T-Theory Applications to Online Algorithms for the Server Problem

Although largely unnoticed by the online algorithms community, T-theory, a field of discrete mathematics, has contributed to the development of several online algorithms for the k-server problem. A brief summary of the k-server problem, and some important application concepts of T-theory, are given. Additionally, a number of known k-server results are restated using the established terminology of T-theory. Lastly, a previously unpublished 3-competitiveness proof, using T-theory, for the Harmonic algorithm for two servers is presented.Comment: 19 figures 38 page

A Fast Algorithm for Online k-servers Problem on Trees

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, where $n$ is a number of nodes in a tree. 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

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

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 $O\left(k^2 + k\cdot \log n\right)$ 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 $O(k^2\sqrt{n}\log n)$. When the number of queries is $o\left(\frac{\sqrt{n}}{k^2\log n}\right)$, 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 complexity $O(\sqrt{m})$. In the particular case of one-sided errors on the input, it has expected time complexity $O(\sqrt{x})$ where $x$ is the position of the first marked element. Compare with previous work, our algorithm can handle errors in the input oracle.Comment: ICTCS202