  Computing shortest paths is one of the central problems in the theory of
distributed computing. For the last few years, substantial progress has been
made on the approximate single source shortest paths problem, culminating in an
algorithm of Becker et al. [DISC'17] which deterministically computes
$(1+o(1))$-approximate shortest paths in $\tilde O(D+\sqrt n)$ time, where $D$
is the hop-diameter of the graph. Up to logarithmic factors, this time
complexity is optimal, matching the lower bound of Elkin [STOC'04].
  The question of exact shortest paths however saw no algorithmic progress for
decades, until the recent breakthrough of Elkin [STOC'17], which established a
sublinear-time algorithm for exact single source shortest paths on undirected
graphs. Shortly after, Huang et al. [FOCS'17] provided improved algorithms for
exact all pairs shortest paths problem on directed graphs.
  In this paper, we present a new single-source shortest path algorithm with
complexity $\tilde O(n^{3/4}D^{1/4})$. For polylogarithmic $D$, this improves
on Elkin's $\tilde{O}(n^{5/6})$ bound and gets closer to the
$\tilde{\Omega}(n^{1/2})$ lower bound of Elkin [STOC'04]. For larger values of
$D$, we present an improved variant of our algorithm which achieves complexity
$\tilde{O}\left( n^{3/4+o(1)}+ \min\{ n^{3/4}D^{1/6},n^{6/7}\}+D\right)$, and
thus compares favorably with Elkin's bound of $\tilde{O}(n^{5/6} +
n^{2/3}D^{1/3} + D ) $ in essentially the entire range of parameters. This
algorithm provides also a qualitative improvement, because it works for the
more challenging case of directed graphs (i.e., graphs where the two directions
of an edge can have different weights), constituting the first sublinear-time
algorithm for directed graphs. Our algorithm also extends to the case of exact
$\kappa$-source shortest paths...
