Randomized algorithm for the k-server problem on decomposable spaces

We study the randomized k-server problem on metric spaces consisting of widely separated subspaces. We give a method which extends existing algorithms to larger spaces with the growth rate of the competitive quotients being at most O(log k). This method yields o(k)-competitive algorithms solving the randomized k-server problem, for some special underlying metric spaces, e.g. HSTs of “small” height (but unbounded degree). HSTs are important tools for probabilistic approximation of metric spaces.

💡 Research Summary

The paper addresses the randomized k‑server problem in metric spaces that consist of well‑separated subspaces, a setting the authors formalize as µ‑decomposable spaces. In such a space the points are partitioned into t blocks B₁,…,B_t; each block has diameter at most δ, while any two points belonging to different blocks are at distance exactly Δ, with the separation ratio µ = Δ/δ satisfying µ ≥ k. This structure captures important families of metrics, notably µ‑hierarchically separated trees (µ‑HSTs) of bounded height, which are central to probabilistic metric embeddings.

The authors start from an arbitrary randomized online k‑server algorithm A that works on a single block and has a performance guarantee of the form

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