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	~~A '''kinetic width''' data structure~~ is a [~~[kinetic data structure~~]~~] which maintains~~ the ~~[[width]] of~~ a ~~set of moving points~~. ~~In 2D, the width~~ of a ~~point set is~~ the ~~minimum distance between two parallel lines that contain~~ the ~~point set in the strip between them~~. ~~For the two dimensional case, the kinetic data structure for [[kinetic convex hull]]~~ can be ~~used to construct a kinetic data structure for~~ the ~~width~~ of ~~a point set that is~~ [~~[Kinetic data structure#Performance\|responsive]], [[Kinetic data structure#Performance\|compact]] and [[Kinetic data structure#Performance\|efficient]]~~.		Background check is carried out by companies and folks to know more about a person and his existing activities, financial transactions, prison records and more. There are a lot of reasons for doing a background examine but it should be executed in detail with [http://tinyurl.com/pch83be ugg boots sale] proper cross verification wherever necessary. <br><br>It helps corporations to trust the person with work and also entrust kids with self-assurance. Imagine a situation where you leave your kid with a babysitter with a felony record. Few hours of leaving your little one with an not known babysitter who has burglary record can turn out to be a nightmare. It helps to know the person before entrusting the kid with her. A babysitter who has been verified can help you be at ease when your child is at the care of an attendant. The same can be [http://search.About.com/?q=applied applied] to patient care when you leave the sick and elderly dependent with an attendant to help them out. A clear background history can go a long way in reducing the risk and ensuring relaxing time [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] for you.<br><br>Why do enterprises go for a background check? organizations thrive on goodwill and workers become the face of the company when they meet the client. It is important for the companies to avoid negligent hiring and take responsibility for the employees they hired. workers who have history of indulging in violence or have records of drug abuse or alcohol addiction can ruin employer's status amongst [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] the clients. It is not challenging to get a suggestions on a possible candidate if you choose to have the applicant verified. Make use of the free background verify offered by companies to cross verify the candidate's credentials for bad [http://tinyurl.com/pch83be ugg boots usa] carry out including rash driving, sexual offenses in add-on to the above mentioned criteria.<br><br>It is possible for you to do a prison background verify oneself as all the felony records are public and can be accessed by anyone. If you want to do it by oneself online or manually, it is a time intensive process. You should know where to look for the information online if you decide to check the person's background. Your efforts can turn out to be nothing if you do not know the advanced search techniques to locate the necessary information. If you plan to do it manually, it involves submitting documents to the concerned departments and may include hanging around time.<br><br>The vital aspect of background check is in verifying the credentials without the person figuring out about it and in a discrete method. When the procedure is [http://tinyurl.com/pch83be ugg boots] done physically, there is more possibility of the information regarding the background verify reaching the concerned person. It may be best to carry out the free background check online where confidentiality is assured and the process is headache free and easy.<br><br>Make sure to confirm that the staff have been verified for their background, criminal records, and credit worthiness before you hire them. It is best to utilize the free background verify systems available on the net and make an informed determination to hire or fire the resource.

	=~~=2D Case==~~
	~~Consider~~ the ~~parallel lines which contain the point set in the strip between~~ them ~~and are of minimal distance apart~~. ~~One of the lines must contain an edge ab of the convex hull, and the other line must~~ go ~~through~~ a ~~point c of~~ the ~~convex hull such that (a,c)~~ and ~~(b,c) are~~ [~~[antipodal pairs]~~]. ab and ~~c are referred to as an antipodal edge-vertex pair.~~
	~~Consider~~ the ~~[[Duality_(projective_geometry)\|dual]]~~ of the ~~point set~~. ~~The points dualize~~ to ~~lines~~ and the ~~convex hull~~ of ~~the points dualizes to the upper and lower envelope~~ of the ~~set of lines~~. ~~The vertices of the upper convex hull dualize~~ to ~~segments~~ on the ~~upper envelope~~. ~~The vertices~~ of the ~~lower convex hull dualize~~ to ~~segments on~~ the ~~lower envelope~~. ~~The range of slopes of the supporting lines of a point~~ on ~~the hull dualize~~ to the ~~x-interval of segment that point dualizes~~ to~~. When viewed in this dualized fashion~~ the ~~antipodal pairs,~~ are ~~pairs of segments, one from the upper envelope, one from the lower, with overlapping x ranges. Now, the upper~~ and ~~lower envelopes~~ can be ~~viewed as two different x-ordered lists of non overlapping intervals~~. If ~~these two lists are merged~~, ~~the antipodal pairs are the overlaps in the merged list. If an pair ab and c~~ is ~~an antipodal edge-vertex pair then the x-interval for~~ a ~~and b must both intersect the x-interval for c~~. ~~This means that the common endpoint of the x intervals~~ for ~~a and b must lie within~~ the ~~x-interval for c.~~

	~~The endpoints of both of~~ the ~~sets of x-intervals can be maintained in a [[kinetic sorted list]]~~. ~~When points swap, the list of antipodal edge-point pairs are updated appropriately. The upper and lower envelopes~~ can be ~~maintained using~~ the ~~standard data structure for [[kinetic convex hull]]. The minimum distance between edge-point pairs can be maintained with a [[kinetic tournament]]. Thus, using kinetic convex hull~~ to ~~maintain~~ the ~~upper and lower envelopes, a kinetic sorted list on these intervals~~ to ~~maintain the antipodal edge-vertex pairs~~, ~~and a kinetic tournament~~ to ~~maintain~~ the ~~pair of minimum distance apart, the diameter of a moving point set can be maintained.~~

	~~This data structure is [[Kinetic data structure#Performance\|responsive]], [[Kinetic data structure#Performance\|compact]]~~ and ~~[[Kinetic data structure#Performance\|efficient]]~~. ~~The data structure uses <math>O(n)~~<~~/math~~> ~~space because the kinetic convex hull, sorted list, and tournament data structures all use~~ <~~math~~>~~O(n)</math> space. In all~~ of the ~~data structures, events, inserts,~~ and ~~deletes can be handled~~ in ~~<math>O(\log^2 n)</math> time, so the data structure are responsive, requiring <math>O(\log^2 n)</math> per event~~. ~~The data structure is efficient because~~ the ~~total number of events~~ is ~~<math>O(n^{2+\epsilon})<~~/~~math> for all <math>\epsilon>0<~~/~~math> and the width of a point set can change <math>\Omega(n^2)<~~/~~math> times~~, ~~even if~~ the ~~points are moving linearly~~. ~~This data structure is not [[Kinetic data structure#Performance\|local]] because one point~~ may be ~~in many antipodal edge-vertex pairs, and thus appear many times in~~ the ~~kinetic tournament.~~

	~~The existence of a local kinetic data structure for width~~ is ~~open.~~

	~~==Higher Dimensions==~~
	~~Efficiently maintaining~~ the ~~kinetic width of a point set in dimensions higher than 2~~ is ~~an open problem. Efficient [[kinetic convex hull]] in dimensions higher than 2 is also an open problem~~.<~~ref~~>~~{{Citation~~
	~~\| first = Leonidas J.~~
	~~\| last = Guibas~~
	~~\| author-link = Leonidas J. Guibas~~
	~~\| contribution = Kinetic Data Structures~~
	~~\| contribution-url = http://graphics.stanford.edu/projects/lgl/papers/g-KDS_DS-Handbook-04/g-KDS_DS-Handbook-04.pdf~~
	~~\| title = Handbook of Data Structures and Applications~~
	~~\| editor-first = Dinesh P.~~
	~~\| editor-last = Mehta~~
	~~\| editor2-first = Sartaj~~
	~~\| editor2-last = Sahni~~
	~~\| publisher = Chapman and Hall/CRC~~
	~~\| isbn = 978-1584884354~~
	~~\| pages = 23-1–23-18~~
	~~\| year = 2001~~
	}}<~~/ref~~>
	~~==Related Problems==~~
	*[[Kinetic diameter]]
	*[[Kinetic minimum box]]

	~~== References ==~~
	~~{{Reflist}}~~

	~~==Further reading==~~
	~~P. K. Agarwal~~, ~~L. J. Guibas~~, J. ~~Hershberger,~~ and ~~E. Verach. Maintaining~~ the ~~extent of a moving set of points~~.

	~~<!--- Categories --->~~
	~~[[Category:Articles created via the Article Wizard]]~~
	~~[[Category:Kinetic data structures]]~~
	~~[[Category:Geometric data structures]]~~

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A '''kinetic width''' data structure is a [[kinetic data structure]] which maintains the [[width]] of a set of moving points. In 2D, the width of a point set is the minimum distance between two parallel lines that contain the point set in the strip between them. For the two dimensional case, the kinetic data structure for [[kinetic convex hull]] can be used to construct a kinetic data structure for the width of a point set that is [[Kinetic data structure#Performance|responsive]], [[Kinetic data structure#Performance|compact]] and [[Kinetic data structure#Performance|efficient]].

==2D Case==
Consider the parallel lines which contain the point set in the strip between them and are of minimal distance apart. One of the lines must contain an edge ab of the convex hull, and the other line must go through a point c of the convex hull such that (a,c) and (b,c) are [[antipodal pairs]]. ab and c are referred to as an antipodal edge-vertex pair.
Consider the [[Duality_(projective_geometry)|dual]] of the point set. The points dualize to lines and the convex hull of the points dualizes to the upper and lower envelope of the set of lines. The vertices of the upper convex hull dualize to segments on the upper envelope. The vertices of the lower convex hull dualize to segments on the lower envelope. The range of slopes of the supporting lines of a point on the hull dualize to the x-interval of segment that point dualizes to. When viewed in this dualized fashion the antipodal pairs, are pairs of segments, one from the upper envelope, one from the lower, with overlapping x ranges. Now, the upper and lower envelopes can be viewed as two different x-ordered lists of non overlapping intervals. If these two lists are merged, the antipodal pairs are the overlaps in the merged list. If an pair ab and c is an antipodal edge-vertex pair then the x-interval for a and b must both intersect the x-interval for c. This means that the common endpoint of the x intervals for a and b must lie within the x-interval for c.

The endpoints of both of the sets of x-intervals can be maintained in a [[kinetic sorted list]]. When points swap, the list of antipodal edge-point pairs are updated appropriately. The upper and lower envelopes can be maintained using the standard data structure for [[kinetic convex hull]]. The minimum distance between edge-point pairs can be maintained with a [[kinetic tournament]]. Thus, using kinetic convex hull to maintain the upper and lower envelopes, a kinetic sorted list on these intervals to maintain the antipodal edge-vertex pairs, and a kinetic tournament to maintain the pair of minimum distance apart, the diameter of a moving point set can be maintained.

This data structure is [[Kinetic data structure#Performance|responsive]], [[Kinetic data structure#Performance|compact]] and [[Kinetic data structure#Performance|efficient]]. The data structure uses <math>O(n)</math> space because the kinetic convex hull, sorted list, and tournament data structures all use <math>O(n)</math> space. In all of the data structures, events, inserts, and deletes can be handled in <math>O(\log^2 n)</math> time, so the data structure are responsive, requiring <math>O(\log^2 n)</math> per event. The data structure is efficient because the total number of events is <math>O(n^{2+\epsilon})</math> for all <math>\epsilon>0</math> and the width of a point set can change <math>\Omega(n^2)</math> times, even if the points are moving linearly. This data structure is not [[Kinetic data structure#Performance|local]] because one point may be in many antipodal edge-vertex pairs, and thus appear many times in the kinetic tournament.

The existence of a local kinetic data structure for width is open.

==Higher Dimensions==
Efficiently maintaining the kinetic width of a point set in dimensions higher than 2 is an open problem. Efficient [[kinetic convex hull]] in dimensions higher than 2 is also an open problem.<ref>{{Citation
| first = Leonidas J.
| last = Guibas
| author-link = Leonidas J. Guibas
| contribution = Kinetic Data Structures
| contribution-url = http://graphics.stanford.edu/projects/lgl/papers/g-KDS_DS-Handbook-04/g-KDS_DS-Handbook-04.pdf
| title = Handbook of Data Structures and Applications
| editor-first = Dinesh P.
| editor-last = Mehta
| editor2-first = Sartaj
| editor2-last = Sahni
| publisher = Chapman and Hall/CRC
| isbn = 978-1584884354
| pages = 23-1–23-18
| year = 2001
}}</ref>
==Related Problems==
*[[Kinetic diameter]]
*[[Kinetic minimum box]]

== References ==
{{Reflist}}

==Further reading==
P. K. Agarwal, L. J. Guibas, J. Hershberger, and E. Verach. Maintaining the extent of a moving set of points.


[[Category:Articles created via the Article Wizard]]
[[Category:Kinetic data structures]]
[[Category:Geometric data structures]]