  The fundamental model of a periodic structure is a periodic point set up to
rigid motion or isometry. Our recent paper in SoCG 2021 defined isometry
invariants (density functions), which are complete in general position and
continuous under perturbations. This work introduces much faster isometry
invariants (average minimum distances), which are also continuous and
distinguish some sets that have identical density functions. We explicitly
describe the asymptotic behaviour of the new invariants for a wide class of
sets including non-periodic. The proposed near linear time algorithm processed
a dataset of hundreds of thousands of real structures in a few hours on a
modest desktop.
