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The Generalized Definitions of the Two-Dimensional Largest Common Substructure Problems

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Abstract

The similarity of two one-dimensional sequences is usually measured by the longest common subsequence (LCS) algorithms. However, these algorithms cannot be directly extended to solve the two or higher dimensional data. Thus, for the two-dimensional data, computing the similarity with an LCS-like approach remains worthy of investigation. In this paper, we utilize a systematic way to give the generalized definition of the two-dimensional largest common substructure (TLCS) problem by referring to the traditional LCS concept. With various matching rules, eight possible versions of TLCS problems may be defined. However, only four of them are shown to be valid. We prove that all of these four TLCS problems are \({\mathcal {NP}}\)-hard and \({\mathcal {APX}}\)-hard. To accomplish the proofs, two of the TLCS problems are reduced from the 3-satisfiability problem, and the other two are reduced from the 3-dimensional matching problem.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan

    Huang-Ting Chan, Hsuan-Tsung Chiu & Chang-Biau Yang

  2. Innovative DigiTech-Enabled Applications and Services Institute Institute for Information Industry, Kaohsiung, Taiwan

    Yung-Hsing Peng

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  1. Huang-Ting Chan
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  2. Hsuan-Tsung Chiu
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  3. Chang-Biau Yang
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  4. Yung-Hsing Peng
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Correspondence to Chang-Biau Yang.

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This research work was partially supported by the Ministry of Science and Technology of Taiwan under Contract MOST 104-2221-E-110-018-MY3. And, it was also partially supported by the “Online and Offline Integrated Smart Commerce Platform (3/4)” of the Institute for Information Industry, which is subsidized by the Ministry of Economy Affairs of Taiwan.

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Chan, HT., Chiu, HT., Yang, CB. et al. The Generalized Definitions of the Two-Dimensional Largest Common Substructure Problems. Algorithmica 82, 2039–2062 (2020). https://doi.org/10.1007/s00453-020-00685-8

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