Knowledge State Algorithms and the 2-Server Problem

Abstract

Abstract We introduce the novel concept of knowledge states. A knowledge state simply states conditional obligations of an adversary, by fixing a work function, and gives a distribution for the algorithm. When a knowledge state algorithm receives a request, it then calculates one or more “subsequent” knowledge states, together with a probability of transition to each. The algorithm then uses randomization to select one of those subsequents to be the new knowledge state. Although the formal definition of knowledge state algorithms appears, as yet, in no publication, many well-known algorithms can be viewed as knowledge state algorithms. We have used optimization techniques to construct a non-trivial knowledge state algorithm for the 2-server problem on the line with competitive ratio 71 36 ≈ 1.972 . As much as one avenue of investigation might be to further improve this result for the line, what we really envision is to use this technique to finally settle the question of whether there exists an online algorithm with competitive ratio better than 2 for general spaces – a notorious open problen in online algorithms. For progress in that direction, we consider the class of metric spaces M 2 , 4 , which consists of all metric spaces where every distance is either 1 or 2, and where the perimeter of every triangle is either 3 or 4. Using the knowledge state approach we are able to obtain the result that there exists a C-competitive randomized online algorithm for the 2-server problem in the space M 2 , 4 with C = 173 + 137 112 ≈ 1.649149106 .