We consider the k-server problem on trees and HSTs. We give an algorithm based on Bregman projections. This algorithm has a competitive ratios that match some of the recent results given by Bubeck et al. (STOC 2018), whose algorithm was based on mirror-descent-based continuous dynamics prescribed via a differential inclusion.
Cite
@article{arxiv.1810.07508,
title = {$k$-Servers with a Smile: Online Algorithms via Projections},
author = {Niv Buchbinder and Anupam Gupta and Marco Molinaro and Joseph and Naor},
journal= {arXiv preprint arXiv:1810.07508},
year = {2018}
}