Near-Optimal O(k)-Robust Geometric Spanners
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For any constants d> 1, ϵ >0, t>1, and any n-point set P⊂R^d, we show that there is a geometric graph G=(P,E) having O(n^4 n n) edges with the following property: For any F⊆ P, there exists F^+⊇ F, |F^+| < (1+ϵ)|F| such that, for any pair p,q∈ P∖ F^+, the graph G-F contains a path from p to q whose (Euclidean) length is at most t times the Euclidean distance between p and q. In the terminology of robust spanners (Bose 2013) the graph G is a (1+ϵ)k-robust t-spanner of P. This construction is more sparse than the most recent work (Buchin, Olàh, and Har-Peled 2018) which proves the existence of (1+ϵ)k-robust t-spanners with n^O(d) n edges.
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