Small Stretch Spanners on Dynamic Graphs
Vol. 10, no. 2, pp. 365-385, 2006. Regular paper.
Abstract We present fully dynamic algorithms for maintaining 3- and 5-spanners of undirected graphs under a sequence of update operations. For unweighted graphs we maintain a 3-spanner or a 5-spanner under insertions and deletions of edges; on a graph with n vertices each operation is performed in O(∆) amortized time over a sequence of Ω(n) updates, where ∆ is the maximum degree of the original graph. The maintained 3-spanner (resp., 5-spanner) has O(n3/2) edges (resp., O(n4/3) edges), which is known to be optimal. On weighted graphs with d different edge cost values, we maintain a 3- or 5-spanner within the same amortized time bounds over a sequence of Ω(d ·n) updates. The maintained 3-spanner (resp., 5-spanner) has O(d ·n3/2) edges (resp., O(d ·n4/3) 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.
Submitted: November 2005.
Revised: June 2006.
Communicated by Giuseppe Liotta
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