Improved distributed Δ-coloring

We present a randomized distributed algorithm that computes a Δ -coloring in any non-complete graph with maximum degree Δ ≥ 4 in O ( log Δ ) + 2 O ( log log n ) rounds, as well as a randomized algorithm that computes a Δ -coloring in O ( ( log log n ) 2 ) rounds when Δ ∈ [ 3 , O ( 1 ) ] . Both these...

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Vydané v:	Distributed computing Ročník 34; číslo 4; s. 239 - 258
Hlavní autori:	Ghaffari, Mohsen, Hirvonen, Juho, Kuhn, Fabian, Maus, Yannic
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2021 Springer Nature B.V
Predmet:	Algorithms Coloring Computer Communication Networks Computer Hardware Computer Science Computer Systems Organization and Communication Networks Lower bounds Software Engineering/Programming and Operating Systems Theory of Computation
ISSN:	0178-2770, 1432-0452
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Abstract	We present a randomized distributed algorithm that computes a Δ -coloring in any non-complete graph with maximum degree Δ ≥ 4 in O ( log Δ ) + 2 O ( log log n ) rounds, as well as a randomized algorithm that computes a Δ -coloring in O ( ( log log n ) 2 ) rounds when Δ ∈ [ 3 , O ( 1 ) ] . Both these algorithms improve on an O ( log 3 n / log Δ ) -round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an Ω ( log log n ) round lower bound of Brandt et al. (STOC’16).
AbstractList	We present a randomized distributed algorithm that computes a Δ -coloring in any non-complete graph with maximum degree Δ ≥ 4 in O ( log Δ ) + 2 O ( log log n ) rounds, as well as a randomized algorithm that computes a Δ -coloring in O ( ( log log n ) 2 ) rounds when Δ ∈ [ 3 , O ( 1 ) ] . Both these algorithms improve on an O ( log 3 n / log Δ ) -round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an Ω ( log log n ) round lower bound of Brandt et al. (STOC’16). We present a randomized distributed algorithm that computes a Δ -coloring in any non-complete graph with maximum degree Δ ≥ 4 in O ( log Δ ) + 2 O ( log log n ) rounds, as well as a randomized algorithm that computes a Δ -coloring in O ( ( log log n ) 2 ) rounds when Δ ∈ [ 3 , O ( 1 ) ] . Both these algorithms improve on an O ( log 3 n / log Δ ) -round algorithm of Panconesi and Srinivasan (STOC'93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an Ω ( log log n ) round lower bound of Brandt et al. (STOC'16).We present a randomized distributed algorithm that computes a Δ -coloring in any non-complete graph with maximum degree Δ ≥ 4 in O ( log Δ ) + 2 O ( log log n ) rounds, as well as a randomized algorithm that computes a Δ -coloring in O ( ( log log n ) 2 ) rounds when Δ ∈ [ 3 , O ( 1 ) ] . Both these algorithms improve on an O ( log 3 n / log Δ ) -round algorithm of Panconesi and Srinivasan (STOC'93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an Ω ( log log n ) round lower bound of Brandt et al. (STOC'16). We present a randomized distributed algorithm that computes a Δ-coloring in any non-complete graph with maximum degree Δ≥4 in O(logΔ)+2O(loglogn) rounds, as well as a randomized algorithm that computes a Δ-coloring in O((loglogn)2) rounds when Δ∈[3,O(1)]. Both these algorithms improve on an O(log3n/logΔ)-round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an Ω(loglogn) round lower bound of Brandt et al. (STOC’16). We present a randomized distributed algorithm that computes a $$\Delta $$ Δ -coloring in any non-complete graph with maximum degree $$\Delta \ge 4$$ Δ ≥ 4 in $$O(\log \Delta ) + 2^{O(\sqrt{\log \log n})}$$ O ( log Δ ) + 2 O ( log log n ) rounds, as well as a randomized algorithm that computes a $$\Delta $$ Δ -coloring in $$O((\log \log n)^2)$$ O ( ( log log n ) 2 ) rounds when $$\Delta \in [3, O(1)]$$ Δ ∈ [ 3 , O ( 1 ) ] . Both these algorithms improve on an $$O(\log ^3 n / \log \Delta )$$ O ( log 3 n / log Δ ) -round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an $$\Omega (\log \log n)$$ Ω ( log log n ) round lower bound of Brandt et al. (STOC’16).
Author	Hirvonen, Juho Kuhn, Fabian Maus, Yannic Ghaffari, Mohsen
Author_xml	– sequence: 1 givenname: Mohsen surname: Ghaffari fullname: Ghaffari, Mohsen organization: ETH Zurich – sequence: 2 givenname: Juho surname: Hirvonen fullname: Hirvonen, Juho organization: Aalto University – sequence: 3 givenname: Fabian orcidid: 0000-0002-1025-5037 surname: Kuhn fullname: Kuhn, Fabian email: kuhn@cs.uni-freiburg.de organization: University of Freiburg – sequence: 4 givenname: Yannic surname: Maus fullname: Maus, Yannic organization: Technion
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References_xml	– reference: VizingVVextex coloring with given colorsMetody Diskretn. Anal.197629310 – reference: BarenboimLElkinMPettieSSchneiderJThe locality of distributed symmetry breakingJ. ACM20166332012045354953010.1145/2903137 – reference: Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM (2000) – reference: Erdős, P., Rubin, A., Taylor, H.: Choosability in graphs. In: Proceedings of West Coast Conference on Combinatorics, Graph Theory and Computing, vol. 26, pp. 125–157. Congressus Numerantium (1979) – reference: Barenboim, L., Elkin, M., Pettie, S., Schneider, J.: The locality of distributed symmetry breaking. In: Proceedings of 53rd Symposium on Foundations of Computer Science (FOCS 2012), pp. 321–330 (2012) – reference: Aboulker, P., Bonamy, M., Bousquet, N., Esperet, L.: Distributed coloring in sparse graphs with fewer colors. 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Snippet	We present a randomized distributed algorithm that computes a Δ -coloring in any non-complete graph with maximum degree Δ ≥ 4 in O ( log Δ ) + 2 O ( log log n... We present a randomized distributed algorithm that computes a $$\Delta $$ Δ -coloring in any non-complete graph with maximum degree $$\Delta \ge 4$$ Δ ≥ 4 in... We present a randomized distributed algorithm that computes a Δ-coloring in any non-complete graph with maximum degree Δ≥4 in O(logΔ)+2O(loglogn) rounds, as... We present a randomized distributed algorithm that computes a Δ -coloring in any non-complete graph with maximum degree Δ ≥ 4 in O ( log Δ ) + 2 O ( log log n...
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SubjectTerms	Algorithms Coloring Computer Communication Networks Computer Hardware Computer Science Computer Systems Organization and Communication Networks Lower bounds Software Engineering/Programming and Operating Systems Theory of Computation
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