Linear-time nearest point algorithms for Coxeter lattices

TL;DR

This paper introduces two new algorithms for efficiently finding the nearest point in Coxeter lattices, achieving linear and near-linear time complexities, which improve computational efficiency for high-dimensional lattice problems.

Contribution

The paper presents two novel algorithms with worst-case complexities of O(n) and O(n log n) for nearest point search in Coxeter lattices, including special cases for $A_n$ and $A_n^*$.

Findings

Algorithms achieve linear and near-linear worst-case complexity.

Special cases for $A_n$ and $A_n^*$ reduce to existing simple algorithms.

Enhanced efficiency for high-dimensional lattice nearest point problems.

Abstract

The Coxeter lattices, which we denote $A_{n/m}$, are a family of lattices containing many of the important lattices in low dimensions. This includes $A_n$, $E_7$, $E_8$ and their duals $A_n^*$, $E_7^*$ and $E_8^*$. We consider the problem of finding a nearest point in a Coxeter lattice. We describe two new algorithms, one with worst case arithmetic complexity $O(n\log{n})$ and the other with worst case complexity O(n) where $n$ is the dimension of the lattice. We show that for the particular lattices $A_n$ and $A_n^*$ the algorithms reduce to simple nearest point algorithms that already exist in the literature.