Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic and Faulty Networks

Barenboim, Leonid

doi:10.1145/2767386.2767410

Cited by 50 publications

(94 citation statements)

References 33 publications

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“…The currently fastest algorithm is by Fraigniaud, Heinrich and Kosowski [10] and uses O( √ ∆ log 2.5 ∆ + log * n). This result improved on Barenboim's algorithm [2], which uses O(∆ 3 4 log ∆ + log * n) rounds and was the first deterministic (∆ + 1)-coloring algorithm which achieved a sublinear in ∆ number of rounds. Faster (∆ + 1)-colorings can be achieved on special graph classes.…”

Section: Introductionmentioning

confidence: 76%

Improved Distributed Algorithms for Coloring Interval Graphs with Application to Multicoloring Trees

Halldórsson

Konrad

2017

Structural Information and Communication Complexity

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We give a distributed (1+)-approximation algorithm for the minimum vertex coloring problem on interval graphs, which runs in the LOCAL model and operates in O(1 log * n) rounds. If nodes are aware of their interval representations, then the algorithm can be adapted to the CON GEST model using the same number of rounds. Prior to this work, only constant factor approximations using O(log * n) rounds were known [12]. Linial's ring coloring lower bound implies that the dependency on log * n cannot be improved. We further prove that the dependency on 1 is also optimal. To obtain our CON GEST model algorithm, we develop a color rotation technique that may be of independent interest. We demonstrate that color rotations can also be applied to obtain a (1 +)-approximate multicoloring of directed trees in O(1 log * n) rounds.

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Section: Introductionmentioning

confidence: 76%

Improved Distributed Algorithms for Coloring Interval Graphs with Application to Multicoloring Trees

Halldórsson

Konrad

2017

Structural Information and Communication Complexity

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“…Using this coloring we run the algorithm from [4,7] on G 2 and simulate one round of it in ∆ rounds of communication on G. That way, we obtain a coloring of…”

Section: Summarizing the Ideas For Theorem 12mentioning

confidence: 99%

“…From coloring to d2-coloring. While most existing distributed coloring algorithms were primarily developed for the LOCAL model, several of them directly also work in the CONGEST model (e.g., the ones in [1,25,24,19,23,5,8,4,7,22]). There is also some recent work, which explicitly studies distributed coloring in the CONGEST model.…”

Section: Introductionmentioning

confidence: 99%

Distance-2 Coloring in the CONGEST Model

Halldórsson¹,

Kühn²,

Maus³

2020

Preprint

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We give efficient randomized and deterministic distributed algorithms for computing a distance-2 vertex coloring of a graph G in the CONGEST model. In particular, if ∆ is the maximum degree of G, we show that there is a randomized CONGEST model algorithm to compute a distance-2 coloring of G with ∆ 2 + 1 colors in O(log ∆ • log n) rounds. Further if the number of colors is slightly increased to (1 + ǫ)∆ 2 for some ǫ > 1/ polylog n, we show that it is even possible to compute a distance-2 coloring deterministically in polylog n time in the CONGEST model. Finally, we give a O(∆ 2 + log * n)-round deterministic CONGEST algorithm to compute distance-2 coloring with ∆ 2 + 1 colors.

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“…Corollary 1.2 is a drastic improvement over the state of the art even for the standard (∆ + 1)-coloring problem: Surprisingly until the beginning of 2019 the best deterministic CONGEST algorithm for the (∆+1)coloring problem was the O(∆ 3/4 + log * n) algorithm by Barenboim [Bar15,BEG18]. Even though the objective of [Bar15] was to optimize the runtime mainly as a function of the maximum degree ∆, the paper also provided the fastest known algorithm if the runtime is solely expressed as a function of the number of nodes, i.e., it provided an O(n 3/4 ) round CONGEST algorithm. Very recently, the runtime for (degree +1)-list coloring was improved to 2 O( [Kuh20].…”

Section: Our Contributions In Congestmentioning

confidence: 99%

Efficient Deterministic Distributed Coloring with Small Bandwidth

Bamberger¹,

Kühn²,

Maus³

2019

Preprint

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We show that the (degree + 1)-list coloring problem can be solved deterministically in O(D • log n • log 2 ∆) rounds in the CONGEST model, where D is the diameter of the graph, n the number of nodes, and ∆ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhoň and Ghaffari [STOC 2020], this implies the first efficient (i.e., poly log n-time) deterministic CONGEST algorithm for the (∆+1)coloring and the (degree+1)-list coloring problem. Previously the best known algorithm required 2 O( √ log n) rounds and was not based on network decompositions. Our techniques also lead to deterministic (degree + 1)-list coloring algorithms for the congested clique and the massively parallel computation (MPC) model. For the congested clique, we obtain an algorithm with time complexity O(log ∆•log log ∆), for the MPC model, we obtain algorithms with round complexity O(log 2 ∆) for the linear-memory regime and O(log 2 ∆+log n) for the sublinear memory regime.Distributed Graph Coloring. Computing a coloring with the optimal number of colors χ(G) was one of the first problems known to be NP-complete [Kar72]. In the distributed setting, one therefore aims for a more relaxed goal and the usual objective is to color a given graph G with ∆ + 1 colors, where ∆ is the maximum degree of G [BE13]. Note that any graph admits such a coloring and it can be computed by a simple sequential greedy algorithm. Despite the simplicity of the sequential algorithm, the question of determining the distributed complexity of computing a (∆ + 1)-coloring has been an extremely challenging question. In particular, while O(log n)-time randomized distributed (∆+1)-coloring algorithms have been known for more than 30 years, the question whether a similarly efficient (i.e., polylogarithmic time) deterministic distributed coloring algorithm exists remained one of the most important open problems in the area [BE13, GKM17] until it was resolved very recently by Rozhoň and Ghaffari [RG19]. In [RG19] the question was answered in the affirmative for the LOCAL model by designing an efficient deterministic distributed method to decompose the communication graph into a logarithmic number of subgraphs consisting of connected components (clusters) of polylogarithmic (weak) diameter, a structure known as a network decomposition [AGLP89]. It was known before that an efficient deterministic algorithm for network decomposition would essentially imply efficient determinstic LOCAL algorithms for all problems for which efficient randomized algorithms exist [Lin92, AGLP89, BE13, GKM17, GHK18].Note that due to the unbounded message size any (solvable) problem can be solved in diameter time in the LOCAL model by simply collecting the whole graph topology at one node, solving the problem locally (potentially using unbounded computational power), and redistributing the solution to the nodes. Thus the small diameter components of a network decomposition almost immediately give rise to an efficient (∆ + 1)coloring algorithm in the...

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Deterministic (Δ + 1)-Coloring in Sublinear (in Δ) Time in Static, Dynamic and Faulty Networks

Cited by 50 publications

References 33 publications

Improved Distributed Algorithms for Coloring Interval Graphs with Application to Multicoloring Trees

Improved Distributed Algorithms for Coloring Interval Graphs with Application to Multicoloring Trees

Distance-2 Coloring in the CONGEST Model

Efficient Deterministic Distributed Coloring with Small Bandwidth

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