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… 
An optimal distributed (Δ+1)-coloring algorithm?
TLDR
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 41 REFERENCES
Fast distributed graph coloring with O(Δ) colors
We consider the problem of deterministic distributed coloring of an n-vertex graph with maximum degree Δ, assuming that every vertex knows a priori only its own label and parameters n and Δ. The aim
A new technique for distributed symmetry breaking
TLDR
Three randomized algorithms for distributed (vertex or edge) coloring improving on previous algorithms and showing a time/color trade-off are presented, obtaining new results for several graph classes.
Weak graph colorings: distributed algorithms and applications
  • F. Kuhn
  • Mathematics, Computer Science
    SPAA '09
  • 2009
TLDR
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
TLDR
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.
A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs
TLDR
The algorithm shows that for computing a MIS, randomization is a viable alternative to distance information and is close to optimal.
Coloring with defect
This paper is concerned with algorithms and complexity results for defective coloring, where a defective (k,d)-coloring is a k coloring of the vertices of a graph such that each vertex is adjacent to
Locality based graph coloring
TLDR
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 Network Decomposition
In this paper, we improve the bounds for computing a network decomposition distributively and deterministically. Our algorithm computes an (n?(n),n?(n))-decomposition innO(?(n))time, whereformula. As
On the complexity of distributed graph coloring
TLDR
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.
A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem
TLDR
A technique due to A. Joffe (1974) is applied and deterministic construction in fast parallel time of various combinatorial objects whose existence follows from probabilistic arguments is obtained.
...
1
2
3
4
5
...