Metrical task systems on trees via mirror descent and unfair gluing

We consider metrical task systems on tree metrics, and present an O(depth × 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 loglog n in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an O((log n)2)-competitive randomized algorithm for every n-point metric space.