Small Candidate Set for Translational Pattern Search

• Published in: Algorithmica Vol. 84; no. 10; pp. 3034 - 3053
• Main Authors: Huang, Ziyun, Feng, Qilong, Wang, Jianxin, Xu, Jinhui
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2022; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Hypercubes; Mathematics of Computing; Pattern matching; Pattern search; Theory of Computation; Translations
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-022-00997-x

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 ( n log 2 n ) in O d , ϵ ( n log 2 n ) 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.