本篇論文針對最長共同子序列問題提出一個新的變型問題，名為具間隙型限制的最長共同子序列問題。給予兩個序列A、B，以及一個帶有間隙限制的限制序列C，其長度分別為m、n與r。本問題會找出一個A與B的最長共同子序列，而C必須為此答案的一個子序列，並且不能違反C的間隙限制。在本篇論文中，我們針對具間隙型限制的最長共同子序列問題以動態規劃的演算法提出兩個時間複雜度分別O(m2n2r)以及O(mnr × min(m ,n)的演算法。

This thesis considers a variant of the classical problem for finding the longest common subsequence (LCS) called longest common subsequence problem with a gapped constraint (LCSGC). Given two sequences A, B, and a constrained sequence C, which is accomplished with a corresponding gapped constraint for each symbol, whose lengths are m, n, and r, respectively, the LCSGC problem is to find an LCS of A and B, such that C is also a subsequence of this LCS and the gapped constraints corresponding to C are satisfied. In this thesis, two algorithms with time complexities O(m2n2r) and O(mnr × min(m, n)) are proposed based on the dynamic programming technique for solving the LCSGC problem.

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 The Longest Common Subsequence Problem . . . . . . . . . . . . . . 4
2.2 The Constrained Longest Common Subsequence Problem . . . . . . . 5
2.3 The Fixed Gapped Longest Common Subsequence Problem . . . . . . 7
2.4 The Variable Gapped Longest Common Subsequence Problem . . . . 7
2.5 The Longest Common Subsequence Problem with a Gapped Constraint 8
Chapter 3. Our Algorithms for the LCSGC Problem . . . . . . . . . 10
3.1 A Straightforward Algorithm . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 An Improved Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 4. Experimental Result . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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