Time Efficient Implementation for Online $k$-server Problem on Trees
<|reference_start|>Time Efficient Implementation for Online $k$-server Problem on Trees: 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.<|reference_end|>
@article{khadiev2024time,
    title={Time Efficient Implementation for Online $k$-server Problem on Trees},
    author={Kamil Khadiev and Maxim Yagafarov},
    journal={TAMC 2024, LNCS 14637},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.14633},
    primaryClass={cs.DS}
}