Making Decisions in Spite of Ignorance: Annotated Course Summary

The introduction to competitive analysis presented in Borodin et al. [1] (Section 1.1) is enlightening and provides a good theoretical foundation to the concepts introduced during the first lecture of the course. Here, useful connections are made between offline and online problems (and their algorithms, which is another useful distinction). In many cases, the offline algorithm can be seen as a reference implementation for how to solve a problem in an optimal way, and is used when assessing an algorithm for the corresponding online problem, which is understood to be (potentially much) harder. In fact, this comparison between the offline and online algorithms is formalized into a way of thinking in the concept of an adversary. Furthermore, online algorithms that perform “well”, i.e. which attain a constant ratio c (c≥ 1 in all cases) compared to the optimal algorithm, are considered to be competitive. Seen from this angle, the concepts are rather intuitive. It is important to keep the relative nature of the competitive ratio measure in mind when faced with algorithms that are competitive in regards to a problem parameter n; such an ncompetitive algorithm does not necessarily degrade in performance when n increases, only in regard to the offline adversary. So, online paging algorithms (see Section 5) that are k-competitive, with k being the page capacity of the cache, do not perform worse as k is increased (quite the opposite), but the adversary will perfor better and better. Since we are dealing with a malicious adversary intent on making life difficult for the hapless searcher, we can silently wonder whether the outcome will be different depending on whether the adversary makes his choice about the input to the algorithm before or during the search. However, if we restrict our views to deterministic algorithms, it is clear that since we are looking for the worst-case scenario, the adversary can do no worse if he waits a while to define the input fully than if he decides the input before the search starts; the searcher will perform a fixed search strategy in any case. As Borodin suggests, this can be thought of as turning the adversary and the optimal offline algorithm into a single entity, the offline player, which is in line with our previous thinking.