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| A '''kinetic convex hull''' data structure is a [[kinetic data structure]] that maintains the [[convex hull]] of a set of continuously moving points.<ref name="BGH99" />
| | What improved tool to help you you stay on track this year than a sharp looking organizer. Hermes purses and scarves are famous, but simple examples of a Brand which has been built from scratch, painstakingly after a while and by being extremely protective of distribution channels for his or her limited production, hand crafted products. Sweet arm candy is dangling from every corner of summer 2011 fashion. Tory Burch is one in the most fashionable ladies fashion brand in America. because we've got the right to do lots of issues,along with all the most thrilling is we are in a position to obtain lots of our favorite items. <br><br><br><br>tory burch wallet saleWear start using machine to try the particular the washing. To get one of the most from heels, it's recommended to practice walking included before putting them on out. Consignment stores really are a great way to be stylish without spending a ton of money. I have experienced my best love before me, I did right treasure it. Vitamin C is really a nutrient which has been demonstrated to use a substantial influence on human health. <br><br>tory burch handbags saleWe have in mind the Tory Burch Handbags are fit to the young people particularly the young ladies and also the teenagers. Associated most with bringing back the charm bracelet, their particular metal and [http://search.huffingtonpost.com/search?q=gemstone+beads&s_it=header_form_v1 gemstone beads] also allow women to customize necklaces, rings, earrings and bracelets. Jimmy Choo Outlet They provide a wide range a fabulous bouquets together with bunches you may choose and maybe they aren't necessarily pricey in anyway. Harcombe allows that vegetables can be a great addition towards the diet ' if served in butter to offer the fat-soluble vitamins they contain ' but. Some type of [http://www.pcs-systems.co.uk/Images/celinebag.aspx http://www.pcs-systems.co.uk/Images/celinebag.aspx] candy like M&M's, mints, Hershey's Kisses (again you'll be able to match your candy color for a theme). <br><br><br><br>tory burch walletscom and Better Homes and Garden have cute crafts that you could make using your children or grandchildren. And although this function has being retained through the modern designers, some improvements are actually made to make garment more [http://www.bbc.Co.uk/search/?q=fashionable fashionable] and appealing to the modern woman. Tory Burch, the famed New York dressmaker, will bring a downtown preppy vibe to South - Park. The stylish sandal, raised high with a rustic-looking cork wedge, is held by rounded straps of various leathers. Bi-fold wallets, for the other hand, only folds in half. <br><br>tory burch saleOlivia has recentl tweaked clearl to make sure ou Secretar about Immunit Walternate (Kim Chevalier) experiments of our own couple of ears best that this There Olivia,Celine Bags Louis Vuitton Shoes alias Bolivia (furthermore Torv). Still, it's best to come to be discreet along with all the embellishments, rather than just produce over-decorating all of the running sneakers. Tory Burch males Leopard City Zip Tote Camel Leopard Print in 2009 are loved by countless layout males also it will last for 2010 despite belonging inside the direction in the easy actuality that within of belonging in the direction of the layout world. It could also aid in posture, forcing the wearer to handle the body more upright and provides them a much more confident stride. The two products that have grown to be the trademark of Burberry are their wide variety of perfumes in addition to their check bags. <br><br>tory burch pursesStill, watch as a possible accessory is demand especially for men who don't prefer to use bracelets. The enterprise might be grown at a pace that is certainly more easily handled by thinly resourced entrepreneurs. A quick search on line for "poems about mom" will arrive an endless selection of selections which you are able to then cut and paste into your individual do-it-yourself book; just be sure to credit the authors. Lucky Brand name Fly from the Night time Leather Hobo: Drawstring bags day back for the 30s whenever they ended up used to handle wine bottles. Sighing, I groped around my Moleskin notebook, my keys, a couple packs of Trident gum and my Mac lipgloss. <br><br>tory burch handbags outletas well as stealing content Web e-mail lists, and looking for Web addresses. This summer months sees the wedge shoe return with stunning colors and sensational towering heights. An peanut butter let alone jam pendant is additionall a interesting equipment to get lovers. You can Lv Monogram Speedy 20 intended for $199 or Louis Vuitton Immediate 25 for $100 tiny websites. Almost any set out to end up getting periodic glimpses involved with Lagoon Athabaska spanning not known wetlands together with crushed stone bars. |
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| ==The 2D case==
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| The best known data structure for the 2 dimensional kinetic convex hull problem is by Basch, [[Leonidas J. Guibas|Guibas]], and Hershberger. This data structure is [[Kinetic data structure#Performance|responsive]], [[Kinetic data structure#Performance|efficient]], [[Kinetic data structure#Performance|compact]] and [[Kinetic data structure#Performance|local]].<ref name="BGH99" />
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| ===The data structure===
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| The [[Duality (projective geometry)|dual]] of a convex hull of a set of points is the upper and lower envelopes of the dual set of lines. Therefore, maintaining the upper and lower envelopes of a set of moving lines is equivalent to maintaining the convex hull of a set of moving points. Computing upper and lower envelopes are equivalent problems, so computing the upper envelope of a set of lines is equivalent to computing the convex hull of a set of moving points.
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| The upper envelope of a set of static lines can be computed using a [[divide and conquer algorithm]] which partitions the lines into two sets of equal size, calls itself recursively on the two sets to find their upper envelopes, and then merges the two resulting upper envelopes. The merge step is performed using a [[Sweep line algorithm|vertical line sweep]]. Call the first set of points blue and the second set of points red.
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| The standard line sweep algorithm for merging upper envelopes sweeps though all of vertices of the red and blue upper envelopes, from left to right. The most recently encountered red and blue points are maintained as the line sweeps. When a point is encountered, the algorithm checks if the point is above or below the segment following the last encountered point of the opposite color. If it is above, that point is added to the merged upper envelope. If it of a different color than the last added point, the red and blue envelopes have crossed, so the intersection point is also added to the merged upper envelope.<ref>John Hershberger, Finding the upper envelope of n line segments in O(n log n) time,Information Processing Letters Volume 33, Issue 4, 21 December 1989, Pages 169–174</ref>
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| The sequence of edges and vertices resulting from this algorithm is only dependent on the ordering of points, and the results of the line-point comparisons. Thus, the result can be certified with the following certificates:
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| *x-certificates (<math><_x</math>) are used to certify the order the vertices of the red and blue envelopes. They are the certificates for a [[kinetic sorted list]] on the set of vertices. Since each point involves 2 lines, and the certificate involves 2 points, each certificate involves 4 lines.
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| *y-certificates (<math><_y</math>) are used certify that a vertex is above or below an edge. The certificates appear for all comparisons that would occur during the algorithm.
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| As long as all of these certificates are hold, the merge steps will be executed identically, so the resulting upper envelope will be the same. A kinetic data structure for upper envelopes can be created by using these certificates to certify the static upper envelope algorithm. However, this scheme is not local, because one line many be involved in many y-certificates if it remains on top or bottom as many points from the other envelope are encountered.
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| Thus, it is necessary to introduce a s-certificates (<math><_s</math>) which certifies that the slope of a line is greater than or less than the slope of another line.
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| Having the following certificates for all points ab is sufficient to certify the sequence of edges and vertices resulting from a merge, with only O(1) certificates per line:<ref name="BGH99" />[[File:Kinetic convex hull, detection of intersections.png|thumb|540px|right|alt=A picture of the certificates in several difference cases|The certificates certify structure of the intersection of the red and blue envelopes by certifying intersections(top left) or the absence of intersections(top right and bottom). The arrows indicate which elements are being compared by the certificate.]]
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| #<math>x[ab]</math>: <math>ab<_x ab.next</math>. <math>ab.next</math> denotes vertex closest to <math>ab</math> on its right. This certificate is stored for all points <math>ab</math> which have a different color than the point, <math>ab.next</math>, which follows them.
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| #<math>yli[ab]</math>: <math>ab<_y ce(ab)</math> or <math>ab>_y ce(ab)</math>. This certificate is stored for all points <math>ab</math> such that <math>b</math> intersects <math>ce(ab)</math>. <math>ce(ab)</math> denotes the contender edge of <math>ab</math>, the edge from the other envelope that is above or below <math>ab</math>.
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| #<math>yri[lb]</math>: <math>ab<_y ce(ab)</math> or <math>ab>_y ce(ab)</math>. This certificate is stored for all points <math>ab</math> such that <math>a</math> intersects <math>ce(ab)</math>.
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| #<math>yt[ab]</math>: <math>ce(ab)<_yab</math>. This certificate is stored for all points <math>ab</math> for which <math>a<_sce(ab)<_s b</math> and <math>ce(ab)<_yab</math>.
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| #<math>srt[ab]</math>: <math>a<_sce(ab)</math>. This certificate is stored for all points <math>ab</math> for which <math>a<_sce(ab)<_s b</math> and <math>ce(ab)<_yab</math>.
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| #<math>srt[ab]</math>: <math>ce(ab)<_s b</math>. This certificate is stored for all points <math>ab</math> for which <math>a<_sce(ab)<_s b</math> and <math>ce(ab)<_yab</math>.
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| #<math>sl[ab]</math>: <math>b<_sce(ab)</math>. This certificate is stored for all points <math>ab</math> for which <math>b<_sce(ab)</math> and <math>ab<_yce(ab)</math>.
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| #<math>sr[ab]</math>: <math>ce(ab)<_sa</math>. This certificate is stored for all points <math>ab</math> for which <math>ce(ab)<_sa</math> and <math>ab<_yce(ab)</math>.
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| The first certificate, <math>x[ab]</math>, certifies the x-ordering of the points in the red and blue envelopes. The second and third certificates, <math>yli[ab]</math> and <math>yri[ab]</math>, certify intersections of the red and blue envelopes. The remaining 5 certificates, <math>yt[ab]</math>, <math>srt[ab]</math>, <math>srt[ab]</math>, <math>sl[ab]</math>, and <math>sr[ab]</math> place edges that are not in the upper envelopes in the sequence of slopes of the edges that are above it. If the slope is at the start or end of the sequence, <math>sl</math> or <math>sr</math> certify this. If it is in the middle of the sequence <math>slt</math>, and <math>srt</math> certify this, and <math>yt</math> certifies that the intersection point of the two lines that the edge's slope is in between, is above it. These one or three certificates suffice to prove that all of the edges are above this line. Unlike the previous scheme all lines are only involved in a constant number of certificates. Whenever of these certificates fail, the merged envelope and certificates can be updated in O(1) time.
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| The kinetic data structure for convex hull is constructed by using these certificates to certify the recursive merge of the upper envelopes. Whenever certificates fail, their merge is updated, and if the envelope resulting from the merge changes, the changes are propagated up through all merges that depend on the result of the merge.<ref name="BGH99" /> | |
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| === Performance ===
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| This data structure is [[Kinetic data structure#Performance|responsive]], [[Kinetic data structure#Performance|local]], [[Kinetic data structure#Performance|compact]], and [[Kinetic data structure#Performance|efficient]]. This is due to the logarithmic depth of the merges used to certify the upper envelope.<ref name="BGH99" />
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| *'''Responsive:''' When a certificate fails, it takes O(1) to fix the merge it certifies. If the resulting envelope changes, the change must be propagated up to all merges that depend on the result of the changed merge. There are O(log n) such mergers, so the update can can be performed in O(log n) time total.
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| *'''Local:''' Each line is involved in a most O(log n) certificates. This is because each line is involved in a constant number of certificates in each merge, and each line is in O(log n) merges total.
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| *'''Compactness:''' The maximum number of certificates used in this data structure is O(n log n). This is because each merge involves a number of certificates linear to the number of lines merged. Certifying an upper envelope of n lines requires certificates for the merge upper envelope of the two subsets of n/2 lines, and certificates for the merge of the envelopes. Thus the number of certificates, C(n), required to certify the upper envelope of n lines satisfies the recurrence C(n)=2C(n/2)+O(n), with C(1)=O(1). By the [[master theorem]] the C(n)=O(n log n).
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| *'''Efficiency:''' The maximum number of events processed by this algorithm on [[Kinetic data structure#Types of Trajectories|algebraic]] or [[Kinetic data structure#Types of Trajectories|pseudo-algebraic]] trajectories is near quadratic, <math>O(n^{2+\epsilon})</math> for all <math>\epsilon>0</math>.<ref name="ASS96">
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| P. K. Agarwal, O. Schwarzkopf, and Micha Sharir. The overlay of lower envelopes and its applications. Discrete Comput. Geom., 15:1–13, 1996.
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| </ref><ref name="Sha94"> | |
| Micha Sharir. Almost tight upper bounds for lower envelopes in higher dimensions. Discrete Comput. Geom., 12:327–345, 1994.
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| </ref> Convex hulls of linearly moving points can change <math>\Omega(n^2)</math> times.<ref name="AGHV01">
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| Pankaj K. Agarwal, Leonidas J. Guibas, John Hershberger, and Eric Veach. Maintaining the extent of a moving point set. Discrete and Computational Geometry, 26(3):353–374, 2001.
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| </ref> Thus this data structure is efficient.
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| ==Higher dimensions==
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| Finding an [[Kinetic data structure#Performance|efficient]] kinetic data structure for maintaining the convex hull of moving points in dimensions higher than 2 is an open problem.<ref name="BGH99">
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| Julien Basch, Leonidas J. Guibas, and John Hershberger. Data structures for mobile data. J. Algorithms, 31(1):1{28, 1999. [http://graphics.stanford.edu/courses/cs268-11-spring/notes/kinetic.pdf]
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| </ref>
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| == Related Problems ==
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| Kinetic convex hull can be used to solve the following related problems:<ref>P. K. Agarwal, L. J. Guibas, J. Hershberger, and E. Verach. Maintaining the extent of a moving set of points.</ref>
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| *[[Kinetic diameter]]
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| *[[Kinetic width]]
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| *[[Kinetic minimum box]]
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| *[[Kinetic smallest enclosing disk]]
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| == References ==
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| {{Reflist}}
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| [[Category:Articles created via the Article Wizard]]
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| [[Category:Kinetic data structures]]
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| [[Category:Convex hull algorithms]]
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