  We present the first $m\,\text{polylog}(n)$ work, $\text{polylog}(n)$ time
algorithm in the PRAM model that computes $(1+\epsilon)$-approximate
single-source shortest paths on weighted, undirected graphs. This improves upon
the breakthrough result of Cohen~[JACM'00] that achieves $O(m^{1+\epsilon_0})$
work and $\text{polylog}(n)$ time. While most previous approaches, including
Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent
developments in \emph{continuous optimization}, studying the shortest path
problem from the lens of the closely-related \emph{minimum transshipment}
problem. To obtain our algorithm, we demonstrate a series of near-linear work,
polylogarithmic-time reductions between the problems of approximate shortest
path, approximate transshipment, and $\ell_1$-embeddings, and establish a
recursive algorithm that cycles through the three problems and reduces the
graph size on each cycle. As a consequence, we also obtain faster parallel
algorithms for approximate transshipment and $\ell_1$-embeddings with
polylogarithmic distortion. The minimum transshipment algorithm in particular
improves upon the previous best $m^{1+o(1)}$ work sequential algorithm of
Sherman~[SODA'17].
  To improve readability, the paper is almost entirely self-contained, save for
several staple theorems in algorithms and combinatorics.
