# Diameter Spanner, Eccentricity Spanner, and Approximating Extremal Graph Distances: Static, Dynamic, and Fault Tolerant

The diameter, vertex eccentricities, and the radius of a graph are some of the most fundamental graph parameters. Roditty and Williams [STOC 2013] gave an O(m√(n)) time algorithm for computing a 1.5 approximation of graph diameter. We present the first non-trivial algorithm for maintaining `< 2'- approximation of graph diameter in dynamic setting. Our algorithm maintain a (1.5+ϵ) approximation of graph diameter that takes amortized update time of O(ϵ^-1n^1.25) in partially dynamic setting. For graphs whose diameter remains bounded by some large constant, the total amortized time of our algorithm is O(ϵ^-2√(n)), which almost matches the best known bound for static 1.5-approximation of diameter. Backurs et al. [STOC 2018] gave an Õ(m√(n)) time algorithm for computing 2-approximation of eccentricities. They also showed that no O(n^2-o(1)) time algorithm can achieve an approximation factor better than 2 for graph eccentricities, unless SETH fails. We present the Õ(m) time algorithm for computing 2-approximation of vertex eccentricities in directed weighted graphs. We also present fault tolerant data-structures for maintaining 1.5-diameter and 2-eccentricities. We initiate the study of Extremal Distance Spanners. Given a graph G=(V,E), a subgraph H=(V,E0) is defined to be a t-diameter-spanner if the diameter of H is at most t times the diameter of G. We show that for any n-vertex directed graph G we can compute a sparse subgraph H which is a (1.5)-diameter-spanner of G and contains at most O(n^1.5) edges. We also show that this bound is tight for graphs whose diameter is bounded by n^1/4-ϵ. We present several other extremal-distance spanners with various size-stretch trade-offs. Finally, we extensively study these objects in the dynamic and fault-tolerant settings.

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