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An Algorithm to Compute the Nearest Point in the Lattice $A_{n}^*$

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 Added by Robert McKilliam
 Publication date 2008
and research's language is English




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The lattice $A_n^*$ is an important lattice because of its covering properties in low dimensions. Clarkson cite{Clarkson1999:Anstar} described an algorithm to compute the nearest lattice point in $A_n^*$ that requires $O(nlog{n})$ arithmetic operations. In this paper, we describe a new algorithm. While the complexity is still $O(nlog{n})$, it is significantly simpler to describe and verify. In practice, we find that the new algorithm also runs faster.



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We consider the closest lattice point problem in a distributed network setting and study the communication cost and the error probability for computing an approximate nearest lattice point, using the nearest-plane algorithm, due to Babai. Two distinct communication models, centralized and interactive, are considered. The importance of proper basis selection is addressed. Assuming a reduced basis for a two-dimensional lattice, we determine the approximation error of the nearest plane algorithm. The communication cost for determining the Babai point, or equivalently, for constructing the rectangular nearest-plane partition, is calculated in the interactive setting. For the centralized model, an algorithm is presented for reducing the communication cost of the nearest plane algorithm in an arbitrary number of dimensions.
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