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

Abstract

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 time complexity O(n) to process each request, where n is the number of nodes. 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 request. 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 request is O(k 2√ n log n). When the number of requests is o √n k2 , we obtain a quantum speed-up on the total runtime compared to our classical algorithm. We also give a simple 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 query complexity O( √ m). In the particular case of one-sided errors on the input, it has expected query complexity O( √ x) where x is the position of the first marked element. Compared with previous work, our algorithm can handle errors in the input oracle.