中文摘要
在電腦最佳化問題及生物領域中，最長共同子序列都是一個非常典型
的問題。在過去數十年中，非常多相關的論文被發表及討論。在這一
篇論文中，我們將做一個完整的相關回顧，包括最長共同子序列以及
其相關問題，並且在每一個問題中介紹一些有效率的演算法。在每個
問題的演算法中，我們都會給相關的時間複雜度及空間複雜度。

ABSTRACT
The longest common subsequence (LCS) problem is a classical problem both in com-
binational optimization and computational biology. During the past few decades,
a considerable number of studies have been focused on LCS and its related prob-
lems. In this thesis, we shall present a complete survey on LCS and its related
problems, and review some e±cient algorithms for solving these problems. We shall
also give the de‾nition to each problem, present some theorems, and illustrate time
complexity and space complexity of the algorithms for solving these problems.

TABLE OF CONTENTS
Page
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. The Longest Common Subsequence Problem . . . . . . . 4
2.1 The Longest Common Subsequence Problem . . . . . . . . . . . . . . 4
2.2 Hunt and Szymanki's Algorithm . . . . . . . . . . . . . . . . . . . . . 7
2.3 Hirschberg's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Some Algorithms of LCS . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Linear Space Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 3. Related Problems of LCS . . . . . . . . . . . . . . . . . . . 30
3.1 The Cyclic String Correction Problem . . . . . . . . . . . . . . . . . 30
3.2 The Longest Increasing Subsequence Problem . . . . . . . . . . . . . 34
3.3 The Constrained Longest Common Subsequence Problem . . . . . . . 37
Chapter 4. The Edit Distance Problem . . . . . . . . . . . . . . . . . . 46
4.1 The Edit Distance Problem . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Other Operations in the Edit Distance Problem . . . . . . . . . . . . 48
Chapter 5. Sequence Alignment . . . . . . . . . . . . . . . . . . . . . . . 53
5.1 Global Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Page
5.2 Local Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Affine Gap Penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Multiple Sequences Alignment . . . . . . . . . . . . . . . . . . . . . . 63
Chapter 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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