Computing treedepth in polynomial space and linear fpt time

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 2O(d2)· nO(1) and the space usage is polynomial in n. Further, by allowing randomization, the time and space complexities can be improved to 2O(d2)· n and dO(1)· n, respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014], which also has time complexity 2O(d2)· n, but uses exponential space.