Keywords and Synonyms
Graph algorithms; Randomized algorithms; Shortest path; Spanner
Problem Definition
A spanner is a sparse subgraph of a given undirected graph that preserves approximate distance between each pair of vertices. More precisely, a t-spanner of a graph \( { G=(V,E) } \) is a subgraph \( { (V,E_S), E_S\subseteq E } \) such that, for any pair of vertices, their distance in the subgraph is at most t times their distance in the original graph, where t is called the stretch factor. The spanners were defined formally by Peleg and Schäffer [14], though the associated notion was used implicitly by Awerbuch [3] in the context of network synchronizers.
Computing a t-spanner of smallest size for a given graph is a well motivated combinatorial problem with many applications. However, computing t-spanner of smallest size for a graph is NP-hard. In fact, for \( { t > 2 } \), it is NP-hard [10] even to approximate the smallest size of a t-spanner of a graph with ratio \( {...
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Notes
- 1.
Ties can be broken arbitrarily. However, it helps conceptually to assume that all weights are distinct.
Recommended Reading
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Baswana, S., Sen, S. (2008). Algorithms for Spanners in Weighted Graphs. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_10
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