A Fast Algorithm for Online k-servers Problem on Trees

TL;DR

This paper introduces a new, more efficient implementation of the online k-server algorithm on trees, reducing query processing time from quadratic or linear to near-linear complexity.

Contribution

The authors present a novel implementation of the online k-server algorithm on trees with significantly improved time complexities for preprocessing and query processing.

Findings

Preprocessing time reduced to O(n)

Query processing time improved to O(k(log n)^2)

Maintains optimal competitive ratio of k

Abstract

We consider online algorithms for the $k$-server problem on trees. There is a $k$-competitive algorithm for this problem, and it is the best competitive ratio. M. Chrobak and L. Larmore provided it. At the same time, the existing implementation has $O(n)$ time complexity for processing a query and $O(n)$ for prepossessing, where $n$ is the number of nodes in a tree. Another implementation of the algorithm has $O(k^2+k\log n)$ time complexity for processing a query and $O(n\log n)$ for prepossessing. We provide a new time-efficient implementation of the algorithm. It has $O(n)$ time complexity for preprocessing and $O\left(k(\log n)^2\right)$ for processing a query.