Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference

A Review of Interference Reduction in Wireless Networks Using Graph Coloring Methods

The interference imposes a significant negative impact on the performance of wireless networks. With the continuous deployment of larger and more sophisticated wireless networks, reducing interference in such networks is quickly being focused upon as a problem in today's world. In this paper we analyze the interference reduction problem from a graph theoretical viewpoint. A graph coloring methods are exploited to model the interference reduction problem. However, additional constraints to graph coloring scenarios that account for various networking conditions result in additional complexity to standard graph coloring. This paper reviews a variety of algorithmic solutions for specific network topologies.Comment: 10 pages, 5 figure

Parallel Maximum Clique Algorithms with Applications to Network Analysis and Storage

We propose a fast, parallel maximum clique algorithm for large sparse graphs that is designed to exploit characteristics of social and information networks. The method exhibits a roughly linear runtime scaling over real-world networks ranging from 1000 to 100 million nodes. In a test on a social network with 1.8 billion edges, the algorithm finds the largest clique in about 20 minutes. Our method employs a branch and bound strategy with novel and aggressive pruning techniques. For instance, we use the core number of a vertex in combination with a good heuristic clique finder to efficiently remove the vast majority of the search space. In addition, we parallelize the exploration of the search tree. During the search, processes immediately communicate changes to upper and lower bounds on the size of maximum clique, which occasionally results in a super-linear speedup because vertices with large search spaces can be pruned by other processes. We apply the algorithm to two problems: to compute temporal strong components and to compress graphs.Comment: 11 page

Dynamic Algorithms for Graph Coloring

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(\Delta+1)$- vertex coloring and $(2\Delta-1)$-edge coloring in a graph with maximum degree $\Delta$. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a $(\Delta+1)$-vertex coloring with $O(\log \Delta)$ expected amortized update time. (2) We present a deterministic algorithm which maintains a $(1+o(1))\Delta$-vertex coloring with $O(\text{poly} \log \Delta)$ amortized update time. (3) We present a simple, deterministic algorithm which maintains a $(2\Delta-1)$-edge coloring with $O(\log \Delta)$ worst-case update time. This improves the recent $O(\Delta)$-edge coloring algorithm with $\tilde{O}(\sqrt{\Delta})$ worst-case update time by Barenboim and Maimon.Comment: To appear in SODA 201