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Here methods like random self-reducibility can be used for some specific problems to show that the worst case is no harder than the average case, or, equivalently, that the average case is no easier than the worst case. On the other hand, some data structures like hash tables have very poor worst-case behaviors, but a well written hash table of ...
The worst-case is when the pivot is always the largest or smallest value in the list (so the list is never divided). In this case, the algorithm takes time O(). If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is (). The best case occurs when each pivoting divides the list ...
Average-case computational hardness assumptions are useful for proving average-case hardness in applications like statistics, where there is a natural distribution over inputs. [22] Additionally, the planted clique hardness assumption has also been used to distinguish between polynomial and quasi-polynomial worst-case time complexity of other ...
Big O notation is a convenient way to express the worst-case scenario for a given algorithm, although it can also be used to express the average-case — for example, the worst-case scenario for quicksort is O(n 2), but the average-case run-time is O(n log n).
To obtain the average-case complexity, given an input distribution, the expected time of an algorithm is evaluated, whereas for the almost-always complexity estimate, it is evaluated that the algorithm admits a given complexity estimate that almost surely holds.
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.
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Amortized analysis requires knowledge of which series of operations are possible. This is most commonly the case with data structures, which have state that persists between operations. The basic idea is that a worst-case operation can alter the state in such a way that the worst case cannot occur again for a long time, thus "amortizing" its cost.