A Fast Algorithm for Online k-servers Problem on Trees

Khadiev, Kamil; Yagafarov, Maxim

doi:10.48550/arxiv.2006.00605

Cited by 1 publication

(2 citation statements)

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“…However, it is too slow to apply it to the case of a tree. In the case of a tree, there exists an algorithm with time complexity O(n) for preprocessing and O k(log n) 2 for each query [22].…”

Section: Introductionmentioning

confidence: 99%

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Fast Classical and Quantum Algorithms for Online $k$-server Problem on Trees

Kapralov¹,

Khadiev²,

Mokut³

et al. 2020

Preprint

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We consider online algorithms for the k-server problem on trees. Chrobak and Larmore proposed a k-competitive algorithm for this problem that has the optimal competitive ratio. However, a naive implementation of their algorithm has O(n) time complexity for processing each query, where n is the number of nodes in the tree. We propose a new time-efficient implementation of this algorithm that has O(n log n) time complexity for preprocessing and O k 2 + k • log n time for processing a query. We also propose a quantum algorithm for the case where the nodes of the tree are presented using string paths. In this case, no preprocessing is needed, and the time complexity for each query is O(k 2 √ n log n). When the number of queries is o√ n k 2 log n , we obtain a quantum speedup on the total runtime compared to our classical algorithm. Our algorithm builds on a result of independent interest: we give a quantum algorithm to find the first marked element in a collection of m objects, that works even in the presence of two-sided bounded errors on the input oracle. It has worst-case complexity O( √ m). In the particular case of one-sided errors on the input, it has expected time complexity O( √ x)where x is the position of the first marked element.

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Section: Introductionmentioning

confidence: 99%

“…It is based on fast algorithms for computing Lowest Common Ancestor (LCA) [9,7] and the binary lifting technique [8]. Compared to [22], the idea of our algorithm is simpler: it has less efficient preprocessing and more efficient processing of a query when k = o (log n) 2 .…”

Section: Introductionmentioning

confidence: 99%