pP

In this paper we define a point set similarity measure that includes both the bottleneck distance and the Hausdorff distance as special cases. This measure relaxes the condition that the mapping must be one-to-one, but guarantees that only a few points are mapped to any point. Using a novel application of Hall's Theorem to reduce the geometric matching problem to a combinatorial matching problem, we present near-linear time approximation schemes for minimising this distance between two point sets in the plane under isometries; we note here that the best known algorithms for congruence under the bottleneck measure run in time .

We also obtain a combinatorial bound on the metric entropy of certain families of geometric objects. This result yields improved algorithms for approximate congruence, and may be of independent interest.