On-Line Algorithms for Robot Navigation and Server Problems

Many classical problems of computer science such as paging, scheduling, and maintaining dynamic data structures are naturally on-line; an algorithm for such a problem is constantly making irrevocable decisions without knowing what its future input will be. The competitive analysis of on-line algorithms was brought into prominence by the work of Sleator and Tarjan in 1985 as a theoretical framework in which to measure the performance of such algorithms. Since then, a variety of on-line problems have been studied from this perspective. We consider two such problems in this setting: robot navigation, and computationally efficient algorithms for the k-server problem of Manasse, McGeoch, and Sleator. For the first of these, we give algorithms for a robot searching for a goal in an unknown simple polygon; our approach can be viewed as an extension of some of the strategies of Baeza-Yates, Culberson, and Rawlins to a more general class of geometric search spaces. We also provide the first competitive analysis of the robot localization problem a fundamental task for an autonomous mobile robot, in which it must determine its location in a known environment. Finally, we address the general question of how the performance of an on-line algorithm varies with the computational resources it has at its disposal. Within the context of the k-server problem, a natural specialization of this question is the following: can an algorithm which uses only constant space, and constant time in processing each request, match the performance of algorithms which are allowed to perform an arbitrary amount of computation at each step? We show a non-trivial lower bound for the class of 2-server balancing algorithms, a subclass of the constant-time algorithms; this represents one of the first lower bounds for an on-line algorithm based solely on its computational resources. Thesis Supervisor: Michel X. Goemans Title: Assistant Professor of Applied Mathematics