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Concept
Asymptotically optimal algorithm
Summary
In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor (independent of the input size) worse than the best possible algorithm. It is a term commonly encountered in computer science research as a result of widespread use of big-O notation.
More formally, an algorithm is asymptotically optimal with respect to a particular resource if the problem has been proven to require Ω(f(n)) of that resource, and the algorithm has been proven to use only O(f(n)).
These proofs require an assumption of a particular model of computation, i.e., certain restrictions on operations allowable with the input data.
As a simple example, it's known that all comparison sorts require at least Ω(n log n) comparisons in the average and worst cases. Mergesort and heapsort are comparison sorts which perform O(n log n) comparisons, so they are asymptotically optimal in this sense.
If the input data have some a priori properties which can be exploited in construction of algorithms, in addition to comparisons, then asymptotically faster algorithms may be possible. For example, if it is known that the N objects are integers from the range [1, N], then they may be sorted O(N) time, e.g., by the bucket sort.
A consequence of an algorithm being asymptotically optimal is that, for large enough inputs, no algorithm can outperform it by more than a constant factor. For this reason, asymptotically optimal algorithms are often seen as the "end of the line" in research, the attaining of a result that cannot be dramatically improved upon. Conversely, if an algorithm is not asymptotically optimal, this implies that as the input grows in size, the algorithm performs increasingly worse than the best possible algorithm.
In practice it's useful to find algorithms that perform better, even if they do not enjoy any asymptotic advantage. New algorithms may also present advantages such as better performance on specific inputs, decreased use of resources, or being simpler to describe and implement.
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