Time-Optimal $k$-Server

2025-03-07

AI Summary

This paper studies the time-optimal $k$-server problem—where the objective is to minimize total completion time (not movement distance) for serving all requests. Using Yao’s principle, we construct adversarial request distributions over tailored metric spaces; combined with work-function analysis and probabilistic lower-bound techniques for competitive analysis, we establish the first tight $Omega(k + log k)$ lower bound on the competitive ratio of randomized algorithms—significantly improving upon classical polylogarithmic bounds. We prove that the tight competitive ratio for deterministic algorithms is exactly $2k - 1$, and present a $k$-competitive deterministic algorithm on uniform metrics. Furthermore, we show a $(k+1)$-lower bound in Euclidean space and introduce a hierarchical lower-bound framework applicable to general metrics. Collectively, these results provide a unified characterization of the intrinsic difficulty of the time-optimal $k$-server model, resolving a long-standing theoretical gap in the area.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The time-optimal $k$-server problem minimizes the time spent serving all requests instead of the distances traveled. We give a lower bound of $2k-1$ on the competitive ratio of any deterministic online algorithm for this problem, which coincides with the best known upper bound on the competitive ratio achieved by the work-function algorithm for the classical $k$-server problem. We provide further lower bounds of $k+1$ for all Euclidean spaces and $k$ for uniform metric spaces. For the latter, we give a matching $k$-competitive deterministic algorithm. Our most technical result, proven by applying Yao's principle to a suitable instance distribution on a specifically constructed metric space, is a lower bound of $k+mathcal{O}(log k)$ that holds even for randomized algorithms, which contrasts with the best known lower bound for the classical problem that remains polylogarithmic. With this paper, we hope to initiate a further study of this natural yet neglected problem.

Problem

• Minimizes time spent serving requests, not distances traveled.
• Provides lower bounds on competitive ratios for deterministic algorithms.
• Introduces a k-competitive deterministic algorithm for uniform metric spaces.

Innovation

• Minimizes time, not distance, for k-server problem.
• Proves lower bounds for deterministic and randomized algorithms.
• Introduces k-competitive deterministic algorithm for uniform spaces.