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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").
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 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).
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.
The Kirkpatrick–Seidel algorithm, proposed by its authors as a potential "ultimate planar convex hull algorithm", is an algorithm for computing the convex hull of a set of points in the plane, with () time complexity, where is the number of input points and is the number of points (non dominated or maximal points, as called in some texts) in the hull.
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 ...
Sequence of probes around the convex hull of a polygon to determine its diameter using Rotating Caliper method. In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points.
For many years the best algorithm of this type was that of Overmars and van Leeuwen (1981), which took time O(log 2 n) per update, but it has since been improved by Timothy M. Chan and others. In a number of applications finding the convex hull is a step in an algorithm for the solution of the overall problem.