過去數十年以來，最長共同子序列問題及其變形都已經有很深的研究。其中，限制最長共同子序列問題為給定 A 與 B 兩條序列以及一條限制序列 C，長度分別為m、n、r，找出 A 與 B 的最長共同子序列，而且此序列必須包含限制序列 C。本論文中提出一個演算法基於Nakatsu等學者提出的對角線概念來得到CLCS的長度。我們的演算法能更有效找到CLCS的長度並且時間複雜度與空間複雜度為O(rL(m-L))跟O(mr)，L 為CLCS長度。如實驗結果所示，我們的演算法表現出比先前發表的演算法還要好的效能。

The longest common subsequence (LCS) problem and its variations have been studied deeply in past decades. In the constrained longest common subsequence (CLCS) problem, given three sequences A, B, and C of lengths m, n, and r, respectively, its goal is to find the LCS of A and B that C is a subsequence contained in the LCS answer. This thesis proposes an algorithm for obtaining the CLCS length based on the diagonal concept for finding the LCS length proposed by Nakatsu et al. Our algorithm can find the CLCS length more efficiently with O(rL(m - L)) time and O(mr) space, where L is the CLCS length. As the experimental results show, our CLCS algorithm outperforms the previously published algorithms.

TABLE OF CONTENTS
Page
VERIFICATION FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
THESIS AUTHORIZATION FORM . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
CHINESE ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 The Longest Common Subsequence Problem . . . . . . . . . . . . . . 5
2.2 The Longest Common Subsequence Algorithm Proposed by Nakatsu
et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 The Constrained Longest Common Subsequence
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 The Constrained Longest Common Subsequence Algorithm
Proposed by Tsai . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 The Constrained Longest Common Subsequence Algorithm
Proposed by Chin et al. . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 The Constrained Longest Common Subsequence Algorithm
Proposed by Arslan and Egecioglu . . . . . . . . . . . . . . . 16
Chapter 3. The Diagonal-based Algorithm . . . . . . . . . . . . . . . . 17
Chapter 4. Experimental Results . . . . . . . . . . . . . . . . . . . . . . 24
Chapter 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Appendixes
A. Miscellaneous Experimental Results . . . . . . . . . . . . . . . . . . 42

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