  The treedepth of a graph $G$ is the least possible depth of an elimination
forest of $G$: a rooted forest on the same vertex set where every pair of
vertices adjacent in $G$ is bound by the ancestor/descendant relation. We
propose an algorithm that given a graph $G$ and an integer $d$, either finds an
elimination forest of $G$ of depth at most $d$ or concludes that no such forest
exists; thus the algorithm decides whether the treedepth of $G$ is at most $d$.
The running time is $2^{O(d^2)}\cdot n^{O(1)}$ and the space usage is
polynomial in $n$. Further, by allowing randomization, the time and space
complexities can be improved to $2^{O(d^2)}\cdot n$ and $d^{O(1)}\cdot n$,
respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014],
which also has time complexity $2^{O(d^2)}\cdot n$, but uses exponential space.
