Geometric properties of the monotonic lagrangian grid algorithm for near neighbor calculations

Abstract

Because spatial coordinates define a natural ordering of positions, it is always possible to associate with a set of randomly located points in a 3D space, grid indices which are ordered according to their relative positions. Such an indexing scheme can be used to construct a “monotonic Lagrangian grid” (MLG), a data structure where adjacent objects in space have close grid indices. Using an MLG to index positions and attributions of objects in computer memory permits a near neighbor algorithm to be based on a “maximum index offset”, Nc, rather than a short range “cutoff” distance Rc. An MLG algorithm removes the necessity of having to test distances or directions to determine adjacency. Further, “close” objects can be indexed via contiguous memory, thus permitting efficient vectorization of computations.