Plotkin, Rao, and Smith (SODA'97) showed that any graph with m edges and
n vertices that excludes Kh as a depth O(ℓlogn)-minor has a
separator of size O(n/ℓ+ℓh2logn) and that such a separator can be
found in O(mn/ℓ) time. A time bound of O(m+n2+ϵ/ℓ) for
any constant ϵ>0 was later given (W., FOCS'11) which is an
improvement for non-sparse graphs. We give three new algorithms. The first has
the same separator size and running time O(\mbox{poly}(h)\ell
m^{1+\epsilon}). This is a significant improvement for small h and ℓ.
If ℓ=Ω(nϵ′) for an arbitrarily small chosen constant
ϵ′>0, we get a time bound of O(\mbox{poly}(h)\ell n^{1+\epsilon}).
The second algorithm achieves the same separator size (with a slightly larger
polynomial dependency on h) and running time O(\mbox{poly}(h)(\sqrt\ell
n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when ℓ=Ω(nϵ′). Our third algorithm has running time
O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when ℓ=Ω(nϵ′). It finds a separator of size O(n/\ell) + \tilde
O(\mbox{poly}(h)\ell\sqrt n) which is no worse than previous bounds when h
is fixed and ℓ=O~(n1/4). A main tool in obtaining our results
is a novel application of a decremental approximate distance oracle of Roditty
and Zwick.Comment: 16 pages. Full version of the paper that appeared at ICALP'14. Minor
fixes regarding the time bounds such that these bounds hold also for
non-sparse graph