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 An, E ₇ , E ₈ and their duals An*, 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(nlogn) and the other with worst case complexity O(n) where n is the dimension of the lattice. We show that for the particular lattices An and An* the algorithms are equivalent to nearest point algorithms that already exist in the literature.