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

Given finite sets P and T of points in the Euclidean space R, the point pattern matching problem studied in this paper is to find all translations f ∈ R 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 ∈ R 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.