Self-Stabilizing Graph Coloring Algorithms

2013 ◽  
pp. 96-110
Author(s):  
Shing-Tsaan Huang ◽  
Chi-Hung Tzeng ◽  
Jehn-Ruey Jiang
Keyword(s):  
Distributed Computing ◽  
Graph Coloring ◽  
Finite Time ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Vertex Coloring ◽  
Transient Faults ◽  
Initial State ◽  
Self Stabilization ◽  
Legitimate State

The concept of self-stabilization in distributed systems was introduced by Dijkstra in 1974. A system is said to be self-stabilizing if (1) it can converge in finite time to a legitimate state from any initial state, and (2) when it is in a legitimate state, it remains so henceforth. That is, a self-stabilizing system guarantees to converge to a legitimate state in finite time no matter what initial state it may start with; or, it can recover from transient faults automatically without any outside intervention. This chapter first introduces the self-stabilization concept in distributed computing. Next, it discusses the coloring problem on graphs and its applications in distributed computing. Then, it introduces three self-stabilizing algorithms. The first two are for vertex coloring and edge coloring on planar graphs, respectively. The last one is for edge coloring on bipartite graphs.

SELF-STABILIZING ALGORITHMS FOR ORDERINGS AND COLORINGS

2005 ◽  
Vol 16 (01) ◽  
pp. 19-36 ◽  
Author(s):  
WAYNE GODDARD ◽  
STEPHEN T. HEDETNIEMI ◽  
DAVID P. JACOBS ◽  
PRADIP K. SRIMANI
Keyword(s):  
Distributed Algorithms ◽  
Polynomial Time ◽  
Graph Coloring ◽  
Planar Graphs ◽  
Special Cases ◽  
Legitimate State

A k-forward numbering of a graph is a labeling of the nodes with integers such that each node has less than k neighbors whose labels are equal or larger. Distributed algorithms that reach a legitimate state, starting from any illegitimate state, are called self-stabilizing. We obtain three self-stabilizing (s-s) algorithms for finding a k-forward numbering, provided one exists. One such algorithm also finds the k-height numbering of graph, generalizing s-s algorithms by Bruell et al. [4] and Antonoiu et al. [1] for finding the center of a tree. Another k-forward numbering algorithm runs in polynomial time. The motivation of k-forward numberings is to obtain new s-s graph coloring algorithms. We use a k-forward numbering algorithm to obtain an s-s algorithm that is more general than previous coloring algorithms in the literature, and which k-colors any graph having a k-forward numbering. Special cases of the algorithm 6-color planar graphs, thus generalizing an s-s algorithm by Ghosh and Karaata [13], as well as 2-color trees and 3-color series-parallel graphs. We discuss how our s-s algorithms can be extended to the synchronous model.

Self-Stabilizing Algorithms for Tree Metrics

1998 ◽  
Vol 08 (01) ◽  
pp. 121-133
Author(s):  
Ajoy K. Datta ◽  
Teofilo F. Gonzalez ◽  
Visalakshi Thiagarajan
Keyword(s):  
Fault Tolerance ◽  
Finite Time ◽  
Initial System ◽  
Transient Faults ◽  
System Configuration ◽  
Tree Metrics ◽  
Self Stabilization

We present algorithms for finding the diameter, centroid(s), and median(s) for tree structured networks subject to transient faults. In our solutions, the system reaches its final correct configuration in a finite time after the faults cease. The fault-tolerance is achieved using Dijkstra's paradigm of self-stabilization. A self-stabilizing algorithm, regardless of the initial system configuration, converges, in finite time, to a set of legitimate configurations.

Graph Coloring

2013 ◽  
pp. 106-126
Author(s):  
Faraz Dadgostari ◽  
Mahtab Hosseininia
Keyword(s):  
Graph Coloring ◽  
Edge Coloring ◽  
Graph Labeling ◽  
Vertex Coloring ◽  
List Coloring ◽  
Simple Type ◽  
Critical Graphs ◽  
Classification Of Graphs ◽  
Edge List

In this chapter a particular type of graph labeling, called graph coloring, is introduced and discussed. In the first part, the simple type of coloring, vertex coloring, is focused. Thus, concerning vertex coloring, some terms and definitions are introduced. Next, some theorems and applying those theorems, some coloring algorithms and applications are introduced. At last, some helpful concepts such as critical graphs, list coloring, and vertex decomposition are presented and discussed. In the second section, edge coloring is focused. Thus, concerning edge coloring, some terms and definitions are described, some important information about edge chromatic number and edge list coloring is presented, and applying them, classification of graphs using the coloring approach is summarized. At last some helpful concepts such as edge list coloring and edge decomposition are illustrated and discussed.

scholarly journals Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles

2021 ◽  
Author(s):  
Qiaojun Shu ◽  
Yong Chen ◽  
Shuguang Han ◽  
Guohui Lin ◽  
Eiji Miyano ◽  
...  
Keyword(s):  
Planar Graphs ◽  
Edge Coloring ◽  
Acyclic Edge Coloring

Locally-iterative Distributed (Δ + 1)-coloring and Applications

2022 ◽  
Vol 69 (1) ◽  
pp. 1-26
Author(s):  
Leonid Barenboim ◽  
Michael Elkin ◽  
Uri Goldenberg
Keyword(s):  
Message Passing ◽  
Edge Coloring ◽  
Iterative Algorithms ◽  
Independent Set ◽  
Local Model ◽  
Vertex Coloring ◽  
Maximal Independent Set ◽  
List Type ◽  
Inherent Limitation ◽  
Running Time

We consider graph coloring and related problems in the distributed message-passing model. Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1-hop-neighborhood . In STOC’93 Szegedy and Vishwanathan showed that any locally-iterative Δ + 1-coloring algorithm requires Ω (Δ log Δ + log * n ) rounds, unless there exists “a very special type of coloring that can be very efficiently reduced” [ 44 ]. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms and to explore other approaches to the coloring problem [ 2 , 3 , 19 , 32 ]. The latter gave rise to faster algorithms, but their heavy machinery that is of non-locally-iterative nature made them far less suitable to various settings. In this article, we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative Δ + 1-coloring algorithm with running time O (Δ + log * n ), i.e., below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing, and bandwidth-restricted settings. This includes the following results: We obtain self-stabilizing distributed algorithms for Δ + 1-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set, and maximal matching with O (Δ + log * n ) time. This significantly improves previously known results that have O(n) or larger running times [ 23 ]. We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O (Δ + log * n ) time and O (Δ)-edge-coloring in the Bit-Round model with O (Δ + log n ) time. The factors of log * n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. We obtain an arbdefective coloring algorithm with running time O (√ Δ + log * n ). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it to compute a proper (1 + ε)Δ-coloring within O (√ Δ + log * n ) time and Δ + 1-coloring within O (√ Δ log Δ log * Δ + log * n ) time. This improves the recent state-of-the-art bounds of Barenboim from PODC’15 [ 2 ] and Fraigniaud et al. from FOCS’16 [ 19 ] by polylogarithmic factors. Our algorithms are applicable to the SET-LOCAL model [ 25 ] (also known as the weak LOCAL model). In this model a relatively strong lower bound of Ω (Δ 1/3 ) is known for Δ + 1-coloring. However, most of the coloring algorithms do not work in this model. (In Reference [ 25 ] only Linial’s O (Δ 2 )-time algorithm and Kuhn-Wattenhofer O (Δ log Δ)-time algorithms are shown to work in it.) We obtain the first linear-in-Δ Δ + 1-coloring algorithms that work also in this model.

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