# The Mobile Server Problem

## TL;DR

This paper introduces the mobile server problem, a variant of page migration, analyzing online algorithms' competitiveness and proposing a simple algorithm with near-optimal competitive ratios under resource augmentation.

## Contribution

The paper formalizes the mobile server problem, proves limitations of online algorithms, and presents a simple, effective algorithm with tight competitive ratios for various problem variants.

## Key findings

- No online algorithm can be competitive without resource augmentation.
- A resource augmentation approach achieves near-optimal competitive ratios.
- A simple deterministic algorithm performs well across multiple variants.

## Abstract

We introduce the mobile server problem, inspired by current trends to move computational tasks from cloud structures to multiple devices close to the end user. An example for this are embedded systems in autonomous cars that communicate in order to coordinate their actions. Our model is a variant of the classical Page Migration Problem. More formally, we consider a mobile server holding a data page. The server can move in the Euclidean space (of arbitrary dimension). In every round, requests for data items from the page pop up at arbitrary points in the space. The requests are served, each at a cost of the distance from the requesting point and the server, and the mobile server may move, at a cost $D$ times the distance traveled for some constant $D$. We assume a maximum distance $m$ the server is allowed to move per round. We show that no online algorithm can achieve a competitive ratio independent of the length of the input sequence in this setting. Hence we augment the maximum movement distance of the online algorithms to $(1+\delta)$ times the maximum distance of the offline solution. We provide a deterministic algorithm which is simple to describe and works for multiple variants of our problem. The algorithm achieves almost tight competitive ratios independent of the length of the input sequence. Our Algorithm also achieves a constant competitive ratio without resource augmentation in a variant where the distance between two consecutive requests is restricted to a constant smaller than the limit for the server.