We study dynamic graphs in the fully dynamic centralized setting. In this setting, the vertex set of size n of a graph G is fixed, and the edge set changes step-by-step, such that each step either adds or removes an edge. Dynamic graphs have various applications in fields such as Communication Networks, Computer Graphics, and VLSI Design. The goal in this setting is maintaining a solution to a certain problem (e.g., maximal matching, edge coloring) after each step, such that each step is executed efficiently. The running time of a step is called update-time. One can think of this setting as a dynamic network that is monitored by a central processor that is responsible for maintaining the solution. Prior to the current work, for several central problems, the best-known deterministic algorithms for general graphs were the naive ones with update-time O(n). This is the case for maximal matching and proper O(Δ)-edge-coloring. The question of existence of sublinear in n update-time deterministic algorithms for dense graphs and general graphs remained wide open. In this article, we address this question by devising sublinear update-time deterministic algorithms for maximal matching in graphs with bounded neighborhood independence o(n/ log2 n), and for proper O(Δ)-edge-coloring in general graphs. The family of graphs with bounded neighborhood independence is a very wide family of dense graphs. In particular, graphs with constant neighborhood independence include line-graphs, claw-free graphs, unit disk graphs, and many other graphs. Thus, these graphs represent very well various types of networks. For graphs with constant neighborhood independence, our maximal-matching algorithm has Õ (n) update-time. Our O(Δ)-edge-coloring algorithms has Õ (Δ) update-time for general graphs. To obtain our results, we employ a novel approach that adapts certain distributed algorithms of the LOCAL setting to the centralized fully dynamic setting. This is achieved by optimizing the work each processor performs and efficiently simulating a distributed algorithm in a centralized setting. The simulation is efficient, thanks to a careful selection of the network parts that the algorithm is invoked on, and by deducing the solution from the additional information that is present in the centralized setting, but not in the distributed one. Our experiments on various network topologies and scenarios demonstrate that our algorithms are highly efficient in practice. We believe that our approach is of independent interest and may be applicable to additional problems.
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A preliminary version of this paper appeared in the International Conference on Computational Sciences, 2017. This research was supported by ISF grant 724/15 and by the Open University of Israel research fund. Part of this work has been performed while the first-named author was affiliated with the Simons institute at UC Berkeley and Weizmann institute of science. Part of this work has been performed while the second-named author was affiliated with the Open University of Israel. Authors’ addresses: L. Barenboim, The Open University of Israel, 1 University Road, P. O. Box 808, Raanana, Israel; email: firstname.lastname@example.org; T. Maimon, Ben-Gurion University of the Negev, P.O.Box 653, Beer-Sheva, Israel; email: email@example.com. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from firstname.lastname@example.org. © 2019 Association for Computing Machinery. 1084-6654/2019/09-ART1.14 $15.00 https://doi.org/10.1145/3338529
© 2019 Association for Computing Machinery.
- Dynamic networks
- Neighborhood independence
- Social networks