Formal Definitions
Formally, suppose that we have a lower-bound theorem showing that a problem requires Ω(f(n)) time to solve for an instance (input) of size n (see big-O notation for the definition of Ω). Then, an algorithm which solves the problem in O(f(n)) time is said to be asymptotically optimal. This can also be expressed using limits: suppose that b(n) is a lower bound on the running time, and a given algorithm takes time t(n). Then the algorithm is asymptotically optimal if:
Note that this limit, if it exists, is always at least 1, as t(n) ≥ b(n).
Although usually applied to time efficiency, an algorithm can be said to use asymptotically optimal space, random bits, number of processors, or any other resource commonly measured using big-O notation.
Sometimes vague or implicit assumptions can make it unclear whether an algorithm is asymptotically optimal. For example, a lower bound theorem might assume a particular abstract machine model, as in the case of comparison sorts, or a particular organization of memory. By violating these assumptions, a new algorithm could potentially asymptotically outperform the lower bound and the "asymptotically optimal" algorithms.
Read more about this topic: Asymptotically Optimal Algorithm
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