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In computer science, best, worst, and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively.Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource.
The average-case performance of algorithms has been studied since modern notions of computational efficiency were developed in the 1950s. Much of this initial work focused on problems for which worst-case polynomial time algorithms were already known. [3]
The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n {\displaystyle n} may be faster to solve than others, we define the following complexities:
Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed as a function of the size of the input.
The order of growth (e.g. linear, logarithmic) of the worst-case complexity is commonly used to compare the efficiency of two algorithms. The worst-case complexity of an algorithm should be contrasted with its average-case complexity, which is an average measure of the amount of resources the algorithm uses on a random input.
Comparison column has the following ranking classifications: "Best", "Average" and "Worst" if the time complexity is given for each case. "Memory" denotes the amount of additional storage required by the algorithm. The run times and the memory requirements listed are inside big O notation, hence the base of the logarithms does not matter.
For non-probabilistic, more specifically deterministic, algorithms, the most common types of complexity estimates are the average-case complexity and the almost-always complexity. 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 ...
Finer computations of the average time complexity yield a worst case of (+ + ()) + for random pivots (in the case of the median; other k are faster). [3] The constant can be improved to 3/2 by a more complicated pivot strategy, yielding the Floyd–Rivest algorithm , which has average complexity of 1.5 n + O ( n 1 / 2 ) {\displaystyle 1.5n ...