A primal–dual online algorithm for the k-server problem on weighted HSTs

In this paper, we show that there is a 52ℓ·ln(1+k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{5}{2}\ell \cdot \ln (1+k)$$\end{document}-competitive randomized algorithm for the k-sever problem on weighted Hierarchically Separated Trees (HSTs) with depth ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} when n=k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=k+1$$\end{document} where n is the number of points in the metric space, which improved previous best competitive ratio 12ℓln(1+4ℓ(1+k))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$12 \ell \ln (1+4\ell (1+k))$$\end{document} by Bansal et al. (FOCS, pp 267–276, 2011).