一個子序列是經由把一個給定的序列不刪或刪掉一些符號得來的。給定兩個序列，最長共同子序列問題就是找出一個共同擁有的子序列中長度最長的。而有條件限制的最長共同子序列問題則是找出一個最長的內含指定序列的子序列。注意一個附加條件限制的最長子序列可能比一般的最長子序列短。
在本論文中我們提出一個動態規劃方法的演算法來解有條件限制的最長共同子序列問題。所需的時間複雜度是O(pmn)，其中m和n為給定的序列的長度，p為要求的序列的長度。我們的演算法也可以應用到附加條件限制的序列比對問題。也就是序列比對問題在額外要求指定的符號對齊在同一行。

A subsequence is obtained by deleting zero or more symbols from a given sequence. For given two sequences, the longest common subsequence problem is to find a common subsequence whose length is the longest. The constrained longest common subsequence (CLCS) problem is to find a longest common subsequence that contains a specific subsequence. Note a CLCS may be shorter than an LCS.
In this thesis, we propose a dynamic programming algorithm for solving the CLCS problem. The time complexity is O(pmn), where m and n are the lengths of the given sequences and p is the length of the constraint sequence. Our algorithm
can also be applied to solve the constrained sequence alignment problem, which is a sequence alignment problem with the requirement that some specific symbols must be aligned together.

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 The Longest Common Subsequence Problem . . . . . . . . . . . . . . . . . . 4
2.2 The Constrained LCS Problem . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 3. An Algorithm for Solving the Constrained LCS Problem . . . . . . .10
3.1 The Constrained LCS Problem . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Algorithm and Complexity . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Proof of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 4. Simplified Algorithm . . . . . . . . . . . . . . . . . . . . . . . 21
4.1 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Algorithm and Complexity . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 5. Extension to the Sequences Alignment Problem . . . . . . . . . . . 23
5.1 Multiple Sequences Alignment . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Multiple Sequences Alignment with Constraints . . . . . . . . . . . . . . 25
Chapter 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

[1] A. Aho, D. Hirschberg, and J. Ullman, “Bounds on the complexity of the longest common subsequence problem,” Journal of the ACM, Vol. 23, No. 1, pp. 1–12, 1976.
[2] A. Apostolico and C. Guerra, “The longest common subsequence problem revisited,” Algorithmica, No. 2, pp. 315–336, 1987.
[3] L. Bergroth, H. Hakonen, and T. Raita, “A survey of longest common subsequence algorithms,” Seventh International Symposium on String Processing
Information Retrieval, pp. 39–48, 2000.
[4] V. Chavatal, D. Klarner, and D. Knuth, “Selected combinatorial research problem,” Tech. Rep. STAN-CS-72-292, Stanford Univ., p. 26, 1972.
[5] D. S. Hirschberg, “Algorithms for the longest common subsequence problem,”
Journal of the ACM, Vol. 24, No. 4, pp. 664–675, 1977.
[6] J. W. Hunt and T. G. Szymanski, “A fast algorithm for computing longest
common subsequences,” Communications of the ACM, Vol. 20, No. 5, pp. 350–353, 1977.
[7] J. Modelevsky, “Computer applications in applied genetic engineering,” Advances in Applied Microbiology, Vol. 30, pp. 169–195, 1984.
[8] E. W. Myers, “An O(nd) difference algorithm and its variations,” Algorithmica, No. 1, pp. 251–266, 1986.
[9] N. Nakatsu, Y. Kambayashi, and S. Yajima, “A longest common subsequence
algorithm suitable for similiar texts,” Acat Informatica, Vol. 18, pp. 171–179, 1982.
[10] C. Rick, “Simple and fast linear space computation of longest common subsequences,” Information Processing Letters, pp. 275–281, 2000.
[11] T. F. Smith and M. S. Waterman, “Comparison of biosequences,” Advances in Applied Mathematics, Vol. 2, pp. 482–489, 1981.
[12] R. A. Wagner and M. J. Fischer, “The string-to-string correction problem,”Journal of the ACM, Vol. 21, No. 1, pp. 168–173, 1975.
[13] S. Wu, U. Manber, G. Myers, and W. Miller, “An O(NP) sequence comparison
algorithm,” Information Processing Letters, Vol. 35, pp. 317–323, 1990.
[14] C. B. Yang and R. C. T. Lee, “Systolic algorithm for the longest common
subsequence problem,” Journal of the Chinese Institue of Engineers, Vol. 10,
No. 6, pp. 691–699, 1987.