Small Candidate Set for Translational Pattern Search

Ziyun Huang, Qilong Feng, Jianxin Wang, Jinhui Xu

Abstract: In this paper, we study the following pattern search problem: Given a pair of point sets A and B in fixed dimensional space $$\mathbb {R}^d$$ R d , with $$|B| = n,|A| = m$$ | B | = n , | A | = m and $$n \ge m$$ n ≥ m , the pattern search problem is to find the translations $$\mathcal {T}$$ T ’s of A such that each of the identified translations induces a matching between $$\mathcal {T}(A)$$ T ( A ) and a subset $$B'$$ B ′ of B with cost no more than some given threshold, where the cost is defined as the minimum bipartite matching cost of $$\mathcal {T}(A)$$ T ( A ) and $$B'$$ B ′ . We present a novel algorithm to produce a small set of candidate translations for the pattern search problem. For any $$B' \subseteq B$$ B ′ ⊆ B with $$|B'| = |A|$$ | B ′ | = | A | , there exists at least one translation $$\mathcal {T}$$ T in the candidate set such that the minimum bipartite matching cost between $$\mathcal {T}(A)$$ T ( A ) and $$B'$$ B ′ is no larger than $$(1+\epsilon )$$ ( 1 + ϵ ) times the minimum bipartite matching cost between A and $$B'$$ 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 $$O_{d,\epsilon }(n \log ^2 n)$$ O d , ϵ ( n log 2 n ) in $$O_{d,\epsilon }(n \log ^2 n)$$ O d , ϵ ( n log 2 n ) time with high probability of success. As a by-product of our construction, we obtain a weak $$\epsilon $$ 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.