Metrical Task Systems and the k-Server Problem on HSTs

• Published in: Automata, Languages and Programming pp. 287 - 298
• Main Authors: Bansal, Nikhil, Buchbinder, Niv, Naor, Joseph (Seffi)
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Allocation Problem; Competitive Algorithm; Competitive Ratio; Dual Variable; Online Algorithm
• ISBN: 3642141641; 9783642141645
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-14165-2_25

We consider the randomized k-server problem, and give improved results for various metric spaces. In particular, we extend a recent result of Coté et al [15] for well-separated binary Hierarchically Separated Trees (HSTs) to well-separated d-ary HSTs for poly-logarithmic values of d. One application of this result is an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rm exp}(O(\sqrt{\log \log k \log n}))$\end{document}-competitive algorithm for k-server on n uniformly spaced points on a line. This substantially improves upon the prior guarantee of O( min (k,n2/3) for this metric [16]. These results are based on obtaining a refined guarantee for the unfair metrical task systems problem on an HST. Prior to our work, such a guarantee was only known for the case of a uniform metric [5,7,18]. Our results are based on the primal-dual approach for online algorithms. Previous primal-dual approaches in the context of k-server and MTS [2,4,3] worked only for uniform or weighted star metrics, and the main technical contribution here is to extend many of these techniques to work directly on HSTs.