Fully Dynamic Randomized Algorithms for Graph Spanners in Centralized as well as Distributed environments∗

Spanner of an undirected graph G = (V, E) is a sub graph which is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with stretch t ∈ N is a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. We present two randomized algorithms for maintaining a sparse t-spanner of an undirected unweighted graph under insertion and deletion of edges. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners in centralized as well as distributed environment. The expected size of the t-spanner maintained at each stage by our algorithms matches the worst case optimal size of a t-spanner up to poly-logarithmic factor, and the expected time required to process an update (insertion/deletion of an edge) is close to optimal. The results of the preliminary version of this article appeared in ESA 2006 and SODA 2008 Dept. of Comp. Sc. and Engg., IIT Kanpur, India. Email : [email protected]. This work was supported by a Young Faculty Chair Award from Research I Foundation, CSE, IIT Kanpur. Department of Comp. Sc., University of Texas at Austin, Texas 78712, USA. Email : [email protected]. This Work was done while the author was an M.Tech student at CSE, IIT Kanpur, India.