  The aspect ratio of a (positively) weighted graph $G$ is the ratio of its
maximum edge weight to its minimum edge weight. Aspect ratio commonly arises as
a complexity measure in graph algorithms, especially related to the computation
of shortest paths. Popular paradigms are to interpolate between the settings of
weighted and unweighted input graphs by incurring a dependence on aspect ratio,
or by simply restricting attention to input graphs of low aspect ratio.
  This paper studies the effects of these paradigms, investigating whether
graphs of low aspect ratio have more structured shortest paths than graphs in
general. In particular, we raise the question of whether one can generally take
a graph of large aspect ratio and reweight its edges, to obtain a graph with
bounded aspect ratio while preserving the structure of its shortest paths. Our
findings are:
  - Every weighted DAG on $n$ nodes has a shortest-paths preserving graph of
aspect ratio $O(n)$. A simple lower bound shows that this is tight.
  - The previous result does not extend to general directed or undirected
graphs; in fact, the answer turns out to be exponential in these settings. In
particular, we construct directed and undirected $n$-node graphs for which any
shortest-paths preserving graph has aspect ratio $2^{\Omega(n)}$.
  We also consider the approximate version of this problem, where the goal is
for shortest paths in $H$ to correspond to approximate shortest paths in $G$.
We show that our exponential lower bounds extend even to this setting. We also
show that in a closely related model, where approximate shortest paths in $H$
must also correspond to approximate shortest paths in $G$, even DAGs require
exponential aspect ratio.
