Well separated simplical decompositions using net-trees

Net trees are data structures which capture the hierarchy of nets for a point set. A net for a point set P
is a subset of the set such that it captures the structure of P at a certain scale. A net-tree behaves like a
compressed quad tree, but its size depends on the doubling dimension of the metric instead of the
ambient dimension. Cech complexes are structures which contain important topological information
about point sets at certain scales. A previous work by Kerber and Sharathkumar uses the compressed
quad tree construction on a point set to form a well separated simplicial decomposition, a
generalization of the well separated pair decomposition. These complexes approximate the
aforementioned Cech complex. The properties of the resulting structure depend on the ambient
dimension of the point set. Our work aims at adapting this construction to use net trees instead of
quad trees. The size of the well separated structure thus generated should depend on the doubling
dimension of the point set, which often is much less than the ambient dimension. This may help in
alleviating the curse of dimensionality for the approximate Cech complex.