傳統的最長共同子序列 (LCS) 問題，欲設法從給定的兩序列間，找出具有最多匹配字元的共同子序列，而並未考慮匹配字元間的連續與否。然而在許多應用領域，即便匹配字元的個數較少，較為連續的序列對齊方式，是比起零碎而分散者來得有意義。於是，我們在此提出新的 LCS 變形問題，稱作「彈性最長共同子序列」(FLCS) 問題。於此論文中，我們設計用以評定序列對齊連續性的評分函數，並發展出有效率解決 FLCS 問題的演算法。藉由動態規劃的方式，我們先是以直覺的想法提出了時間複雜度為 O(n^3) 的演算法，其中 n 為輸入序列中長度較長者的長度。接著，我們導入優勢清單的概念，以減少格子間冗餘的計算。最後，我們提出了有效率的演算法, 能夠在 O(mn) 的時間複雜度內解決 FLCS 問題，m 與 n 分別代表兩輸入序列的長度。

Given two sequences, the traditional longest common subsequence (LCS) problem is to obtain the common subsequence with the maximum number of matches, without considering the continuity of the matched characters. However, in many applications, the alignment results with higher continuity are more meaningful than the sparse ones, even if the number of matched characters is a little lower. Accordingly, we define a new variant of the LCS problem, called the flexible longest common subsequence (FLCS) problem. In this thesis, we design a scoring function to estimate the continuity of an alignment between two strings, and develop efficient algorithms for solving the FLCS problem. We first propose a straightforward method with the dynamic programming approach, which requires O(n^3) time, where n denotes the longer length of the input sequences. Then, we apply the concept of dominant lists to reduce the redundant computation in each lattice cell. Finally, we propose an efficient algorithm for solving the FLCS problem with O(mn) time, where m and n denote the lengths of the two input sequences.

LIST OF FIGURES ........................................ iii
ABSTRACT ................................................. v
Chapter 1. Introduction .................................. 1
Chapter 2. Preliminaries ................................. 5
2.1 Notations ............................................ 5
2.2 The Longest Common Subsequence Problem ............... 5
2.3 Affine Gap Penalty ................................... 7
2.4 Dynamic Time Warping ................................. 9
Chapter 3. The Scoring Function ..........................12
Chapter 4. The Algorithms for the FLCS .................. 15
4.1 The Algorithm with Straightforward Dynamic Programing 15
4.2 The Algorithm with the Dominant Strategy ............ 18
4.3 The Efficient Dominating Method ..................... 23
4.4 The O(mn)-time Algorithm ............................ 30
Chapter 5. Conclusion ................................... 37
BIBLIOGRAPHY ............................................ 38
Appendixes
A. The Intersection Point When γ = 4 ................... 41
B. The Intersection Point When γ = 5 ................... 42

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