Marginal distribution - Revision history

152.19.32.133: /* Real-world example */ There was an error in the real-world example (joint probability distributions in the table were calculated with different probabilities for red, yellow, and green light state than what was used in the text).

2014-06-02T21:46:58Z

Real-world example: There was an error in the real-world example (joint probability distributions in the table were calculated with different probabilities for red, yellow, and green light state than what was used in the text).

← Older revision		Revision as of 23:46, 2 June 2014
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140.180.255.46: /* Continuous variables */

2014-03-02T21:48:51Z

Continuous variables

← Older revision		Revision as of 23:48, 2 March 2014
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	~~{{unsolved\|computer science\|Is there an algorithm to solve 3SUM problem faster than <math>O(n^2)</math> time?}}~~		Eusebio is the name women and men use to call my home and I think it's sounds quite good when you say it. I worn to be unemployed yet still now I am a huge cashier. My house is now in South Carolina and I don't plan onto [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=changing&btnI=lucky changing] it. It's not a common application but what I like for example doing is bottle blouses collecting and now I do have time to think about on new things. I'm not high-quality at webdesign but retailers . want to check my website: http://circuspartypanama.com<br><br>my web site; [http://circuspartypanama.com clash of clans hack no survey download]
	~~In [[computational complexity theory]],~~ the ~~'''3SUM''' problem asks if a given set of <math>n</math> integers, each with absolute value bounded by some polynomial in <math>n</math>, contains three elements that sum~~ to ~~zero. The generalized version, rSUM, asks the same question of r elements. 3SUM can be easily solved in <math>O(n^2)</math> time,~~ and ~~matching <math>\Omega(n^{r/2})</math> lower bounds are known in some specialized [[models of computation]] {{harv\|Erickson\|1999}}.~~ Slightly faster randomized algorithms are known that exploit [[model of computation\|computational-model]] parallelism on a [[Random Access Machine\|RAM]] and in the external-memory and [[cache-oblivious]] models {{harv\|Baran\|Demaine\|Pǎtraşcu\|2008}}. ~~When the integers are in the range <math>[-u, \dots, u]</math>, 3SUM can~~ be ~~solved in <math>O(n + u\log u)</math> time by representing the input set <math>S</math> as~~ a bit vector, computing the set <math>S+S</math> of all pairwise sums as a [[Convolution#Discrete_convolution\|discrete convolution]] using the [[Fast Fourier transform]], and finally comparing this set to <math>-S</math>.

	~~==Quadratic algorithm==~~

	~~Suppose the input array~~ is ~~<math>S[0..n-1]</math>. 3SUM can be solved~~ in ~~<math>O(n^2)</math> time by inserting each number <math>S[i]</math> into a hash table,~~ and ~~then for each index <math>i</math> and <math>j</math>, checking whether the hash table contains the integer <math>-S[i]-S~~[~~j]</math>.~~

	Alternatively, the algorithm below first sorts the input array and then tests all possible pairs in a careful order that avoids the need to [[binary search]] for the pairs in the sorted list, again achieving <math>O(n^2)</math> time, as follows.<ref>[http://~~www.ti.inf~~.~~ethz~~.ch/~~ew/courses/CG09/materials/v12.pdf Visibility Graphs and 3-Sum] by Michael Hoffmann</ref>~~
	~~sort(S);~~
	~~'''for''' i=0 '''to''' n-3 '''do'''~~
	a = ~~S[i];~~
	k = ~~i+1;~~
	l = ~~n-1;~~
	~~'''while''' (k<l) '''do'''~~
	b = ~~S[k];~~
	c = ~~S[l~~];
	'~~''if''' (~~a~~+b+c == 0) '''then'''~~
	~~'''output''' a, b, c;~~
	~~exit;~~
	~~'''else''' '''if''' (a+b+c > 0) '''then'''~~
	~~l = l - 1;~~
	~~'''else'''~~
	~~k = k + 1;~~
	~~'''end'''~~
	~~'''end'''~~
	~~'''end'''~~

	~~The following~~ example ~~shows this algorithm's execution on a small sorted array. Current values of '''a''' are shown in bold, values of '''b'''~~ and ~~'''c''' are shown in red.~~
	~~'''-25''' <span style="color:red">-10</span> -7 -3 2 4 8 <span style="color:red">10</span> (a+b+c==-25)~~
	~~'''-25''' -10 <span style="color:red">-7</span> -3 2 4 8 <span style="color:red">10</span> (a+b+c==-22)~~
	~~. . .~~
	~~'''-25''' -10 -7 -3 2 4 <span style="color:red">8</span> <span style="color:red">10</span> (a+b+c==-7)~~
	~~-25 '''-10''' <span style="color:red">-7</span> -3 2 4 8 <span style="color:red">10</span> (a+b+c==-7)~~
	~~-25 '''-10''' -7 <span style="color:red">-3</span> 2 4 8 <span style="color:red">10</span> (a+b+c==-3)~~
	~~-25 '''-10''' -7 -3 <span style="color:red">2</span> 4 8 <span style="color:red">10</span> (a+b+c==2)~~
	~~-25 '''-10''' -7 -3 <span style="color:red">2</span> 4 <span style="color:red">8</span> 10 (a+b+c==0)~~

	~~The correctness of the algorithm can be seen as follows. Suppose we~~ have ~~a solution a + b + c = 0. Since the directions of the points only move in one direction, we can run the algorithm until the leftmost pointer points~~ to ~~a. Run the algorithm until either one of the remaining pointers points to a or b, whichever occurs first~~. ~~Then the algorithm will run until the last pointer points to the remaining term, giving the affirmative solution.~~

	~~==3SUM-hardness==~~

	~~A problem is called ''~~'~~3SUM~~-~~hard''' if solving it in [[subquadratic time]] implies a subquadratic-time [[algorithm]] for 3SUM~~. The concept of 3SUM-hardness was introduced by {{harvtxt\|Gajentaan\|Overmars\|1995}} in [[analysis of algorithms]] in [[computational geometry]]. By now there are a multitude of problems that fall into this category.

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~
	*{{citation
	~~\| last1 = Baran \| first1 = Ilya~~
	~~\| last2 = Demaine \| first2 = Erik D. \| author2-link = Erik Demaine~~
	~~\| last3 = Pătraşcu \| first3 = Mihai \| author3-link = Mihai Pătraşcu~~
	~~\| doi = 10.1007/s00453-007-9036-3~~
	~~\| issue = 4~~
	~~\| journal = Algorithmica~~
	~~\| pages = 584–596~~
	~~\| title = Subquadratic algorithms for 3SUM~~
	~~\| url = http~~:~~//erikdemaine.org/papers/3SUM_Algorithmica/~~
	~~\| volume = 50~~
	~~\| year = 2008}}.~~
	*{{citation
	~~\| last1 = Demaine \| first1 = Erik D. \| author1-link = Erik Demaine~~
	~~\| last2 = Mitchell \| first2 = Joseph S. B. \| author2-link = Joseph S. B. Mitchell~~
	~~\| last3 = O'Rourke \| first3 = Joseph \| author3-link = Joseph O'Rourke (professor)~~
	~~\| date = July 2005~~
	~~\| publisher = [http://maven.smith.edu/~orourke/TOPP/Welcome.html The Open Problems Project]~~
	~~\| title = Problem 11: 3SUM Hard Problems~~
	~~\| url = http://cs.smith.edu/~orourke/TOPP/P11.html}}.~~
	*{{citation
	~~\| last1 = Erickson \| first1 = Jeff~~
	~~\| year = 1999~~
	~~\| journal = Chicago Journal of Theoretical Computer Science~~
	~~\| volume = 1999~~
	~~\| publisher = MIT Press~~
	~~\| title = Lower bounds for linear satisﬁability problems~~
	~~\| url =~~ http://~~cjtcs.cs.uchicago.edu/articles/1999/8/contents.html}}.~~
	*{{citation
	~~\| last1 = Gajentaan \| first1 = Anka~~
	~~\| last2 = Overmars \| first2 = Mark H. \| author2-link = Mark Overmars~~
	~~\| doi = 10~~.~~1016/0925-7721(95)00022-2~~
	~~\| issue = 3~~
	~~\| journal = Computational Geometry: Theory and Applications~~
	~~\| pages = 165–185~~
	~~\| title = On a class of O(''n''~~<~~sup~~>2<~~/sup~~>~~) problems in computational geometry~~
	~~\| volume = 5~~
	~~\| year = 1995}}.~~
	*{{citation
	~~\| last = King \| first = James~~
	~~\| title = A survey of 3SUM-hard problems~~
	~~\| url =~~ http://~~www~~.~~cs.mcgill.ca/~jking/papers/3sumhard.pdf~~
	~~\| year = 2004}}.~~

	~~== See also ==~~
	* [[Subset sum problem]]

	~~[[Category:Computational geometry]]~~
	~~[[Category:Polynomial-time problems]]~~
	~~[[Category:Unsolved problems in computer science]~~]

en>Saviourmachine: /* Real-world example */ There is not only 1 marginal probability, so added P(H)

2014-01-29T14:32:35Z

Real-world example: There is not only 1 marginal probability, so added P(H)

Show changes

en>Thehelpfulbot: r2.6.5) (Robot: Modifying fr:Loi de probabilité marginale

2012-05-21T16:59:00Z

r2.6.5) (Robot: Modifying fr:Loi de probabilité marginale

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