傳統最長共同子序列問題是在於在兩個序列之中找出最多合乎順序的配對，其中兩個一維序列的相似度可以利用最長共同子序列演算法來獲得，而此演算法已經被大量研究，然而利用類似於最長共同子序列的方法來計算二維資料的相似度仍值得我們研究探討。在本論文中，我們利用傳統最長共同子序列的概念給予二維最大共同子結構問題更加通用的定義，利用不同的配對條件，我們定義四種二維最大共同子結構的問題，我們也提出不同的轉換方法來證明兩個二維最大共同子結構問題屬於NP-hard，接著我們提出一些方法來解決二維最大共同子結構問題，首先我們提出了兩個利用整數線性規劃的公式來解決二維最大共同子結構問題，接著我們提供兩個啟發式演算法來有效率的找尋次佳解。

The traditional longest common subsequence (LCS) problem is to find the maximum number of ordered matches in two sequences. The similarity of two one-dimensional sequences can be measured by the LCS algorithms, which have been extensively studied. However, for the two-dimensional data, computing the similarity with an LCS-like approach remains worthy of investigation. In this thesis, we give the more generalized definition of the two-dimensional largest common substructure (TLCS) problem by referring to the traditional LCS concept. With different matching rules, we thus define four versions of TLCS problems. We also show that two of the TLCS problems are NP-hard by another proof way. Then, we develop some methods for solving the TLCS problem. We first provide two integer linear programming formulas to solve the TLCS problem. Furthermore, we devise two heuristic algorithms for finding a sub-optimal solution efficiently.

VERIFICATION FORM i
THESIS AUTHORIZATION FORM iii
ACKNOWLEDGMENTS iv
ABSTRACT v
ENGLISH ABSTRACT vi
LIST OF FIGURES ix
LIST OF TABLES xii
1 Introduction 1
2 Preliminary 4
2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The Longest Common Subsequence Problem . . . . . . . . . . . . . . 5
2.3 The k-Clique Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Integer Linear Programming . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 The Picture Retrieval Problem . . . . . . . . . . . . . . . . . . . . . . 11
2.7 The Two-Dimensional Largest Common Substructure Problem . . . . 13
3 Problem De nitions and Proofs of NP-hardness 19
3.1 Operator De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Problem De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Proofs of NP-hardness for P(ENE) and P(ENL) . . . . . . . . . . 33
vii
4 Algorithms for the TLCS Problems 40
4.1 Algorithm for Optimal Solutions in P(LNA) . . . . . . . . . . . . . . 40
4.2 Integer Linear Programming for TLCS Problem . . . . . . . . . . . . 43
4.2.1 ILP with a Straightforward Method . . . . . . . . . . . . . . . 45
4.2.2 ILP with the Matching Set Method . . . . . . . . . . . . . . . 51
4.3 Heuristic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 The First Heuristic Algorithm for P(ENE) . . . . . . . . . . 55
4.3.2 The Second Heuristic Algorithm for P(ENE) . . . . . . . . . 56
4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Conclusion 61
BIBLIOGRAPHY 63
A The pyomo code of the ILP formulas for P(ENL) 68
A.1 The Straightforward Method . . . . . . . . . . . . . . . . . . . . . . . 68
A.2 The Matching Set Method . . . . . . . . . . . . . . . . . . . . . . . . 73
B The pseudo code of ILP formulas for P(ENE) 76
C The pyomo code of ILP formulas for P(ENE) 78
C.1 The Straightforward Method . . . . . . . . . . . . . . . . . . . . . . . 78
C.2 The Matching Set Method . . . . . . . . . . . . . . . . . . . . . . . . 81
viii

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