Abstract
We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we identify a broad class of online problems for which the existence of a randomized online algorithm with constant expected competitive ratio r implies the existence of a randomized online algorithm that has a competitive ratio of (1+varepsilon )rwith high probability, measured with respect to the optimal profit or cost, respectively. The class of problems includes some of the well-studied online problems such as paging, k-server, and metrical task systems on finite metric spaces.
Highlights
In online computation, we face the challenge of designing algorithms that work in environments where parts of the input are unknown while parts of the output already need to be provided
The standard way of evaluating the quality of online algorithms is by means of competitive analysis, where one compares the outcome of an online algorithm to the optimal solution constructed by a hypothetical optimal offline algorithm
These algorithms base their computations on the outcome of a random source; for a detailed introduction to online problems we refer the reader to the literature [3,9]
Summary
We face the challenge of designing algorithms that work in environments where parts of the input are unknown while parts of the output already need to be provided. Our approach allows us to design randomized online algorithms that up to a factor of (1+ε) have the expected performance with a probability tending to 1 with a growing size of the optimal profit or cost We show that this technique is applicable for a wide range of online problems. 2.3, we provide a formal definition of the problem properties that we have identified to be crucial for designing online algorithms with high probability guarantees This enables us to provide a precise statement of the two main theorems which state that, for every problem that fulfills certain natural conditions, it is possible to transform an algorithm Rand with constant expected competitive ratio r to an algorithm Rand having a competitive ratio of (1 + ε)r with high probability – with respect to the profit (cost) of an optimal solution. Task systems offer a general view of online problems including classical problems such as paging [2,5,14], k-server [10,13], and the list update problem [14]
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