# Distributed Graph Coloring Made Easy

@article{Maus2021DistributedGC, title={Distributed Graph Coloring Made Easy}, author={Yannic Maus}, journal={Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures}, year={2021} }

In this paper we present a deterministic CONGEST algorithm to compute an O(kΔ)-vertex coloring in O(Δ/k)+łog^* n rounds, where Δ is the maximum degree of the network graph and 1łeq kłeq O(Δ) can be freely chosen. The algorithm is extremely simple: Each node locally computes a sequence of colors and then it "tries colors" from the sequence in batches of size k. Our algorithm subsumes many important results in the history of distributed graph coloring as special cases, including Linial's color…

## 6 Citations

Distributed ∆-coloring plays hide-and-seek

- Computer Science, MathematicsSTOC
- 2022

Lower bounds as a function of Δ are proved for a large class of distributed symmetry breaking problems, which can all be solved by a simple sequential greedy algorithm, including the maximal independent set (MIS) in trees.

Distributed $\Delta$-Coloring Plays Hide-and-Seek

- Computer Science, Mathematics
- 2021

Lower bounds as a function of ∆ are proved for a large class of distributed symmetry breaking problems, which can all be solved by a simple sequential greedy algorithm.

Improved Distributed Fractional Coloring Algorithms

- Mathematics, Computer ScienceOPODIS
- 2021

The fractional coloring problem can be approximated arbitrarily well by an efficient algorithm in the LOCAL model, and it is shown that in regular grids of bounded dimension, a fractional (2 + ε )-coloring can be computed in time O (log ∗ n ).

Brief Announcement: Fault Tolerant Coloring of the Asynchronous Cycle

- Computer SciencePODC
- 2022

A wait-free algorithm for proper coloring the n ≥ 3 nodes of the asynchronous cycle Cn, where each crash-prone node starts with its (unique) identifier as input, and each node terminates upon completing at most O(log*n) write-read-compute steps.

Fault Tolerant Coloring of the Asynchronous Cycle

- Computer ScienceArXiv
- 2022

The model coincides with the shared-memory model whenever 𝑛 = 3, and the minimum number of names for which renaming is possible in 3-process shared- memory systems is 5, which is optimal thanks to a known matching lower bound.

Linial for lists

- Computer Science, ArtDistributed Computing
- 2022

It is shown that when nodes are restricted to choose their color from a list of allowed colors: given an <jats:italic>m-coloring in a directed graph of maximum outdegree, the result is the same as when node selection is restricted.

## References

SHOWING 1-10 OF 57 REFERENCES

Efficient Deterministic Distributed Coloring with Small Bandwidth

- Computer SciencePODC
- 2020

We show that the (degree + 1)-list coloring problem can be solved deterministically in O(D · log n · log2 Δ) rounds in the CONGEST model, where D is the diameter of the graph, n the number of nodes,…

Efficient randomized distributed coloring in CONGEST

- Computer Science, MathematicsSTOC
- 2021

This work presents a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an n-node graph G, and where the nodes of G operate in synchronous communication rounds in which they can exchange O(logn)-bit messages over all the edges of G.

Weak graph colorings: distributed algorithms and applications

- Computer Science, MathematicsSPAA '09
- 2009

A faster deterministic algorithm for the standard vertex coloring problem on graphs with moderate degrees is obtained, it is shown that in time O(Δ+log*n), a (Γ+1)-coloring can be computed, a task for which the best previous algorithm required time O (Δ*log(Γ) + log*n).

Improved distributed algorithms for coloring and network decomposition problems

- Computer Science, MathematicsSTOC '92
- 1992

It is shown that A-coloring G is reducible in 0(log3 n/log A) time to (A+ I)-vertex coloring G in a distributed model, which leads to fast distributed algorithms, and a linear–processor NC algorithm, for Acoloring.

Distributed Edge Coloring in Time Quasi-Polylogarithmic in Delta

- Mathematics, Computer SciencePODC
- 2020

The edge coloring problem in the distributed LOCAL model is studied and it is shown that the (degree + 1)-list edge coloringProblem, and thus also the (2Δ − 1)-edge coloring problem, can be solved deterministically in time 2O (log2 log Δ)+O(log* n).

Deterministic Distributed Vertex Coloring: Simpler, Faster, and without Network Decomposition

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

An improved deterministic algorithm based on an improved variant of the network decomposition of Rozhoň and Ghaffari leads to an improvement in the complexity of randomized algorithms for ($\Delta +1$)-coloring, now reaching the bound of $O(\text{log}^{3}\text{ log}\ n)$ rounds.

Local Conflict Coloring

- Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

This work introduces conflict coloring as a general symmetry-breaking task that includes all the aforementioned tasks as specific instantiations - conflict coloring includes all locally checkable labeling tasks from [Naor & Stockmeyer, STOC 1993] and yields an LCA which requires a smaller number of probes than the previously best known algorithm for vertex-coloring.

Locality based graph coloring

- Computer ScienceSTOC '93
- 1993

A randomized algorithm for the problem of locality based graph coloring is designed and an upper bound of O(A. 2A log log n) is proved and lower bounds that match the upper bounds within a factor that is poly-logarithmic are obtained.

On the complexity of distributed graph coloring

- Computer SciencePODC '06
- 2006

This paper proves new strong lower bounds for two special kinds of coloring algorithms, and proves a time lower bound of Ω(Δ/log<sup>2</sup> Δ+ log*<i>m</i>) to obtain an <i>O</i>(Δ)-coloring.

Distributed Graph Coloring: Fundamentals and Recent Developments

- Computer ScienceDistributed Graph Coloring: Fundamentals and Recent Developments
- 2013

The objective of this monograph is to provide a treatise on theoretical foundations of distributed symmetry breaking in the message-passing model of distributed computing and to stimulate further progress in this exciting area.