Metrical task systems on trees via mirror descent and unfair gluing

We consider metrical task systems on tree metrics, and present an $O(\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}\times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log \log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O((\log n)^2)$-competitiverandomizedalgorithmforevery$n$-point metric space.