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. One such example is the quicksort algorithm, which sorts an array of elements. 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" that deliberately chooses difficult data, to maximize the ratio of the cost of the algorithm being studied and some optimal algorithm. Adversaries range in power from the oblivious adversary, which has no knowledge of the random choices made by the algorithm pitted against it, to the adaptive adversary that has full knowledge of how an algorithm works and its internal state at any point during its execution. Note that this distinction is only meaningful for randomized algorithms. For a deterministic algorithm, 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 algorithm chooses one element, called the "pivot", that is, on average, not too far from the center value of the data being sorted. Quicksort then separates the data into two piles, one of which contains all elements with value less than the value of the pivot, and the other containing the rest of the elements. If quicksort chooses the pivot in some deterministic fashion (for instance, always choosing the first element in the list), then it is easy for an adversary to arrange the data beforehand so that quicksort will perform in worst-case time. If, however, quicksort chooses some random element to be the pivot, then an adversary without knowledge of what random numbers are coming up cannot arrange the data to guarantee worst-case execution time for quicksort.

...
Wikipedia