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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 ...
Worst-case complexity: This is the complexity of solving the problem for the worst input of size . The order from cheap to costly is: Best, average (of discrete uniform distribution), amortized, worst. For example, the deterministic sorting algorithm quicksort addresses the problem of sorting a list of integers. The worst-case is when the pivot ...
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).
A linear search runs in linear time in the worst case, and makes at most n comparisons, where n is the length of the list. If each element is equally likely to be searched, then linear search has an average case of n+1 / 2 comparisons, but the average case can be affected if the search probabilities for each element vary.
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.
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.
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Finding an item in an unsorted list or a malformed tree (worst case) or in an unsorted array; Adding two n-bit integers by ripple carry. () linearithmic, loglinear, or quasilinear: Performing a Fast Fourier transform; heapsort, quicksort (best and average case), or merge sort quadratic