Lloyd's algorithm: Difference between revisions

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In [[computer science]] and [[electrical engineering]], '''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. However,In Lloyd'sthis algorithmsetting, differsthe frommean ''k''-meansoperation clusteringis inan thatintegral its input isover a continuous geometric region rather than a discrete set of points. Thusspace, when re-partitioningand the input,nearest Lloyd'spoint algorithmoperation usesresults in [[Voronoi diagram]]s rather than simply determining the nearest center to each of a finite set of points as the ''k''-means algorithm does.

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]].