Fully-Dynamic Graph Algorithms with Sublinear Time Inspired by Distributed Computing

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. 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. Currently, for several central problems, the best-known deterministic algorithms for general graphs are the naive ones which have 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. 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 bounded neighborhood independence is a very wide family of dense graphs that represent very well various 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. In order 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 processors 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.