Small stretch spanners on dynamic graphs

We present fully dynamic algorithms for maintaining 3- and 5-spanners of undirected graphs. For unweighted graphs we maintain a 3-or 5-spanner under insertions and deletions of edges in O(n) amortized time per operation over a sequence of Omega(n) updates. The maintained 3-spanner (resp., 5-spanner) has O(n(3/2)) edges (resp., O(n(4/3)) edges), which is known to be optimal. On weighted graphs with d different edge cost values, we maintain a 3- or 5-spanner in O(n) amortized time per operation over a sequence of Omega(d(.)n) updates. The maintained 3-spanner (resp., 5-spanner) has O(d.n(3/2)) edges (resp., O(d(.)n(4/3)) edges). The same approach can be extended to graphs with real-valued edge costs in the range [1, C]. All our algorithms are deterministic and are substantially faster than recomputing a spanner from scratch after each update.