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{{Short description|Algorithm used for points in euclidean space}}
In [[electrical engineering]] and [[computer science]], '''Lloyd's algorithm''', also known as '''Voronoi iteration''' or relaxation, is an algorithm named after Stuart P. Lloyd for finding evenly spaced sets of points in subsets of [[Euclidean space]]s and partitions of these subsets into well-shaped and uniformly sized convex cells.<ref name="l82"/> Like the closely related [[k-means clustering|''k''-means clustering]] algorithm, it repeatedly finds the [[centroid]] of each set in the partition and then re-partitions the input according to which of these centroids is closest. In this setting, the mean operation is an integral over a region of space, and the nearest centroid operation results in [[Voronoi diagram]]s.▼
Although the algorithm may be applied most directly to the [[Euclidean plane]], similar algorithms may also be applied to higher-dimensional spaces or to spaces with other [[Non-Euclidean geometry|non-Euclidean]] metrics. Lloyd's algorithm can be used to construct close approximations to [[centroidal Voronoi tessellation]]s of the input,<ref name="dfg99"/> which can be used for [[Quantization (signal processing)|quantization]], [[dithering]], and [[stippling]]. Other applications of Lloyd's algorithm include smoothing of [[triangle mesh]]es in the [[finite element method]].▼
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| header = Example of Lloyd's algorithm.
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| caption4 = Iteration 15
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▲In [[electrical engineering]] and [[computer science]], '''Lloyd's algorithm''', also known as '''Voronoi iteration''' or relaxation, is an algorithm named after Stuart P. Lloyd for finding evenly spaced sets of points in subsets of [[Euclidean space]]s and partitions of these subsets into well-shaped and uniformly sized convex cells.<ref name="l82"/> Like the closely related [[k-means clustering|''k''-means clustering]] algorithm, it repeatedly finds the [[centroid]] of each set in the partition and then re-partitions the input according to which of these centroids is closest. In this setting, the mean operation is an integral over a region of space, and the nearest centroid operation results in [[Voronoi diagram]]s.
▲Although the algorithm may be applied most directly to the [[Euclidean plane]], similar algorithms may also be applied to higher-dimensional spaces or to spaces with other [[Non-Euclidean geometry|non-Euclidean]] metrics. Lloyd's algorithm can be used to construct close approximations to [[centroidal Voronoi tessellation]]s of the input,<ref name="dfg99"/> which can be used for [[Quantization (signal processing)|quantization]], [[dithering]], and [[stippling]]. Other applications of Lloyd's algorithm include smoothing of [[triangle mesh]]es in the [[finite element method]].
==History==
The algorithm was first proposed by Stuart P. Lloyd of [[Bell Labs]] in 1957 as a technique for [[pulse-code modulation]]. Lloyd's work became widely circulated but remained unpublished until 1982.<ref name="l82"/> A similar algorithm was developed independently by Joel Max and published in 1960,<ref name="m60"/> which is why the algorithm is sometimes referred as the Lloyd-Max algorithm.
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** Trivially, a set of [[Tetrahedron|tetrahedra]] is obtained by connecting triangles of the cell's hull with the cell's site.
Integration of a cell and computation of its [[
* Two dimensions:
** For a triangle the centroid can be easily computed, e.g. using [[Centroid#By geometric decomposition|cartesian coordinates]].
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==Applications==
Lloyd's method was originally used for scalar quantization, but it is clear that the method extends for [[vector quantization]] as well. As such, it is extensively used in [[data compression]] techniques in [[information theory]]. Lloyd's method is used in computer graphics because the resulting distribution has [[blue noise]] characteristics (see also [[Colors of noise]]), meaning there are few low-frequency components that could be interpreted as artifacts. It is particularly well-suited to picking sample positions for [[dithering]]. Lloyd's algorithm is also used to generate dot drawings in the style of [[stippling]].<ref name="dhos00"/> In this application, the centroids can be weighted based on a reference image to produce stipple illustrations matching an input image.<ref name="s02"/>
In the [[finite element method]], an input ___domain with a [[complex geometry]] is partitioned into elements with simpler shapes; for instance, two-dimensional domains (either subsets of the Euclidean plane or surfaces in three dimensions) are often partitioned into triangles. It is important for the convergence of the finite element methods that these elements be well shaped; in the case of triangles, often elements that are nearly equilateral triangles are preferred. Lloyd's algorithm
can be used to smooth a mesh generated by some other algorithm, moving its vertices and changing the connection pattern among its elements in order to produce triangles that are more closely equilateral.<ref name="dg02"/> These applications typically use a smaller number of iterations of Lloyd's algorithm, stopping it to convergence, in order to preserve other features of the mesh such as differences in element size in different parts of the mesh. In contrast to a different smoothing method, [[Laplacian smoothing]] (in which mesh vertices are moved to the average of their neighbors' positions), Lloyd's algorithm can change the topology of the mesh, leading to more nearly equilateral elements as well as avoiding the problems with tangling that can arise with Laplacian smoothing. However, Laplacian smoothing can be applied more generally to meshes with non-triangular elements.
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| volume = 19
| year = 2000| s2cid = 142991
| citeseerx = 10.1.1.233.5810
}}.</ref>
<ref name="dew10">{{citation
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| pages = 13–22
| title = Proc. 7th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2010)
| year = 2010| isbn = 978-1-4244-7606-0 | s2cid = 15971504 }}.</ref>
<ref name="dej06">{{citation
| last1 = Du | first1 = Qiang | author1-link = Qiang Du
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| title = Least squares quantization in PCM
| volume = 28
| year = 1982|
}}.</ref>
<ref name="sg86">{{citation
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| publisher = [[ACM SIGGRAPH]]
| title = Proceedings of the Symposium on Non-Photorealistic Animation and Rendering (NPAR)
| year = 2002|
| s2cid = 12153589
}}.</ref>
<ref name="h01">{{citation
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| pages = 573–580
| contribution = Simulating decorative mosaics
| year = 2001|
| s2cid = 7188986
}}.</ref>
<ref name="m60">{{citation
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