Diameter Spanners, Eccentricity Spanners, and Approximating Extremal Distances

The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a natural question to ask how much can we sparsify a graph and still guarantee that its diameter remains preserved within an approximation $t$. This property is captured by the notion of extremal-distance spanners. Given a graph $G=(V,E)$, a subgraph $H=(V,E_H)$ 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 and $m$-edges directed graph $G$, we can compute a sparse subgraph $H$ that is a $(1.5)$-diameter spanner of $G$, such that $H$ contains at most $\tilde O(n^{1.5})$ edges. We also show that the stretch factor cannot be improved to $(1.5-\epsilon)$. For a graph whose diameter is bounded by some constant, we show the existence of $\frac{5}{3}$-diameter spanner that contains at most $\tilde O(n^{\frac{4}{3})$} edges. We also show that this bound is tight. Additionally, we present other types of extremal-distance spanners, such as $2$-eccentricity spanners and $2$-radius spanners, both contain only $\tilde O(n)$ edges and are computable in $\tilde O(m)$ time. Finally, we study extremal-distance spanners in the dynamic and fault-tolerant settings. An interesting implication of our work is the first $\tilde O(m)$-time algorithm for computing $2$-approximation of vertex eccentricities in general directed weighted graphs. Backurs et al. [STOC 2018] gave an $\tilde O(m\sqrt{n})$ time algorithm for this problem, and 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; this shows that our approximation factor is essentially tight.