On Randomized Memoryless Algorithms for the Weighted $k$-server Problem

TL;DR

This paper establishes tight bounds for randomized memoryless algorithms solving the weighted k-server problem on uniform spaces, introducing a framework for analyzing their competitive ratios and demonstrating the first such algorithms for k>2.

Contribution

It provides the first memoryless algorithms with proven competitive ratios for the weighted k-server problem on uniform spaces and introduces a novel potential function analysis framework.

Findings

Existence of an α_k-competitive memoryless algorithm for the problem.

Proved no memoryless algorithm can have a ratio better than α_k.

Harmonic algorithm has a competitive ratio of kα_k.

Abstract

The weighted $k$-server problem is a generalization of the $k$-server problem in which the cost of moving a server of weight $\beta_i$ through a distance $d$ is $\beta_i\cdot d$. The weighted server problem on uniform spaces models caching where caches have different write costs. We prove tight bounds on the performance of randomized memoryless algorithms for this problem on uniform metric spaces. We prove that there is an $\alpha_k$-competitive memoryless algorithm for this problem, where $\alpha_k=\alpha_{k-1}^2+3\alpha_{k-1}+1$; $\alpha_1=1$. On the other hand we also prove that no randomized memoryless algorithm can have competitive ratio better than $\alpha_k$. To prove the upper bound of $\alpha_k$ we develop a framework to bound from above the competitive ratio of any randomized memoryless algorithm for this problem. The key technical contribution is a method for working with potential functions defined implicitly as the solution of a linear system. The result is robust in the sense that a small change in the probabilities used by the algorithm results in a small change in the upper bound on the competitive ratio. The above result has two important implications. Firstly this yields an $\alpha_k$-competitive memoryless algorithm for the weighted $k$-server problem on uniform spaces. This is the first competitive algorithm for $k>2$ which is memoryless. Secondly, this helps us prove that the Harmonic algorithm, which chooses probabilities in inverse proportion to weights, has a competitive ratio of $k\alpha_k$.