  We prove that any $n$-node graph $G$ with diameter $D$ admits shortcuts with
congestion $O(\delta D \log n)$ and dilation $O(\delta D)$, where $\delta$ is
the maximum edge-density of any minor of $G$. Our proof is simple, elementary,
and constructive - featuring a $\tilde{\Theta}(\delta D)$-round distributed
construction algorithm. Our results are tight up to $\tilde{O}(1)$ factors and
generalize, simplify, unify, and strengthen several prior results. For example,
for graphs excluding a fixed minor, i.e., graphs with constant $\delta$, only a
$\tilde{O}(D^2)$ bound was known based on a very technical proof that relies on
the Robertson-Seymour Graph Structure Theorem.
  A direct consequence of our result is that many graph families, including any
minor-excluded ones, have near-optimal $\tilde{\Theta}(D)$-round distributed
algorithms for many fundamental communication primitives and optimization
problems including minimum spanning tree, minimum cut, and shortest-path
approximations.
