The focus of this monograph is on symmetry breaking problems in the message-passing model of distributed computing. In this model a communication network is represented by a n-vertex graph G = (V,E), whose vertices host autonomous processors. The processors communicate over the edges of G in discrete rounds. The goal is to devise algorithms that use as few rounds as possible. A typical symmetry-breaking problem is the problem of graph coloring. Denote by [delta] the maximum degree of G. While coloring G with [delta]+ 1 colors is trivial in the centralized setting, the problem becomes much more challenging in the distributed one. One can also compromise on the number of colors, if this allows for more efficient algorithms. Other typical symmetry-breaking problems are the problems of computing a maximal independent set (MIS) and a maximal matching (MM). The study of these problems dates back to the very early days of distributed computing. The founding fathers of distributed computing laid firm foundations for the area of distributed symmetry breaking already in the eighties. In particular, they showed that all these problems can be solved in randomized logarithmic time. Also, Linial showed that an O([delta]2)-coloring can be solved very efficiently deterministically. However, fundamental questions were left open for decades. In particular, it is not known if the MIS or the ([delta] + 1)-coloring can be solved in deterministic polylogarithmic time. Moreover, until recently it was not known if in deterministic polylogarithmic time one can color a graph with significantly fewer than [delta]2 colors. Additionally, it was open (and still open to some extent) if one can have sublogarithmic randomized algorithms for the symmetry breaking problems. Recently, significant progress was achieved in the study of these questions. More efficient deterministic and randomized ([delta] + 1)-coloring algorithms were achieved. Deterministic [delta]1 + o(1)-coloring algorithms with polylogarithmic running time were devised. Improved (and often sublogarithmic-time) randomized algorithms were devised. Drastically improved lower bounds were given. Wide families of graphs in which these problems are solvable much faster than on general graphs were identified. The objective of our monograph is to cover most of these developments, and as a result to provide a treatise on theoretical foundations of distributed symmetry breaking in the message-passing model. We hope that our monograph will stimulate further progress in this exciting area. Table of Contents: Acknowledgments / Introduction / Basics of Graph Theory / Basic Distributed Graph Coloring Algorithns / Lower Bounds / Forest-Decomposition Algorithms and Applications / Defective Coloring / Arbdefective Coloring / Edge-Coloring and Maximal Matching / Network Decompositions / Introduction to Distributed Randomized Algorithms / Conclusion and Open Questions / Bibliography / Authors' Biographies

https://www.cs.bgu.ac.il/~elkinm/book.pdf

Distributed Graph Coloring: Fundamentals and Recent Developments

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 algorithms from a wide family of locally iterative algorithms are unlikely to achieve running time smaller than Θ(Δ log Δ). We present a deterministic (Δ + 1)-coloring distributed algorithm with running time O(Δ) + 1/2 log* n. We also present a tradeoff between the running time and the number of colors, and devise an O(Δ • t)-coloring algorithm with running time O(Δ / t + log* n), for any parameter t, 1

/pdf/distributed-d-1-coloring-in-linear-in-d-time-2e80k0idhu.pdf

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

https://hal.inria.fr/inria-00634981/document

Minimum k-path vertex cover

It is widely believed that diversity in operating systems, software packages, and hardware platforms will decrease the virulence of worms and the effectiveness of repeated applications of single attacks. Research efforts in the field have focused on introducing diversity using a variety of techniques on a system-by-system basis. This paper, on the other hand, assumes the availability of diverse software packages for each system and then seeks to increase the intrinsic value of available diversity by considering the entire computer network. We present several distributed algorithms for the assignment of distinct software packages to individual systems and analyze their performance. Our goal is to limit the ability of a malicious node to use a single attack to compromise its neighboring nodes, and by extension, the rest of the nodes in the network. The algorithms themselves are analyzed for attack tolerance, and strategies for improving the security of the individual software assignment schemes are presented. We present a comparative analysis of our algorithms using simulation results on a topology obtained from e-mail traffic logs between users at our institution. We find that hybrid versions of our algorithms incorporating multiple assignment strategies achieve better attack tolerance than any given assignment strategy. Our work thus shows that diversity must be introduced at all levels of system design, including any scheme that is used to introduce diversity itself.

/pdf/on-achieving-software-diversity-for-improved-network-221tasvx4r.pdf

On achieving software diversity for improved network security using distributed coloring algorithms

We study the edge-coloring problem in the message-passing model of distributed computing. This is one of the most fundamental problems in this area. Currently, the best-known deterministic algorithms for (2Δ-1)-edge-coloring requires O(Δ) + log* n time [23], where Δ is the maximum degree of the input graph. Also, recent results of [5] for vertex-coloring imply that one can get an O(Δ)-edge-coloring in O(Δµ" log n) time, and an O(Δ1 + µ)-edge-coloring in O(log Δ log n) time, for an arbitrarily small constant µ > 0.In this paper we devise a significantly faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Δ)-edge-coloring in O(Δµ) + log* n time, and an O(Δ1 + µ)-edge-coloring in O(log Δ) + log* n time. This result improves the state-of-the-art running time for deterministic edge-coloring with this number of colors in almost the entire range of maximum degree Δ. Moreover, it improves it exponentially in a wide range of Δ, specifically, for 2©(log* n) ≤ Δ ≤ polylog(n). In addition, for small values of Δ (up to log1 - ?n, for some fixed Δ > 0) our deterministic algorithm outperforms all the existing randomized algorithms for this problem.On our way to these results we study the vertex-coloring problem on graphs with bounded neighborhood independence. This is a large family of graphs, which strictly includes line graphs of r-hypergraphs (i.e., hypergraphs in which each hyperedge contains r or less vertices) for r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to general graphs.Our main technical contribution is a subroutine that computes an O(Δ/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p2) + log* n time, for a parameter p, 1 d p d Δ. In all previous efficient distributed routines for m-defective p-coloring the product m A p is super-linear in Δ. In our routine this product is linear in Δ, and this enables us to speed up the coloring drastically.

/pdf/distributed-deterministic-edge-coloring-using-bounded-47a4phtz04.pdf

Distributed deterministic edge coloring using bounded neighborhood independence

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 at most d-self-colored neighbors. First, (2,d) coloring is shown NP-complete for d <= 1, even for planar graphs, and (3,1) coloring is also shown NP-complete for planar graphs (while there exists a quadratic algorithm to (3,2)-color any planar graph). A reduction from ordinary vertex coloring then shows (X,d) coloring NP-complete for any X <= 3, d <= 0, as well as hardness of approximation results. Second, a generalization of Delta + 1 coloring defects is explored for graphs of maximum degree Delta. Based on a theorem of Lovasz, we obtain an O(Delta E) algorithm to (k, \1floor (Delta/k \rfloor) color any graph; this yields an O(E) algorithm to (2,1)-color 3-regular graphs, and (3,2)-color 6-regular graphs. The generalization of Delta + 1 coloring is used in turn to generalize the polynomial-time approximate 3- and k-coloring algorithms of Widgerson and Karger-Motwani-Sudan to allow defects. For approximate 3-coloring, we obtain an O(Delta E) time algorithm to $(\lceil({8n \over d})^{.5}\rceil,d)$ color, and a polynomial time algorithm to $(O((\frac{n} {d})^{.387}), d)$ color any 3-colorable graph.

Coloring with defect

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Coloring with defect