Dynamic Graph Algorithms with Applications

Abstract

First we review amortized fully-dynamic polylogarithmic algorithms for connectivity, minimum spanning trees (MST), 2-edge-and biconnectivity. Second we discuss how they yield improved static algorithms: connectivity for constructing a tree from homeomorphic subtrees, 2-edge connectivity for finding unique matchings in graphs, and MST for packing spanning trees in graphs. The application of MST for spanning tree packing is new and when boot-strapped, it yields a fully-dynamic polylogarithmic algorithm for approximating general edge connectivity within a factor 2 + o(1). Finally, on the more practical side, we will discuss how output sensitive algorithms for dynamic shortest paths have been applied successfully to speed up local search algorithms for improving routing on the internet, roughly doubling the capacity. 1 Dynamic Graph Algorithms In this talk, we will discuss some simple dynamic graph algorithms and their applications within static graph problems. As a new result, we will derive a fully dynamic polylogarithmic algorithm approximating the edge connectivity λ within a factor 2 + o(1), that is, the algorithm will output a value between λ/ 2 + o(1) and λ × 2 + o(1). The talk is not intended as a general survey of dynamic graph algorithms and their applications. Rather its goal is just to present a few nice illustrations of the potent relationship between dynamic graph algorithms and their applications in static graph problems, showing contexts in which dynamic graph algorithms play a role similar to that played by priority queues for greedy algorithms. In a fully dynamic graph problem, we are considering a graph G over a fixed vertex set V , |V | = n. The graph G may be updated by insertions and deletions of edges. Unless otherwise stated, we assume that we start with an empty edge set. We will review the fully dynamic graph algorithms of Holm et al. [11] for connec-tivity, minimum spanning trees (MST), 2-edge, and biconnectivity in undirected graphs. For the connectivity type problems, the updates may be interspersed by queries on (2-edge-/bi-) connectivity of the graph or between specified vertices. For MST, the fully dynamic algorithm should update the MST in connection with each update to the graph: an inserted edge might have to go into the MST, and if an MST edge is deleted, we should replace with the lightest edge possible.