Linear-Time Nearest Point Algorithms for Coxeter Lattices

The Coxeter lattices are a family of lattices containing many of the important lattices in low dimensions. This includes, 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 are equivalent to nearest point algorithms that already exist in the literature.