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| In [[computational geometry]], the '''gift wrapping algorithm''' is an [[algorithm]] for computing the [[convex hull]] of a given set of points.
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| ==Planar case==
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| In the two-dimensional case the algorithm is also known as '''Jarvis march''', after R. A. Jarvis, who published it in 1973; it has [[Big O notation|O]](''nh'') [[time complexity]], where ''n'' is the number of points and ''h'' is the number of points on the convex hull. Its real-life performance compared with other convex hull algorithms is favorable when n is small or h is expected to be very small with respect to n. In general cases the algorithm is outperformed by many others.
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| ==Algorithm==
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| [[Image:Jarvis march convex hull algorithm diagram.svg|thumb|280px|right|Jarvis's march computing the convex hull.]]
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| For the sake of simplicity, the description below assumes that the points are in [[general position]], i.e., no three points are [[collinear]]. The algorithm may be easily modified to deal with collinearity, including the choice whether it should report only [[extreme point]]s (vertices of the convex hull) or all points that lie on the convex hull. Also, the complete implementation must deal with [[degenerate case]]s when the convex hull has only 1 or 2 vertices, as well as with the issues of limited [[arithmetic precision]], both of computer computations and input data.
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| The gift wrapping algorithm begins with ''i''=0 and a point ''p<sub>0</sub>'' known to be on the convex hull, e.g., the leftmost point, and selects the point ''p<sub>i+1</sub>'' such that all points are to the right of the line ''p<sub>i</sub> p<sub>i+1</sub>''. This point may be found in O(''n'') time by comparing [[Coordinates (elementary mathematics)#Polar coordinates|polar angle]]s of all points with respect to point ''p<sub>i</sub>'' taken for the center of [[polar coordinates]]. Letting ''i''=''i''+1, and repeating with until one reaches ''p<sub>h</sub>''=''p<sub>0</sub>'' again yields the convex hull in ''h'' steps. In two dimensions, the gift wrapping algorithm is similar to the process of winding a string (or wrapping paper) around the set of points.
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| The approach can be extended to higher dimensions.
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| ==Pseudocode==
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| jarvis(S)
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| pointOnHull = leftmost point in S
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| i = 0
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| repeat
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| P[i] = pointOnHull
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| endpoint = S[0] // initial endpoint for a candidate edge on the hull
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| for j from 1 to |S|
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| if (endpoint == pointOnHull) or (S[j] is on left of line from P[i] to endpoint)
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| endpoint = S[j] // found greater left turn, update endpoint
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| i = i+1
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| pointOnHull = endpoint
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| until endpoint == P[0] // wrapped around to first hull point
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| ==Complexity==
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| The inner loop checks every point in the set ''S'', and the outer loop repeats for each point on the hull. Hence the total run time is <math>O(nh)</math>. The run time depends on the size of the output, so Jarvis's march is an [[output-sensitive algorithm]].
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| However, because the running time depends [[linear time|linearly]] on the number of hull vertices, it is only faster than <math>O(n \log n)</math> algorithms such as [[Graham scan]] when the number ''h'' of hull vertices is smaller than log ''n''. [[Chan's algorithm]], another convex hull algorithm, combines the logarithmic dependence of Graham scan with the output sensitivity of the gift wrapping algorithm, achieving an asymptotic running time <math>O(n \log h)</math> that improves on both Graham scan and gift wrapping.
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| ==References==
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| {{refbegin}}
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| * {{Introduction to Algorithms|2|chapter=33.3: Finding the convex hull|pages=pp. 955–956}}
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| *{{cite journal
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| | author = Jarvis, R. A.
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| | title = On the identification of the convex hull of a finite set of points in the plane
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| | journal = [[Information Processing Letters]]
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| | volume = 2
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| | year = 1973
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| | pages = 18–21
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| | doi = 10.1016/0020-0190(73)90020-3}}
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| {{refend}}
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| ==External links==
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| * [http://www.cse.unsw.edu.au/~lambert/java/3d/giftwrap.html Gift wrapping in 2 and higher dimensions]
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| * [http://michal.is/projects/convex-hull-gift-wrapping-method/ Gift wrapping algorithm in C#]
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| [[Category:Polytopes]]
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| [[Category:Convex hull algorithms]]
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