  We present a framework that unifies directed buy-at-bulk network design and
directed spanner problems, namely, buy-at-bulk spanners. The goal is to find a
minimum-cost routing solution for network design problems that capture
economies at scale, while satisfying demands and distance constraints for
terminal pairs. A more restricted version of this problem was shown to be
$O(2^{{\log^{1-\varepsilon} n}})$-hard to approximate, where $n$ is the number
of vertices, under a standard complexity assumption, due to Elkin and Peleg
(Theory of Computing Systems, 2007).
  To the best of our knowledge, our results are the first sublinear factor
approximation algorithms for directed buy-at-bulk spanners. Furthermore, these
results hold even when we allow the edge lengths to be negative, unlike the
previous literature for spanners. Our approximation ratios match the
state-of-the-art ratios in special cases, namely, buy-at-bulk network design by
Antonakopoulos (WAOA, 2010) and weighted spanners by Grigorescu, Kumar, and Lin
(APPROX 2023).
  Our results are based on new approximation algorithms for the following two
problems that are of independent interest: minimum-density distance-constrained
junction trees and resource-constrained shortest path with negative
consumption.
