|
|
| Line 1: |
Line 1: |
| {{For|the theorem in probability theory|Law of the iterated logarithm}}
| | <br><br>I woke up yesterday [http://www.banburycrossonline.com luke bryan concert] and realized - I've been single for a little while today and after much intimidation [http://lukebryantickets.pyhgy.com luke bryan tour with] from buddies I today find myself signed-up for internet dating. They promised me that there are lots of regular, pleasant and [https://Www.google.com/search?hl=en&gl=us&tbm=nws&q=enjoyable+individuals&btnI=lucky enjoyable individuals] to meet up, therefore here goes the pitch!<br>My household and buddies are awesome and spending some time with them at pub gigabytes or dishes is always essential. As I find that one may not own a [https://www.Vocabulary.com/dictionary/nice+dialog nice dialog] using the sound I have never been into night clubs. Additionally, I got 2 very adorable and definitely cheeky canines who are almost always excited to meet new people.<br>I strive to stay as physically fit as possible staying at the fitness center several times a week. I love my athletics and try to perform or see while many a possible. I will often at Hawthorn suits being wintertime. Note: I've seen luke bryan on Sale dates - [http://www.museodecarruajes.org http://www.museodecarruajes.org] - the carnage of wrestling matches at stocktake revenue, In case that you would contemplated purchasing a sport I really do not mind.<br><br>My homepage :: [http://minioasis.com luke bryan concerts 2014] |
| In [[computer science]], the '''iterated logarithm''' of ''n'', written {{log-star}} ''n'' (usually read "'''log star'''"), is the number of times the [[logarithm]] function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition is the result of this recursive function:
| |
| | |
| :<math>
| |
| \log^* n :=
| |
| \begin{cases}
| |
| 0 & \mbox{if } n \le 1; \\
| |
| 1 + \log^*(\log n) & \mbox{if } n > 1
| |
| \end{cases} | |
| </math>
| |
| | |
| On the positive real numbers, the continuous [[super-logarithm]] (inverse [[tetration]]) is essentially equivalent:
| |
| :<math>\log^* n = \lceil \text{slog}_e(n) \rceil</math> | |
| but on the negative real numbers, log-star is 0, whereas <math>\lceil \text{slog}_e(-x)\rceil = -1</math> for positive ''x'', so the two functions differ for negative arguments.
| |
| | |
| [[Image:Iterated logarithm.png|right|300px|thumb|'''Figure 1.''' Demonstrating lg* 4 = 2]]
| |
| In computer science, {{lg-star}} is often used to indicate the binary iterated logarithm, which iterates the [[binary logarithm]] instead. The iterated logarithm accepts any positive [[real number]] and yields an [[integer]]. Graphically, it can be understood as the number of "zig-zags" needed in Figure 1 to reach the interval [0, 1] on the ''x''-axis.
| |
| | |
| Mathematically, the iterated logarithm is well-defined not only for base 2 and base ''e'', but for any base greater than <math>e^{1/e}\approx1.444667</math>.
| |
| | |
| ==Analysis of algorithms==
| |
| The iterated logarithm is useful in [[analysis of algorithms]] and [[computational complexity theory|computational complexity]], appearing in the time and space complexity bounds of some algorithms such as:
| |
| | |
| * Finding the [[Delaunay triangulation]] of a set of points knowing the [[Euclidean minimum spanning tree]]: randomized [[Big-O notation|O]](''n'' {{log-star}} ''n'') time<ref>Olivier Devillers, "Randomization yields simple O(n log* n) algorithms for difficult ω(n) problems.". ''International Journal of Computational Geometry & Applications'' '''2''':01 (1992), pp. 97–111.</ref>
| |
| * [[Fürer's algorithm]] for integer multiplication: O(''n'' log ''n'' 2<sup>{{lg-star}} ''n''</sup>)
| |
| * Finding an approximate maximum (element at least as large as the median): {{lg-star}} ''n'' − 4 to {{lg-star}} ''n'' + 2 parallel operations<ref>Noga Alon and Yossi Azar, "Finding an Approximate Maximum". ''SIAM Journal of Computing'' '''18''':2 (1989), pp. 258–267.</ref>
| |
| * Richard Cole and [[Uzi Vishkin]]'s [[Graph coloring#Parallel_and_distributed_algorithms|distributed algorithm for 3-coloring an ''n''-cycle]]: ''O''({{log-star}} ''n'') synchronous communication rounds.<ref>Richard Cole and Uzi Vishkin: "Deterministic coin tossing with applications to optimal parallel list ranking", Information and Control 70:1(1986), pp. 32–53.</ref><ref>{{Introduction to Algorithms|1}} Section 30.5.</ref>
| |
| | |
| The iterated logarithm grows at an extremely slow rate, much slower than the logarithm itself. For all values of ''n'' relevant to counting the running times of algorithms implemented in practice (i.e., ''n'' ≤ 2<sup>65536</sup>, which is far more than the atoms in the known universe), the iterated logarithm with base 2 has a value no more than 5.
| |
| | |
| {|class=wikitable
| |
| ! ''x'' !! {{lg-star}} ''x''
| |
| |-
| |
| | (−∞, 1] || 0
| |
| |-
| |
| | (1, 2] || 1
| |
| |-
| |
| | (2, 4] || 2
| |
| |-
| |
| | (4, 16] || 3
| |
| |-
| |
| | (16, 65536] || 4
| |
| |-
| |
| | (65536, 2<sup>65536</sup>] || 5
| |
| |}
| |
| | |
| Higher bases give smaller iterated logarithms. Indeed, the only function commonly used in complexity theory that grows more slowly is the [[Ackermann function#Inverse|inverse Ackermann function]].
| |
| | |
| ==Other applications==
| |
| The iterated logarithm is closely related to the generalized logarithm function used in [[symmetric level-index arithmetic]]. It is also proportional to the additive [[persistence of a number]], the number of times one must replace the number by the sum of its digits before reaching its [[digital root]].
| |
| | |
| Santhanam<ref>[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.2392 On Separators, Segregators and Time versus Space]</ref> shows that [[DTIME]] and [[NTIME]] are distinct up to <math>n\sqrt{\log^*n}.</math>
| |
| | |
| ==Notes==
| |
| | |
| {{reflist}}
| |
| | |
| ==References==
| |
| | |
| *{{Introduction to Algorithms|2|chapter=3.2: Standard notations and common functions|pages=55–56}}
| |
| | |
| [[Category:Asymptotic analysis]]
| |
| [[Category:Logarithms]] | |
I woke up yesterday luke bryan concert and realized - I've been single for a little while today and after much intimidation luke bryan tour with from buddies I today find myself signed-up for internet dating. They promised me that there are lots of regular, pleasant and enjoyable individuals to meet up, therefore here goes the pitch!
My household and buddies are awesome and spending some time with them at pub gigabytes or dishes is always essential. As I find that one may not own a nice dialog using the sound I have never been into night clubs. Additionally, I got 2 very adorable and definitely cheeky canines who are almost always excited to meet new people.
I strive to stay as physically fit as possible staying at the fitness center several times a week. I love my athletics and try to perform or see while many a possible. I will often at Hawthorn suits being wintertime. Note: I've seen luke bryan on Sale dates - http://www.museodecarruajes.org - the carnage of wrestling matches at stocktake revenue, In case that you would contemplated purchasing a sport I really do not mind.
My homepage :: luke bryan concerts 2014