最長共同子序列問題為資訊科學與分子生物學之經典問題。共同子序列可明確指出多序列間的相似部份藉以了解序列間之關係。本論文著重在k條序列間之最長共同子序列問題 (k-LCS problem)、融合最長共同子序列問題 (merged LCS problem)與嵌合最長共同子序列問題(mosaic LCS problem)之研究。此三問題之目的分別為：找出多序列間之共同子序列；找出目標序列與由兩序列組成之融合序列間的序列交錯關係；找出目標序列與給定之序列集合中之序列嵌合關係。

給定k條序列，k條序列間之最長共同子序列問題之目標為找出這k條序列中之最長共同子序列。關於此問題，本論文提出了兩個保證最佳解至多為找出解之sigma倍之近似解演算法，其時間複雜度分別為O(sigma k n)與O(sigma^2 k n + sigma^3 n)，其中sigma為字母集合之大小、n為序列長度。由於此演算法所需之時間與空間複雜度較低，故可於資料庫搜尋時負責篩選出候選序列，並可應用於多序列之對齊與親緣樹之重建。

給定目標序列T與待融合序列A與B，融合最長共同子序列問題為找出T與由A與B交錯組成之最佳融合序列兩者間之最長共同子序列。其目標是找出A與B之間如何融合藉以了解T、A與B三序列間之交錯關係。我們首先提出了時間複雜度為O(n^3)的演算法解決此問題，其中n為序列長度。接著我們考慮生物序列之區塊資訊並提出相對應之區塊融合最長共同子序列問題。為解決此新問題，我們提出時間複雜度為O(n^2 m_b)之演算法，其中mb為序列中區塊之個數。利用S-table之資料結構與技巧，我們接著提出時間複雜度為O(n^2 + n m_b^2)之演算法以改進原來的演算法。

另外，為希望了解目標序列與序列集合間之嵌合關係，我們提出了嵌合最長共同子序列問題。此問題針對給定之目標序列T與序列集合S，從序列集合S中選擇k條可重複序列藉以組合出最佳嵌合序列C，使得T與C之間的共同子序列最長。考慮序列T上之可能斷點，我們提出了時間複雜度為O(n^2 m |S|+n^3 log k)之演算法，其中n為序列T之長度，m為序列集合S中最長序列之長度。利用前處理的技巧與S-table資料結構，我們亦提出時間複雜度為O(n(m+k)|S|)之演算法解決此問題。

The longest common subsequence (LCS) problem is a famous and classical problem in computer science and molecular biology. The common subsequence of multiple sequences shows the identical and similar parts in these sequences. This dissertation pays attention to the approximate algorithms for finding the LCS of $k$ input sequence ($k$-LCS problem), the merged LCS problem, and the mosaic LCS problem. These three problems try to hunt out the identical relationships among the $k$ sequences, the interleaving relationship between a target sequence and a merged sequence of a pair of sequences, and the mosaic relationship between a target sequence and a set of sequences, respectively.

Given $k$ input sequences, the $k$-LCS problem is to find the LCS which is common in all sequences. We first propose two $sigma$-approximate algorithms for the $k$-LCS problem with time complexities $O(sigma k n)$ and $O(sigma^{2} k n + sigma^{3} n)$ respectively, where $sigma$ and $n$ are the alphabet size and length of sequences, respectively. Experimental results show that our algorithms for 2-LCS could be a good filter to select the candidate sequences in database searching.

Given a target sequence $T$ and a pair of merging sequences $A$ and $B$, the merged LCS problem is to find the LCS of $T$ and the optimally merged sequence by merging $A$ and $B$ alternately. Its goal is to find a merging way for understanding the interleaving relationship of sequences. We first propose an algorithm with $O(n^{3})$ time for solving the problem, where $n$ is the sequence length. We further add the block information of input sequences in the blocked merged LCS problem. To solve the latter problem, we propose an algorithm with time complexity $O(n^{2}m_{b})$, where $m_{b}$ is the number of blocks. Based on the S-table technique, we can design an improved algorithm with $O(n^{2} + nm_{b}^{2})$ time.

Additionally, we desire to obtain the relationship between one sequence and a set of sequences. Given a target sequence $T$ and a set $S$ of source sequences, the mosaic LCS problem is to find the LCS of $T$ and a mosaic sequence $C$, composed of repeatable $k$ sequences in $S$. Based on the concept of break points in $T$, a divide and conquer algorithm is proposed with $O(n^2m|S|+ n^3log k)$ time, where $n$ and $m$ are the lengths of $T$ and the maximal length of sequences in $S$, respectively. Again, based on the S-table technique, an improved algorithm with $O(n(m+k)|S|)$ time is proposed by applying an efficient preprocessing.

1 Introduction .......... 1
2 Preliminaries .......... 8
2.1 Notations ......... 9
2.2 The Longest Common Subsequence Problem ......... 10
2.2.1 Dynamic Programming Algorithm for 2-LCS ......... 12
2.2.2 Linear Space Algorithms for 2-LCS ......... 14
2.3 The Sequence Alignment Problem ......... 16
2.4 Phylogeny Reconstruction ......... 21
2.4.1 Maximum Parsimony Tree ......... 23
2.4.2 Maximum Likelihood Tree ......... 27
2.5 Longest Common Subsequence Problems with Constraints ... 29
2.5.1 The Constrained Longest Subsequence Problem ..... 29
2.5.2 The Longest Increasing Subsequence Problem ..... 30
2.6 S-table and the Linear Time Merging Algorithm of LCS ... 31
3 Finding a Common Subsequence of Multiple Sequences 38
3.1 Motivation .................. 39
3.2 Related Works .................. 41
3.2.1 The Long Run Algorithm ......... 41
3.2.2 The Expansion Algorithm ......... 42
3.2.3 The Best Next Algorithm ......... 43
3.3 Our Algorithms for k-LCS ......... 44
3.4 Experimental Results ......... 48
3.5 Multiple Sequence Alignment Based on k-LCS ......... 51
3.6 Phylogeny Reconstruction Based on k-LCS ......... 56
3.7 Summary ......... 58
4 The Merged Longest Common Subsequence Problem 60
4.1 Background ......... 61
4.2 The Merged LCS Problem ......... 64
4.3 The Blocked Merged LCS Problem ......... 68
4.4 Result and Discussion ......... 74
4.5 Summary ......... 77
5 The Mosaic Longest Common Subsequence Problem ... 80
5.1 Background ......... 81
5.2 Algorithms for the Chimeric and Mosaic Alignment Problems ... 84
5.2.1 The 2-chimeric Alignment Problem ......... 84
5.2.2 The 2-mosaic Alignment Problem ......... 88
5.2.3 The k-mosaic Alignment Problem ......... 90
5.3 The Mosaic LCS Problem ......... 92
5.4 Algorithms for the Mosaic LCS Problem ......... 92
5.5 Summary ......... 99
6 Conclusion ......... 100
BIBLIOGRAPHY ......... 103
INDEX ......... 115

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