Time Efficient Implementation for Online $k$-server Problem on Trees

TL;DR

This work addresses the online $k$-server problem on trees, aiming to reduce per-query time without sacrificing the optimal deterministic competitive ratio. It leverages virtual-tree compression and fast Lowest Common Ancestor (LCA) and Level Ancestor (LA) techniques to guide server movements. The authors establish an $O(n)$ preprocessing cost and an $O(k \log k)$ per-query cost, with the crucial property that query time is independent of the tree size $n$. These results yield a practical, faster implementation compared to prior $O(k^2 + k log n)$ and $O(k log^2 n)$ schemes, while leaving open the possibility of achieving $O(k)$ per-query time.

Abstract

We consider online algorithms for the $k$-server problem on trees of size $n$. Chrobak and Larmore proposed a $k$-competitive algorithm for this problem that has the optimal competitive ratio. However, the existing implementations have $O\left(k^2 + k\cdot \log n\right)$ or $O\left(k(\log n)^2\right)$ time complexity for processing a query, where $n$ is the number of nodes. We propose a new time-efficient implementation of this algorithm that has $O(n)$ time complexity for preprocessing and $O\left(k\log k\right)$ time for processing a query. The new algorithm is faster than both existing algorithms and the time complexity for query processing does not depend on the tree size.

Figures (2)

• Figure 1: Subtrees of the $par=LCA(u,v)$ node.
• Figure 2: The virtual tree $T^{virt}$ for the tree $T$ and set $V^s=\{3,5,6,13,14\}$.

Theorems & Definitions (14)

• lemma 1: bfc2000
• lemma 2: ah2000bf2004d1991
• theorem 1
• proof
• lemma 3
• proof
• lemma 4
• proof
• lemma 5
• proof