Geometric Point Pattern Matching in the Knuth-Morris-Pratt Way

Given finite sets P and T of points in the Euclidean space Rd, the point pattern matching problem studied in this paper is to find all translations f ∈ Rd such that P + f ⊆ T. A fast search algorithm with some variants is presented for point patterns P that have regular grid-like geometric shape. The algorithm is analogous to the Knuth-Morris-Pratt algorithm of string matching. The time requirement of the search is O(r|T|) where r is the grid dimension of P. Pattern P has grid dimension r = 1 if it consists of evenly spaced points on a line. In general, a pattern P is an r-dimensional grid if it has for some p ∈ P and e1, ... , er ∈ Rd and positive integers m1, ... , mr a representation P = {p + i1e1 + ⋅⋅⋅ + irer | 0 ≤ ij ≤ mj} where the ij's are integers. Both P and T are given to the search algorithm in the lexicographic order.