k-LCS問題是為了找出k條序列的最長共同子序列(LCS)，當序列的數量很大時，此問題是困難的。以往，研究者都著重於尋找兩條序列的最長共同子序列 (2-LCS) 之研究，然而對於找出k條序列的最長共同子序列，從古至今仍未有好的演算法能夠找到最佳解。為了解決k-LCS問題，在本論文中，我們首先提出一個混合式的演算法，它結合啟發式演算法、基因演算法(GA)和螞蟻演算法(ACO)。
此外，我們也提出一個增強式螞蟻演算法(EACO)，它以螞蟻演算法為主體結合啟發式演算法和最佳配對演算法(MPA)。在我們實驗中，我們的演算法與expansion演算法、best next for maximal available symbol演算法、基因演算法和螞蟻演算法相互比較，對於多組基因序列與蛋白質序列的資料集實驗，我們的演算法(EACO)的結果都優於其他的演算法。

The k-LCS problem is to find the longest common subsequence (LCS) of k input sequences. It is difficult while the number of input sequences is large.
In the past, researchers focused on finding the LCS of two sequences (2-LCS). However, there is no good algorithm for finding the optimal solution of k-LCS up to now. For solving the k-LCS problem, in this thesis, we first propose a mixed algorithm, which is a combination of a heuristic algorithm, genetic algorithm (GA) and ant colony optimization (ACO) algorithm.
Then, we propose an enhanced ACO (EACO) algorithm, composed of the heuristic algorithm and matching pair algorithm (MPA). In our experiments, we compare our algorithms with expansion algorithm, best next for maximal available symbol algorithm, GA and ACO algorithm. The experimental results on several sets of DNA and protein sequences show that our EACO algorithm outperforms other algorithms in the lengths of solutions.

LIST OF FIGURES ii
LIST OF TABLES iii
ABSTRACT v
1 Introduction 1
2 Preliminaries 4
2.1 The Longest Common Subsequence Problem . . . . . . . . . . . . . . . 4
2.2 Dynamic Programming Algorithm for 2-LCS . . . . . . . . . . . . . . . 5
2.3 Algorithms for k-LCS . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 The Expansion Algorithm . . . . . . . . . . . . . . . . . . 9
2.3.2 The Best Next for Maximal Available Symbols . . . . . . . . 10
2.3.3 Genetic Algorithm for k-LCS . . . . . . . . . . . . . . . . 11
2.3.4 Ant Colony Optimization Algorithm for k-LCS . . . . . . . . 13
3 Hybrid Algorithms for k-LCS 17
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Mixed Algorithm with ACO and GA . . . . . . . . . . . . . . . . . . . 18
3.3 Enhanced ACO Algorithm with MPA . . . . . . . . . . . . . . . . . . . 26
4 Experimental Results 31
5 Conclusion 50
BIBLIOGRAPHY 52

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