Disjoint-set data structure - Revision history

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	~~{{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''&~~nbsp;log~~&~~nbsp;''n''~~&~~nbsp;2<sup>{{lg-star}}~~&~~nbsp;''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]~~]

en>Rjwilmsi: Journal cites, added 1 DOI using AWB (9887)

2014-01-25T15:59:17Z

Journal cites, added 1 DOI using AWB (9887)

← Older revision		Revision as of 17:59, 25 January 2014
Line 1:		Line 1:
			{{For\|the theorem in probability theory\|Law of the iterated logarithm}}
			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>

	~~In order to invest loads of cash~~ on ~~things like controls and~~ for ~~memory cards~~, ~~appear~~ the ~~net~~ for ~~a secondhand variations~~. ~~Occasionally a store will probably become out of~~ used~~-game hardware~~, which ~~could be quite affordable. Make sure you look recorded at a web-based seller's feedback prior to making~~ the [~~http://Www~~.~~twitpic.com/tag/purchase purchase~~] ~~so a few seconds~~ . ~~whether you are procuring what you covered.<br><br>Inside your are a parent or gaurdian of any hardly any one who appreciates participating in part in video games~~, be ~~familiar with multi~~-~~player and on-line video contests options. These aspects give your kid~~ in the ~~market to interact with many some gamers throughout~~ the ~~world~~. ~~As good as~~ the is for ~~your young children in order to communicate with others~~, ~~you might not know who person on the other closing is.~~<br><br>~~Small business inside your games when you find yourself succesfully done playing them~~. ~~If you beloved this article and you simply would like to collect more info concerning~~ [~~http://circuspartypanama.com hack clash~~ of ~~clans 2014~~] ~~nicely visit our own webpage. Quite retailers provide discount rates or credit score if you want to your next buy each~~ and ~~every~~ time ~~you business your clash~~ of ~~clans sur pc tlcharger during. You can receive~~ the ~~next online game you would like intended~~ for ~~the affordable price once you try this~~. ~~All things considered~~, ~~you don~~'~~t need the dvd games as soon~~ for ~~the reason that you defeat them.~~<br><br>~~Doing now~~, ~~there exists no social options / functions with this game my spouse and i~~.~~e. there is not any chat, having difficulties to team track~~ of ~~all friends~~, ~~etc but nonetheless we could expect this to improve soon as Boom Beach continues to stay in their Beta Mode~~.<br>~~<br>Have a look at your child~~'s ~~xbox play enjoying~~. ~~Video~~ and ~~also are now rated solely like films~~ and ~~this also can help~~. ~~This enables you~~ to ~~maintain your~~ an ~~eye on an information your kids is exposed to~~. ~~Conditional upon your child~~'~~s age, continue~~ to ~~keep he or she clear~~ of ~~[http://Www~~.~~videogames~~.~~net~~/ ~~video games] that happen to you should be meant for people~~ the ~~people that are~~ more ~~fully learned~~ than ~~him~~.<br><br>~~Many of our world can be influenced by supply and shopper demand~~. ~~We shall look during~~ the the ~~Greek~~-~~Roman model~~. ~~Using special care that can highlight~~ the ~~role regarding clash~~ of ~~clans compromise tool no survey inside~~ of a the ~~vast framework and it usually this provides~~.<br><br>~~And any our options~~ are ~~looked into and approved from the greatest virus recognition software and~~ / ~~or anti-virus in the sell~~ to ~~ensure a security-level as large as you can, in might you fear for the security of your computer or your cellular device, no situations~~. ~~In case you nevertheless have any sort of doubts, take a glance at the movie~~ and ~~you'll perceive it operates and it's not 100% secure! It takes merely a few moments of the!~~		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]]

en>Dekart: added Category:Search algorithms using HotCat

2012-09-02T08:34:41Z

added Category:Search algorithms using HotCat

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