) rounds of communication, and for chordal graphs, we need O(ω²Δ²log n) rounds to get one. Our techniques also improve classic coloring algorithms. Namely, we get ω+1-colorings of interval graphs in O(ωlog*n) rounds and of chordal graphs in O(ωlog n) rounds, which improves on previous known algorithms that use ω+2 colors for the same running times.
Cite as

Nicolas Bousquet, Laurent Feuilloley, Marc Heinrich, and Mikaël Rabie. Distributed Recoloring of Interval and Chordal Graphs. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bousquet_et_al:LIPIcs.OPODIS.2021.19,
  author =	{Bousquet, Nicolas and Feuilloley, Laurent and Heinrich, Marc and Rabie, Mika\"{e}l},
  title =	{{Distributed Recoloring of Interval and Chordal Graphs}},
  booktitle =	{25th International Conference on Principles of Distributed Systems (OPODIS 2021)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-219-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{217},
  editor =	{Bramas, Quentin and Gramoli, Vincent and Milani, Alessia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.19},
  URN =		{urn:nbn:de:0030-drops-157941},
  doi =		{10.4230/LIPIcs.OPODIS.2021.19},
  annote =	{Keywords: Distributed coloring, distributed recoloring, interval graphs, chordal graphs, intersection graphs}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.29
Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

Authors: Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, and Zoltán Vidnyánszky

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We study connections between three different fields: distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics. We focus on two central questions: Can we apply techniques from one of the areas to obtain results in another? Can we show that complexity classes coming from different areas contain precisely the same problems? We give an affirmative answer to both questions in the context of local problems on regular trees: 1) We extend the Borel determinacy technique of Marks [Marks - J. Am. Math. Soc. 2016] coming from descriptive combinatorics and adapt it to the area of distributed computing, thereby obtaining a more generally applicable lower bound technique in descriptive combinatorics and an entirely new lower bound technique for distributed algorithms. Using our new technique, we prove deterministic distributed Ω(log n)-round lower bounds for problems from a natural class of homomorphism problems. Interestingly, these lower bounds seem beyond the current reach of the powerful round elimination technique [Brandt - PODC 2019] responsible for all substantial locality lower bounds of the last years. Our key technical ingredient is a novel ID graph technique that we expect to be of independent interest; in fact, it has already played an important role in a new lower bound for the Lovász local lemma in the Local Computation Algorithms model from sequential computing [Brandt, Grunau, Rozhoň - PODC 2021]. 2) We prove that a local problem admits a Baire measurable coloring if and only if it admits a local algorithm with local complexity O(log n), extending the classification of Baire measurable colorings of Bernshteyn [Bernshteyn - personal communication]. A key ingredient of the proof is a new and simple characterization of local problems that can be solved in O(log n) rounds. We complement this result by showing separations between complexity classes from distributed computing, finitary factors, and descriptive combinatorics. Most notably, the class of problems that allow a distributed algorithm with sublogarithmic randomized local complexity is incomparable with the class of problems with a Borel solution. We hope that our treatment will help to view all three perspectives as part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn - arXiv 2004.04905].
Cite as

Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, and Zoltán Vidnyánszky. Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 29:1-29:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brandt_et_al:LIPIcs.ITCS.2022.29,
  author =	{Brandt, Sebastian and Chang, Yi-Jun and Greb{\'\i}k, Jan and Grunau, Christoph and Rozho\v{n}, V\'{a}clav and Vidny\'{a}nszky, Zolt\'{a}n},
  title =	{{Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{29:1--29:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.29},
  URN =		{urn:nbn:de:0030-drops-156259},
  doi =		{10.4230/LIPIcs.ITCS.2022.29},
  annote =	{Keywords: Distributed Algorithms, Descriptive Combinatorics}
}
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