Editing Analysis of algorithms (section)

==Constant factors==
Analysis of algorithms typically focuses on the asymptotic performance, particularly at the elementary level, but in practical applications constant factors are important, and real-world data is in practice always limited in size. The limit is typically the size of addressable memory, so on 32-bit machines 2<sup>32</sup> = 4 GiB (greater if [[segmented memory]] is used) and on 64-bit machines 2<sup>64</sup> = 16 EiB. Thus given a limited size, an order of growth (time or space) can be replaced by a constant factor, and in this sense all practical algorithms are {{math|''O''(1)}} for a large enough constant, or for small enough data.

This interpretation is primarily useful for functions that grow extremely slowly: (binary) [[iterated logarithm]] (log<sup>*</sup>) is less than 5 for all practical data (2<sup>65536</sup> bits); (binary) log-log (log log ''n'') is less than 6 for virtually all practical data (2<sup>64</sup> bits); and binary log (log ''n'') is less than 64 for virtually all practical data (2<sup>64</sup> bits). An algorithm with non-constant complexity may nonetheless be more efficient than an algorithm with constant complexity on practical data if the overhead of the constant time algorithm results in a larger constant factor, e.g., one may have <math>K > k \log \log n</math> so long as <math>K/k > 6</math> and <math>n < 2^{2^6} = 2^{64}</math>.

For large data linear or quadratic factors cannot be ignored, but for small data an asymptotically inefficient algorithm may be more efficient. This is particularly used in [[hybrid algorithm]]s, like [[Timsort]], which use an asymptotically efficient algorithm (here [[merge sort]], with time complexity <math>n \log n</math>), but switch to an asymptotically inefficient algorithm (here [[insertion sort]], with time complexity <math>n^2</math>) for small data, as the simpler algorithm is faster on small data.