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The lower bound on worst-case running time of output-sensitive convex hull algorithms was established to be Ω(n log h) in the planar case. [1] There are several algorithms which attain this optimal time complexity. The earliest one was introduced by Kirkpatrick and Seidel in 1986 (who called it "the ultimate convex hull algorithm").
Input = a set S of n points Assume that there are at least 2 points in the input set S of points function QuickHull(S) is // Find convex hull from the set S of n points Convex Hull := {} Find left and right most points, say A & B, and add A & B to convex hull Segment AB divides the remaining (n − 2) points into 2 groups S1 and S2 where S1 are points in S that are on the right side of the ...
The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from Isaac Newton to Henry Oldenburg in 1676. [71] The term "convex hull" itself appears as early as the work of Garrett Birkhoff , and the corresponding term in German appears earlier, for instance in Hans Rademacher's review of Kőnig .
A demo of Graham's scan to find a 2D convex hull. Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O(n log n). It is named after Ronald Graham, who published the original algorithm in 1972. [1] The algorithm finds all vertices of the convex hull ordered along its boundary.
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 points (vertices of the convex hull) or all points that lie on the convex hull [citation needed].
In computational geometry, Chan's algorithm, [1] named after Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set of points, in 2- or 3-dimensional space. The algorithm takes O ( n log h ) {\displaystyle O(n\log h)} time, where h {\displaystyle h} is the number of vertices of the output (the convex ...
A particularly simple algorithm for this problem was published by Graham & Yao (1983) and Lee (1983). Like the Graham scan algorithm for convex hulls of point sets, it is based on a stack data structure. The algorithm traverses the polygon in clockwise order, starting from a vertex known to be on the convex hull (for instance, its leftmost point).
Toussaint (1986), who provided an efficient algorithm for the construction of the relative convex hull for finite sets of points inside a simple polygon. [3] With subsequent improvements in the time bounds for two subroutines, finding shortest paths between query points in a polygon, [4] and polygon triangulation, [5] this algorithm takes time (+ (+)) on an input with points in a polygon ...