Efficient extraction of canonical spatial relationships using a recursive enumeration of k-subsets

The spatial arrangement of a chemical compound plays an important role regarding the related properties or activities. A straightforward approach to encode the geometry is to enumerate pairwise spatial relationships between k substructures, like functional groups or subgraphs. This leads to a combinatorial explosion with the number of features of interest and redundant information. The goal of this work is to compute all possible k-subsets of spatial points and to extract a single canonical descriptor for each subset in sub-polynomial computation time. More precisely, the problem is to reduce the complexity of nk = n·(n 1)...(n k) possible relationships (patterns or descriptors) for n features and k-point relationships. We propose a two-step algorithm to solve this problem. A modified algorithm for the computation of the binomial coefficient computes the k-subsets [1] containing the possible combinations of the n relevant features. If a k-subset is completed in the inner recursion, the algorithm computes a canonical representation for it. By defining a natural order by means of the geometrical center of gravity of the k points, we extract k patterns that describe the distance to the center of gravity and type of the spatial feature k Î F. Then, the algorithm returns a