  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.
