  Let $A/\overline{\mathbb{F}}\_p$ and $A'/\overline{\mathbb{F}}\_p$ be
supersingular principally polarized abelian varieties of dimension $g>1$. For
any prime $\ell \ne p$, we give an algorithm that finds a path $\phi \colon A
\rightarrow A'$ in the $(\ell, \dots , \ell)$-isogeny graph in
$\widetilde{O}(p^{g-1})$ group operations on a classical computer, and
$\widetilde{O}(\sqrt{p^{g-1}})$ calls to the Grover oracle on a quantum
computer. The idea is to find paths from $A$ and $A'$ to nodes that correspond
to products of lower dimensional abelian varieties, and to recurse down in
dimension until an elliptic path-finding algorithm (such as Delfs--Galbraith)
can be invoked to connect the paths in dimension $g=1$. In the general case
where $A$ and $A'$ are any two nodes in the graph, this algorithm presents an
asymptotic improvement over all of the algorithms in the current literature. In
the special case where $A$ and $A'$ are a known and relatively small number of
steps away from each other (as is the case in higher dimensional analogues of
SIDH), it gives an asymptotic improvement over the quantum claw finding
algorithms and an asymptotic improvement over the classical van
Oorschot--Wiener algorithm.
