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

TL;DR

This paper introduces a new, simpler, and faster algorithm for finding the nearest point in the $A_n^*$ lattice, maintaining the same theoretical complexity as previous methods.

Contribution

The paper presents a novel algorithm for nearest point computation in $A_n^*$ lattice that is simpler to understand and verify, with improved practical performance.

Findings

Algorithm runs faster in practice

Maintains $O(n ext{log}n)$ complexity

Simpler to describe and verify

Abstract

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 $O(n\log{n})$ arithmetic operations. In this paper, we describe a new algorithm. While the complexity is still $O(n\log{n})$, it is significantly simpler to describe and verify. In practice, we find that the new algorithm also runs faster.