Marginal distribution: Difference between revisions

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{{unsolved|computer science|Is there an algorithm to solve 3SUM problem faster than <math>O(n^2)</math> time?}}
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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]]

Latest revision as of 22:46, 2 June 2014

Considering wedding rings of jewelry, there undoubtedly are a staggering quantity of people that do not understand why people wear bracelet charms or pendants on necklaces. Even fewer people understand rings that aren't wedding important. Jewelry exists, and is particularly worn inside of western world for one simple reason.



Sports enthusiasts would as becoming traditional pub sign that has a motor-bike, sports car, or baseball motif. Undertake it ! also gift one by using a poker, basketball, or football design.

Women wear necklace on their own neck. Women are wearing Pendants and necklaces since years. However, since the inception, necklaces have changed a much. The pattern, models of have transformed. Now Pendants are set up of many materials like gold, silver and yellow metal. In today's economy, the prices of gold are soaring and thus, many women are deciding upon silver accessory. One will obtain the best designs in a lot of other styles in the market. Moreover, the associated with pendant associated with silver is comparatively new. A silver pendant is somewhat piece of ornament, which an individual hangs all over neck from a chain. An individual solitaire is reasonably popular your market and females love put on it.

When you set these sort of pieces in sterling silver it really makes them look a great deal more casual. At the same time, it will give it more in regards to a designer look just because you'll own something a lot of people haven't seen before. Precisely what people even choose a pendant with a faux hammered finish might possibly just possess a few gem beads or dangles into it. This will really play inside stone will be important if you're working with something that's very expensive, hard to uncover or just a color that you aren't too excited about. Then the focus will be on the setting and it'll relate to be able to all the additional jewelry in which you own.

The lustre and brilliance of this green panna stone is really a huge hit amongst the elite. Its transparency and properties is accountable for its mass appeal. Probably the most effective time to wear an panna is spring when its powers are doubled. Will be the benefactor of 'eternal spring' or life to honest many.

When you a bit of jewelery, find out about the jeweler's insurance approach. Look for a good policy that permits you to return damaged items. You may be find some jewelers that willing to insure it against loss and stealing.

Hats are always in trend in a bitterly cold winter season. Huge interwoven beanies and oversized Moroccan furry hats are great to have this the winter months. It has never been via style as winter style.

Surely, the previously mentioned tips in no way make in order to regret. From means, you can purchase pendants, chain, earrings along with other fashion jewellery online in India.

If you cherished this article and you would like to be given more info pertaining to Quantum Pendants generously visit the site.

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