Survivable Networks with Bounded Delay: The Edge Failure Case
ISAAC '99 Proceedings of the 10th International Symposium on Algorithms and Computation
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A spanning subgraph $G'$ of a simple undirected graph $G$ is a t-spanner of $G$ if every pair of vertices that are adjacent in $G$ are at distance at most $t$ in $G'$. The parameter $t$ is called the dilation of the spanner. Spanners with small dilations have many applications, such as their use as low-cost approximations of communication networks with only small degradations in performance. In this paper, we derive spanners with small dilations for four closely related bounded-degree approximations of hypercubes: butterflies, cube-connected cycles, binary de Bruijn graphs, and shuffle-exchange graphs. We give both direct constructions and methods for deriving spanners for one class of graphs from spanners for another class. We prove that most of our spanners are minimum in the sense that spanners with fewer edges have larger dilations.