Randomized $k$-server in polynomial time

2026-05-02

AI Summary

This work addresses the challenge that existing randomized $k$-server algorithms rely on exponentially large configuration distributions, rendering them impractical for explicit implementation. The authors propose a black-box derandomization framework based on hierarchical separating trees, which compresses any randomized algorithm to use only $O(\log k)$ random bits. By sparsifying the configuration distribution, they further transform the algorithm into an efficient variant that maintains only a polynomial number of configurations. This approach yields the first randomized $k$-server algorithm with polynomial time complexity and a polylogarithmic competitive ratio on arbitrary $n$-point metric spaces. The method substantially reduces both the randomness required and the configuration complexity, thereby advancing the study of advice complexity in online algorithms.

Abstract

We study the design of computationally efficient randomized algorithms for the $k$-server problem. Existing randomized algorithms with the best known competitive ratios are, on the one hand, inherently implicit and, on the other hand, employ a rounding scheme that maintains a distribution over exponentially many configurations. In this work, we introduce a derandomization framework that transforms any randomized $k$-server algorithm on a hierarchically separated tree into one that uses only $O(\log k)$ random bits for request sequences of arbitrary length; hence maintaining a distribution over only polynomially many server configurations. Leveraging this black-box derandomization, we obtain the first polynomial-time randomized $k$-server algorithm on arbitrary $n$-point metrics with a polylogarithmic competitive ratio. Our results also have implications for the advice complexity of the $k$-server problem.

Problem

• k-server problem
• randomized algorithms
• computational efficiency
• competitive ratio
• derandomization

Innovation

• k-server problem
• derandomization
• hierarchically separated tree
• polynomial-time algorithm
• competitive ratio