Fully dynamic randomized algorithms for graph spanners

Spanner of an undirected graph <i>G</i> = (<i>V,E</i>) is a subgraph that is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with <i>stretch</i> <i>t</i> ∈ ℕ is a subgraph (<i>V,E_S</i>), <i>E_S</i> ⊆ <i>E</i> such that the distance between any two vertices in the subgraph is at most <i>t</i> times their distance in <i>G</i>. Though <i>G</i> is trivially a <i>t</i>-spanner of itself, the research as well as applications of spanners invariably deal with a <i>t</i>-spanner that has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a <i>t</i>-spanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a <i>t</i>-spanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is <i>close</i> to optimal.