  Computing routing schemes that support both high throughput and low latency
is one of the core challenges of network optimization. Such routes can be
formalized as $h$-length flows which are defined as flows whose flow paths are
restricted to have length at most $h$. Many well-studied algorithmic primitives
-- such as maximal and maximum length-constrained disjoint paths -- are special
cases of $h$-length flows. Likewise the optimal $h$-length flow is a
fundamental quantity in network optimization, characterizing, up to poly-log
factors, how quickly a network can accomplish numerous distributed primitives.
  In this work, we give the first efficient algorithms for computing $(1 -
\epsilon)$-approximate $h$-length flows. We give deterministic algorithms that
take $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}))$ parallel time and
$\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}) \cdot 2^{O(\sqrt{\log n})})$
distributed CONGEST time. We also give a CONGEST algorithm that succeeds with
high probability and only takes $\tilde{O}(\text{poly}(h, \frac{1}{\epsilon}))$
time.
  Using our $h$-length flow algorithms, we give the first efficient
deterministic CONGEST algorithms for the maximal length-constrained disjoint
paths problem -- settling an open question of Chang and Saranurak (FOCS 2020)
-- as well as essentially-optimal parallel and distributed approximation
algorithms for maximum length-constrained disjoint paths. The former greatly
simplifies deterministic CONGEST algorithms for computing expander
decompositions. We also use our techniques to give the first efficient
$(1-\epsilon)$-approximation algorithms for bipartite $b$-matching in CONGEST.
Lastly, using our flow algorithms, we give the first algorithms to efficiently
compute $h$-length cutmatches, an object at the heart of recent advances in
length-constrained expander decompositions.
