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
(
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
An
and
An
* the algorithms are equivalent to nearest point algorithms that already exist in the literature.