An Online Algorithm for the 2-Server Problem On The Line with Improved Competitiveness

In this thesis we present a randomized online algorithm for the 2-server problem on the line, named R-LINE (for Randomized Line). This algorithm achieves the lowest competitive ratio of any known randomized algorithm for the 2-server problem on the line. The competitiveness of R-LINE is less than 1.901. This result provides a significant improvement over the previous known competitiveness of 155/78 (approximately 1.987), by Bartal, Chrobak, and Larmore, which was the first randomized algorithm for the 2-server problem one the line with competitiveness less than 2. Taking inspiration from this algorithm,we improve this result by utilizing ideas from T-theory, game theory, and linear programming

A randomized algorithm for two servers in cross polytope spaces

It has been a long-standing open problem to determine the exact randomized competitiveness of the 2-server problem, that is, the minimum competitiveness of any randomized online algorithm for the 2-server problem. For deterministic algorithms the best competitive ratio that can be obtained is 2 and no randomized algorithm is known that improves this ratio for general spaces. For the line, Bartal et al. [2] give a 155 78 competitive algorithm, but their algorithm is specific to the geometry of the line. We consider here the 2-server problem over Cross Polytope Spaces M2,4. We obtain an algorithm with competitive ratio of 19 12, and show that this ratio is best possible. This algorithm gives the second non-trivial example of metric spaces with better than 2 competitive ratio. The algorithm uses a design technique called the knowledge state technique – a method not specific to M2,4