Editing Big O notation (section)

=== Extensions to the Bachmann–Landau notations ===
Another notation sometimes used in computer science is ''[[Õ]]'' (read ''soft-O''), which hides polylogarithmic factors. There are two definitions in use: some authors use ''f''(''n'')&nbsp;=&nbsp;''Õ''(''g''(''n'')) as [[shorthand]] for {{nowrap|1=''f''(''n'') = ''O''(''g''(''n'') [[Polylogarithmic function|log<sup>''k''</sup> ''n'']])}} for some ''k'', while others use it as shorthand for {{nowrap|1=''f''(''n'') = ''O''(''g''(''n'') log<sup>''k''</sup> ''g''(''n''))}}.<ref>{{Cite book |last1=Cormen |first1=Thomas H. |url=https://mitpress.mit.edu/9780262046305/introduction-to-algorithms/ |title=Introduction to Algorithms |last2=Leiserson |first2=Charles E. |last3=Rivest |first3=Ronald L. |last4=Stein |first4=Clifford |publisher=The MIT Press |year=2022 |isbn=9780262046305 |edition=4th |location=Cambridge, Mass. |pages=74–75 |oclc= |url-access=}}</ref> When {{Nowrap|''g''(''n'')}} is polynomial in ''n'', there is no difference; however, the latter definition allows one to say, e.g. that <math>n2^n = \tilde O(2^n)</math> while the former definition allows for <math>\log^k n = \tilde O(1)</math> for any constant ''k''. Some authors write ''O''<sup>*</sup> for the same purpose as the latter definition.<ref>{{cite journal | url=https://www.cs.helsinki.fi/u/mkhkoivi/publications/sicomp-2009.pdf | author=Andreas Björklund and Thore Husfeldt and Mikko Koivisto | title=Set partitioning via inclusion-exclusion | journal=[[SIAM Journal on Computing]] | volume=39 | number=2 | pages=546&ndash;563 | year=2009 | doi=10.1137/070683933 | access-date=2022-02-03 | archive-date=2022-02-03 | archive-url=https://web.archive.org/web/20220203095918/https://www.cs.helsinki.fi/u/mkhkoivi/publications/sicomp-2009.pdf | url-status=live }} See sect.2.3, p.551.</ref> Essentially, it is big ''O'' notation, ignoring [[Polylogarithmic function|logarithmic factors]] because the [[Asymptotic analysis|growth-rate]] effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since log<sup>''k''</sup>&nbsp;''n'' is always ''o''(''n''<sup>ε</sup>) for any constant ''k'' and any {{nowrap|''ε'' > 0}}).

Also, the [[L-notation|''L'' notation]], defined as
:<math>L_n[\alpha,c] = e^{(c + o(1))(\ln n)^\alpha(\ln\ln n)^{1-\alpha}},</math>
is convenient for functions that are between [[Time complexity#Polynomial time|polynomial]] and [[Time complexity#Exponential time|exponential]] in terms of {{nowrap|<math>\ln n</math>.}}