Fully Dynamic Graph Algorithms Inspired by Distributed Computing

Abstract

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 / log 2 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.