Competitive Analysis (online Algorithm)

Competitive analysis is a method invented for analyzing online algorithms, in which the performance of an online algorithm (which must satisfy an unpredictable sequence of requests, completing each request without being able to see the future) is compared to the performance of an optimal ''offline algorithm'' that can view the sequence of requests in advance. An algorithm is ''competitive'' if its ''competitive ratio''—the ratio between its performance and the offline algorithm's performance—is bounded. Unlike traditional worst-case analysis, where the performance of an algorithm is measured only for "hard" inputs, competitive analysis requires that an algorithm perform well both on hard and easy inputs, where "hard" and "easy" are defined by the performance of the optimal offline algorithm.

For many algorithms, performance is dependent not only on the size of the inputs, but also on their values. For example, sorting an array of elements varies in difficulty depending on the initial order. Such data-dependent algorithms are analysed for average-case and worst-case data. Competitive analysis is a way of doing worst case analysis for on-line and randomized algorithms, which are typically data dependent.

In competitive analysis, one imagines an "adversary" which deliberately chooses difficult data, to maximize the ratio of the cost of the algorithm being studied and some optimal algorithm. When considering a randomized algorithm, one must further distinguish between an ''oblivious adversary'', which has no knowledge of the random choices made by the algorithm pitted against it, and an ''adaptive adversary'' which has full knowledge of the algorithm's internal state at any point during its execution. (For a deterministic algorithm, there is no difference; either adversary can simply compute what state that algorithm must have at any time in the future, and choose difficult data accordingly.)

For example, the quicksort algor