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Small Candidate Set for Translational Pattern Search

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Abstract

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

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Authors and Affiliations

  1. Department of Computer Science and Software Engineering, Penn State Erie, The Behrend College, Erie, USA

    Ziyun Huang

  2. School of Computer Science and Engineering, Central South University, Changsha, People’s Republic of China

    Qilong Feng & Jianxin Wang

  3. Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, USA

    Jinhui Xu

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  1. Ziyun Huang
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  2. Qilong Feng
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  3. Jianxin Wang
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  4. Jinhui Xu
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Correspondence to Ziyun Huang.

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The research of the first and last authors was supported in part by NSF through grant CCF-1716400. The research of the last author was also supported by NSF through grant IIS-1910492. A preliminary version of this paper has appeared in the Proceedings of the 30th International Symposium on Algorithms and Computation(ISAAC 2019)

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Huang, Z., Feng, Q., Wang, J. et al. Small Candidate Set for Translational Pattern Search. Algorithmica 84, 3034–3053 (2022). https://doi.org/10.1007/s00453-022-00997-x

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