| Summary: | 碩士 === 國立中山大學 === 資訊工程學系研究所 === 106 === The similarity of one-dimensional data is usually measured by the longest common subsequence (LCS) algorithms. However, these algorithms cannot be directly extended to solve the case with two or higher dimensional data. The two-dimensional largest common substructure (TLCS) problem was therefore proposed to compute the similarity of two-dimensional data. In 2016, Chan et al. defined eight different versions of the TLCS problem, and four of them were shown to be valid for pattern matching, while the other four are invalid. In addition, Chan et al. showed that two versions of them are NP-hard, and left a conjecture that the other two are also NP-hard. In this thesis, we prove that the remaining two versions of the TLCS problem are NP-hard, showing the correctness of Chan''s conjecture. Moreover, we prove that the four valid versions are all APX-hard.
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