給予A跟B這對來源序列和一條目標序列T，融合最長共同子序列 (MLCS) 這問題是找出E(A, B)跟T的最長共同子序列 (LCS)，E(A, B)可以經由融合A跟B序列得出。在這篇論文中，我們首先提出一個O(L(r-L+1)m)時間複雜度的演算法來解決融合最長共同子序列的問題，其中r跟L分別表示T序列的長度跟MLCS序列的長度，m表示A跟B兩序列較短的序列長度。從時間複雜度來看，當T跟E(A, B)非常相似時，我們的演算法會很有效率。在另一方面，對於不相似的情況也會很有效率。我們的演算法只要經過稍微的修改，也可以用O(L(r-L+1)m)的時間複雜度來解決融合最長共同子序列的變形問題 (區塊融合最長共同子序列)。實驗結果顯示當序列的相似度非常高，我們的演算法會比目前已知的MLCS跟BMLCS演算法快。

Given a pair of merging sequences A and B and a target sequence T, the merged longest common subsequence (MLCS) problem is to find out a longest common subsequence (LCS) between sequences E(A, B) and T, where E(A, B) is obtained from merging two subsequences of A and B. In this thesis, we first propose an algorithm for solving the MLCS problem in O(L(r-L+1)m) time, where r and L denote the lengths of T and MLCS length, respectively, and m denotes the minimum length of A and B. From the time complexity, it is clear that our algorithm is extremely efficient when T and E(A, B) are very similar. On the other hand, it is also efficient when the similarity is extremely low. With slight modification, our algorithm can also solve another variant merged LCS problem, the block-merged LCS problem, in O(L(r-L+1)m) time. Experimental results show that our algorithms are faster than other previously published MLCS and BMLCS algorithms for sequences with high similarity.

TABLE OF CONTENTS

論文審定書 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

DISSERTATION VERIFICATION FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

謝誌 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 The Longest Common Subsequence Problem . . . . . . . . . . . . . . . . . 5
2.2 The LCS Algorithm of Nakatsu and Yajima . . . . . . . . . . . . . . . . . . . 6
2.3 The Merged Longest Common Subsequence Problem . . . . . . . . . . . 9
2.4 The Block-Merged Longest Common Subsequence Problem . . . . .10

Chapter 3. Our Merged LCS and Block-Merged LCS Algorithms . . . . . . . . . 13

3.1 Our Merged LCS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Our Block-Merged LCS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 4. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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