Small Candidate Set for Translational Pattern Search

In this paper, we study the following pattern search problem: Given a pair of point sets A and B in fixed dimensional space R^d, with | B| = n, | A| = m and n≥ m, the pattern search problem is to find the translations T’s of A such that each of the identified translations induces a matching between T(A) and a subset B^′ of B with cost no more than some given threshold, where the cost is defined as the minimum bipartite matching cost of T(A) and B^′. We present a novel algorithm to produce a small set of candidate translations for the pattern search problem. For any B^′⊆ B with | B^′| = | A| , there exists at least one translation T in the candidate set such that the minimum bipartite matching cost between T(A) and B^′ is no larger than (1 + ϵ) times the minimum bipartite matching cost between A and B^′ under any translation (i.e., the optimal translational matching cost). We also show that there exists an alternative solution to this problem, which constructs a candidate set of size O_d,ϵ(nlog ²n) in O_d,ϵ(nlog ²n) time with high probability of success. As a by-product of our construction, we obtain a weak ϵ-net for hypercube ranges, which significantly improves the construction time and the size of the candidate set. Our technique can be applied to a number of applications, including the translational pattern matching problem.