# Distributed (δ+1)-coloring in linear (in δ) time

@inproceedings{Barenboim2009DistributedI, title={Distributed ($\delta$+1)-coloring in linear (in $\delta$) time}, author={Leonid Barenboim and Michael Elkin}, booktitle={STOC '09}, year={2009} }

The distributed (Δ + 1)-coloring problem is one of most fundamental and well-studied problems in Distributed Algorithms. Starting with the work of Cole and Vishkin in 86, there was a long line of gradually improving algorithms published. The current state-of-the-art running time is O(Δ log Δ + log* n), due to Kuhn and Wattenhofer, PODC'06. Linial (FOCS'87) has proved a lower bound of 1/2 log* n for the problem, and Szegedy and Vishwanathan (STOC'93) provided a heuristic argument that shows that…

## 125 Citations

An optimal distributed (Δ+1)-coloring algorithm?

- Mathematics, Computer ScienceSTOC
- 2018

This paper presents a new algorithm for (Δ+1)-list coloring in the randomized LOCAL model running in O(log∗n + Detd(poly logn)) time, where Det d(n′) is the deterministic complexity of (deg+1-list coloring) on n′-vertex graphs.

Distributed (Δ +1)-Coloring in Sublogarithmic Rounds

- MathematicsJournal of the ACM
- 2018

We give a new randomized distributed algorithm for (Δ +1)-coloring in the LOCAL model, running in O(√ log Δ)+ 2O(√log log n) rounds in a graph of maximum degree Δ. This implies that the (Δ…

Deterministic distributed edge-coloring with fewer colors

- Mathematics, Computer ScienceSTOC
- 2018

This work presents a deterministic distributed algorithm, in the LOCAL model, that computes a (1+o(1))Δ-edge-coloring in polylogarithmic-time, so long as the maximum degree Δ=Ω(logn), which are the first deterministic algorithms to go below the natural barrier of 2Δ−1 colors.

(2Δ - l)-Edge-Coloring is Much Easier than Maximal Matching in the Distributed Setting

- Computer Science, MathematicsSODA
- 2015

It is shown that a (2Δ − 1)-edge-coloring can be computed in time smaller than loge n for any e > 0, specifically, in eO([EQUATION]log log n) rounds.

Distributed deterministic edge coloring using bounded neighborhood independence

- Computer SciencePODC
- 2011

A significantly faster deterministic edge-coloring algorithm that outperforms all the existing randomized algorithms for this problem and improves it exponentially in a wide range of Δ, specifically, for 2©(log* n) ≤ Δ ≤ polylog(n).

Distributed Coloring Depending on the Chromatic Number or the Neighborhood Growth

- Mathematics, Computer ScienceSIROCCO
- 2011

The greedy algorithm improves the state-of-the-art Δ + 1 coloring algorithms for a large class of graphs, e.g. graphs of moderate neighborhood growth, and is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account.

Distributed Edge Coloring and a Special Case of the Constructive Lovász Local Lemma

- Mathematics, Computer ScienceACM Trans. Algorithms
- 2020

A new distributed Lovász local lemma algorithm for tree-structured dependency graphs, which arise naturally from O(1)-round probabilistic algorithms run on trees, is developed and shown that the Ω (logΔ log n) lower bound can be nearly matched on trees.

Locally-Iterative Distributed (Δ+ 1): -Coloring below Szegedy-Vishwanathan Barrier, and Applications to Self-Stabilization and to Restricted-Bandwidth Models

- Computer SciencePODC
- 2018

It is demonstrated that Szegedy-Vishwanathan barrier is not an inherent limitation for locally-iterative algorithms, and significant improvements for dynamic, self-stabilizing and bandwidth-restricted settings are achieved.

Fast Distributed Coloring Algorithms for Triangle-Free Graphs

- Mathematics, Computer ScienceICALP
- 2013

New distributed algorithms to find (Δ/k)-coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees, where k is at most k, and o(1) is a function of Δ, are given.

Towards the locality of Vizing’s theorem

- Computer Science, MathematicsSTOC
- 2019

The approach is to reduce the distributed edge-coloring problem into an online and restricted version of balls-into-bins problem, and shows how to achieve ℓ = 1 with randomization and ™ = O(logn / loglogn) without randomization.

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