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The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for ...
The complexity of an algorithm is usually taken to be its worst-case complexity unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms . To show an upper bound T ( n ) {\displaystyle T(n)} on the time complexity of a problem, one needs to show only that there is a particular algorithm with ...
Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, () below stands in for the complexity of the chosen multiplication algorithm.
In addition to the supervised learning setting, sample complexity is relevant to semi-supervised learning problems including active learning, [7] where the algorithm can ask for labels to specifically chosen inputs in order to reduce the cost of obtaining many labels.
As stated above, the complexity of finding a convex hull as a function of the input size n is lower bounded by Ω(n log n). However, the complexity of some convex hull algorithms can be characterized in terms of both input size n and the output size h (the number of points in the hull). Such algorithms are called output-sensitive algorithms.
Gaussian complexity is a similar complexity with similar physical meanings, and can be obtained from the Rademacher complexity using the random variables instead of , where are Gaussian i.i.d. random variables with zero-mean and variance 1, i.e. (,). Gaussian and Rademacher complexities are known to be equivalent up to logarithmic factors.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations N as the result of input size n for each function. In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm.
Algorithmic information theory principally studies complexity measures on strings (or other data structures).Because most mathematical objects can be described in terms of strings, or as the limit of a sequence of strings, it can be used to study a wide variety of mathematical objects, including integers.