Slaying Hydrae: Improved Bounds for Generalized k-Server in Uniform Metrics

Authors Marcin Bienkowski , Łukasz Jeż , Paweł Schmidt

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Author Details

Marcin Bienkowski
  • Institute of Computer Science, University of Wrocław, Poland
Łukasz Jeż
  • Institute of Computer Science, University of Wrocław, Poland
Paweł Schmidt
  • Institute of Computer Science, University of Wrocław, Poland

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Marcin Bienkowski, Łukasz Jeż, and Paweł Schmidt. Slaying Hydrae: Improved Bounds for Generalized k-Server in Uniform Metrics. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The generalized k-server problem is an extension of the weighted k-server problem, which in turn extends the classic k-server problem. In the generalized k-server problem, each of k servers s_1, ..., s_k remains in its own metric space M_i. A request is a tuple (r_1,...,r_k), where r_i in M_i, and to service it, an algorithm needs to move at least one server s_i to the point r_i. The objective is to minimize the total distance traveled by all servers. In this paper, we focus on the generalized k-server problem for the case where all M_i are uniform metrics. We show an O(k^2 * log k)-competitive randomized algorithm improving over a recent result by Bansal et al. [SODA 2018], who gave an O(k^3 * log k)-competitive algorithm. To this end, we define an abstract online problem, called Hydra game, and we show that a randomized solution of low cost to this game implies a randomized algorithm to the generalized k-server problem with low competitive ratio. We also show that no randomized algorithm can achieve competitive ratio lower than Omega(k), thus improving the lower bound of Omega(k / log^2 k) by Bansal et al.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • k-server
  • generalized k-server
  • competitive analysis


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