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Plotkin, Rao, and Smith (SODA'97) showed that any graph with $m$ edges and$n$ vertices that excludes $K_h$ as a depth $O(\ell\log n)$-minor has aseparator of size $O(n/\ell + \ell h^2\log n)$ and that such a separator can befound in $O(mn/\ell)$ time. A time bound of $O(m + n^{2+\epsilon}/\ell)$ forany constant $\epsilon > 0$ was later given (W., FOCS'11) which is animprovement for non-sparse graphs. We give three new algorithms. The first hasthe same separator size and running time $O(\mbox{poly}(h)\ellm^{1+\epsilon})$. This is a significant improvement for small $h$ and $\ell$.If $\ell = \Omega(n^{\epsilon'})$ for an arbitrarily small chosen constant$\epsilon' > 0$, 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 largerpolynomial dependency on $h$) and running time $O(\mbox{poly}(h)(\sqrt\elln^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2}))$ when $\ell =\Omega(n^{\epsilon'})$. Our third algorithm has running time$O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon})$ when $\ell =\Omega(n^{\epsilon'})$. It finds a separator of size $O(n/\ell) + \tildeO(\mbox{poly}(h)\ell\sqrt n)$ which is no worse than previous bounds when $h$is fixed and $\ell = \tilde O(n^{1/4})$. A main tool in obtaining our resultsis a novel application of a decremental approximate distance oracle of Rodittyand Zwick.

Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles