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In [[graph theory]], the '''metric ''k''-center''', is a [[combinatorial optimization]] problem studied in [[theoretical computer science]]. Given ''n'' cities with specified distances, one wants to build ''k'' warehouses in different cities and minimize the maximum distance of a city to a warehouse. In graph theory this means finding a set of ''k'' vertices for which the largest distance of any point to its closest vertex in the ''k''-set is minimum. The vertices must be in a metric space, or in other words a [[complete graph]] that satisfies the [[triangle inequality]].


==Formal definition==
Given a complete undirected graph ''G''&nbsp;=&nbsp;(''V'',&nbsp;''E'') with distances ''d''(''v''<sub>''i''</sub>,&nbsp;''v''<sub>''j''</sub>)&nbsp;&isin;&nbsp;''N'' satisfying the triangle inequality, find a subset ''S''&nbsp;&sube;&nbsp;''V'' with |''S''|&nbsp;=&nbsp;''k'' while minimizing:


: <math>\max_{v \in V} \min_{s \in S} d(v,s)</math>
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==Computational complexity==
If we sort the edges in nondecreasing order of the distances: ''d''(''e''<sub>1</sub>)&nbsp;&le;&nbsp;''d''(''e''<sub>2</sub>)&nbsp;&le;&nbsp;&hellip;&nbsp;&le;&nbsp;''d''(''e''<sub>m</sub>) and let ''G''<sub>i</sub>&nbsp;=&nbsp;(''V''<sub>i</sub>,&nbsp;''E''<sub>i</sub>), where ''E''<sub>i</sub>&nbsp;=&nbsp;{''e''<sub>1</sub>,&nbsp;''e''<sub>2</sub>,&nbsp;&hellip;,&nbsp;''e''<sub>i</sub>}. The ''k''-center problem is equivalent to finding the smallest index ''i'' such that ''G''<sub>i</sub> has a [[dominating set]] of size at most ''k''.<ref>{{harvnb|Vazirani|2003|pp=47&ndash;48}}.</ref> Dominating set is [[NP-complete]] and therefore the ''k''-center problem is also NP-complete.
 
==Approximations==
A simple [[greedy algorithm| greedy]] [[approximation algorithm]] that achieves an approximation factor of 2 builds S in ''k'' iterations. The first iteration chooses an arbitrary vertex and adds it to ''S''. Each subsequent iteration chooses a vertex ''v'' for which ''d''(S,&nbsp;v) is maximized and adds ''v'' to ''S''. The running time of the algorithm is ''O''(''nk'').<ref>{{harvnb|Gonzalez|1985|pp=293&ndash;306}}.</ref>
 
Another algorithm with the same approximation factor takes advantage of the fact that the ''k''-center problem is equivalent to finding the smallest index ''i'' such that ''G''<sub>i</sub> has a dominating set of size at most ''k'' and computes a maximal [[Independent set (graph theory)|independent set]] on the of ''G''<sub>i</sub>, looking for the smallest index ''i'' that has a maximal independent set with a size of at least ''k''.<ref>{{harvnb|Hochbaum|Shmoys|1986|pp=533&ndash;550}}.</ref>
 
It is not possible to find an approximation algorithm with an approximation factor of 2&nbsp;&minus;&nbsp;&epsilon; for any &epsilon;&nbsp;>&nbsp;0.<ref>{{harvnb|Hochbaum|1997|pp=346&ndash;398}}.</ref> Furthermore, the distances of all edges in G must satisfy the triangle inequality if the ''k''-center problem is to be approximated unless P = NP.<ref>{{harvnb|Crescenzi|Kann|Halldórsson|Karpinski|2000}}.</ref>
 
==See also==
* [[Traveling salesman problem]]
* [[Minimum k-cut]]
* [[Dominating set]]
* [[Independent set (graph theory)]]
 
==Notes==
<references />
 
==References==
* {{citation
  | last = Vazirani
  | first = Vijay V.
  | authorlink = Vijay Vazirani
  | title = Approximation Algorithms
  | publisher = Springer
  | year = 2003
  | location = Berlin
  | isbn = 3-540-65367-8 }}
* {{citation
  | last = Gonzalez
  | first = Teofilo F.
  | contribution = Clustering to minimize the maximum intercluster distance
  | title = Theoretical Computer Science
  | publisher = Elsevier Science B.V.
  | year = 1985
  | volume = 38
  | pages=293&ndash;306
  | doi = 10.1016/0304-3975(85)90224-5}}
* {{citation
  | last1 = Hochbaum
  | first1 = Dorit S.
  | last2 = Shmoys
  | first2 = David B.
  | contribution = A Best Possible Heuristic for the k-Center Problem
  | title = Mathematics of Operations Research
  | year = 1985
  | volume = 10
  | issue = 2
  | pages = 180&ndash;184}}
* {{citation
  | last1 = Hochbaum
  | first1 = Dorit S.
  | last2 = Shmoys
  | first2 = David B.
  | authorlink2 = David Shmoys
  | contribution = A unified approach to approximation algorithms for bottleneck problems
  | title = Journal of the ACM (JACM)
  | volume = 33
  | issue = 3
  | year = 1986
  | issn = 0004-5411
  | pages=533&ndash;550}}
* {{citation
  | last = Hochbaum
  | first = Dorit S.
  | title = Approximation Algorithms for NP-Hard problems
  | publisher = PWS Publishing Company
  | year = 1997
  | location = Boston
  | isbn = 0-534-94968-1
  | pages=346&ndash;398}}
* {{citation
  | last1=Crescenzi | first1=Pierluigi
| last2=Kann | first2=Viggo
| last3=Halldórsson | first3=Magnús
| last4=Karpinski | first4=Marek | authorlink4=Marek Karpinski
| last5=Woeginger | first5=Gerhard
| title=A Compendium of NP Optimization Problems
| contribution=Minimum k-center
| url=http://www.csc.kth.se/~viggo/wwwcompendium/node128.html
| year=2000 }}
 
[[Category:NP-complete problems]]
[[Category:Combinatorial optimization]]
[[Category:Computational problems in graph theory]]
[[Category:Approximation algorithms]]

Latest revision as of 04:33, 7 January 2015


Autism is not a medical ailment, but instead a developmental sickness that occurs during childhood.
It has many different symptoms, and no two autistic people are exactly alike. A topic that is always being studied cheap ugg boots and debated about by scientists and medical researchers are the root causes of autism. There are no signs that autism is triggered by just one thing, as discount ugg boots this disease has been related to a number of different things.
The determinants of autism we will be focusing on in this article are the ones that several people have faith in, although there is no accord as of yet.

The question of whether autism is caused by vaccines is very controversial, and there is still quite a bit of debate about this. Many people believe that certain vaccines, especially those that contain a mercury compound called thiomersal, are responsible for the apparent increase in autism in recent years.

This belief, however, is not supported by evidence according to many scientists. The belief that giving young children vaccines increases the risk of autism is still held by many in the alternative health community. No matter your view on this hotly debated issues, it has yet to been proven conclusively for either side.

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Some suggest that allergies in the children may play a role in the development of autism. So those children, as part of their treatment, are not allowed any dairy or gluten products which would make sense anyway regardless of the presence of autism. Anyone considering such measures for their own children ugg boots sale should make sure that they understand their children's nutritional requirements so that adequate replacements for these possibly detrimental foods can be found.

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Some are under the impression that the herpes virus might be one reason that autism happens, but this has not been proven in its entirety. For the most part, a majority of scientists think that autism has many different causes, and viruses might be one of them. Since it is well known that a child's genes play a large role in autism, it might be that viruses can set off autism in children who are already genetically predisposed to this sickness.

There are many possible causes of autism, and it will probably be some time before there is any general agreement about what brings about this condition. The grounds for autism reviewed above cannot, however be thought of as cheap ugg boots the final word on this issue, however they are each worth being looked into further.

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