本篇論文研究生物序列近似比對之演算法。生物序列比對問題向來為生物資訊學門之重要問題，旨在利用生物序列的比對來得出各生物在序列間的相同與相異性，以期更為了解生物體內各蛋白質之結構形成或其演化關係。
然生物序列比對之準確性問題由來便頗為生物學家所詬病，其原因主要在於比對時所時用的計分方式不全然正確所致。故而本篇論文先以基於計分方式為正確的狀況下、去求得同分之序列排列中較具生物意義的排列。次者再擴充問題為非同分但近似高分的排列中、是否有哪些排列較具生物意義？最後、有見於計分方式的分岐可能導致使用者的無所適從，本篇論文另外討論當多種計分方式同時使用時、當如何取得所謂的最佳排序？
本篇論文為促進生物序列排列的比對正確性，循序漸進地慢慢將問題擴大並討論及解決。多數問題在平方時間內解決、剩餘亦在三方時間內解決。未來方向可研究降低其時間與空間之需求，或推而廣之就其準確性做更深入的探討。

With the improvement of biological techniques, the amount of biosequences
data, such as DNA, RNA and protein sequences, are growing explosively.
It is almost impossible to handle such huge amount of data purely by manpower.
Thus the requirement of the great computing power is essential.
There are some ways to treat biosequence data, finding identical biosequences,
searching similar biosequences, or mining the signature of biosequences.
All of these are based on the same problems, the biosequence alignment
problems.
In this dissertation, we shall study the biosequence alignment problems to
raise the biological meaning of the optimal or near-optimal alignments since the
biologists and computer scientists sometimes argue
the biological meaning of the mathematically optimal alignment
obtained based on some scoring functions.

We first study the methods to improve the optimal alignment of two given
biosequences. Since usually the optimal alignment is not unique, there
should exist the best one among the optimal alignments, and we try to
extract this by defining some other criteria to judge the goodness of
the alignments when the traditional methods cannot decide which is the better one.

Two algorithms are proposed for solving the newly defined biosequence
alignment problems, the smoothest optimal alignment and the most
conserved optimal alignment problems. Some other criteria are also discussed
since most of them can be solved in a similar way.

Then we notice that the most biologically meaningful alignment may not
be the optimal one since there is no perfect scoring matrix. We address
our candidates in those near-optimal alignments, and present a tracing
marking function to get all near-optimal alignments and use the criterion
"the most conserved" to filter it, which is named as the
near-optimal block alignment (NBA) problem.

Finally, as everybody knows that existing scoring matrices are not
perfect at all, we try to figure out how we choose the winner
when multiple scoring matrices are applied. We define some
reasonable schemes to decide the winner alignment.

In this dissertation, we solve and discuss the algorithms for near-optimal
alignment problems on biosequences.
In the future, we would like to do some experiments to support
or reject these concepts.

TABLE OF CONTENTS
Page
LIST OF FIGURES iii
LIST OF TABLES v
LIST OF SYMBOLS viii
LIST OF ABBREVIATION x
ABSTRACT xi
1 Introduction 1
2 Preliminaries 5
2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Longest Common Subsequence Problem . . . . . . . . . . 7
2.2.1 Dynamic Programming Algorithm for 2-LCS . . . . . . 9
2.2.2 Linear Space Algorithm for 2-LCS . . . . . . . . . . . . 18
2.3 The Sequence Alignment Problem . . . . . . . . . . . . . . . . 20
2.3.1 Near-optimal Alignment . . . . . . . . . . . . . . . . . 22
2.3.2 Sequence Alignment Problem with Multiple Scoring
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 The Better Alignment among the Output Alignments 27
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 The Newly Defined Biosequence Alignment Problems . . . . . 29
3.2.1 The Smoothest Optimal Alignment . . . . . . . . . . . 29
i
3.2.2 The Most Conserved Optimal Alignment . . . . . . . . 32
3.2.3 The Miscellaneous Reasonable Optimal Alignments . . 37
3.3 The Algorithms for Solving the Newly Defined Problems . . . 40
3.3.1 An Algorithm for the Smoothest Optimal Alignment . 40
3.3.2 An Algorithm for the Most Conserved Optimal Align-
ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 54
4 Near-optimal Block Alignment 56
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Tracings in the Alignment Lattice . . . . . . . . . . . . . . . . 58
4.3 An Algorithm for Near-optimal Block Alignment . . . . . . . . 65
4.4 Comparing NBA with Affine Gap Penalty . . . . . . . . . . . 79
4.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 81
5 Finding Winner Alignments with Multiple Scoring Matrices 82
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Losing Score Lattice . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Finding the Winner Alignment . . . . . . . . . . . . . . . . . 87
5.4 Variants of the Comparing Function . . . . . . . . . . . . . . . 98
5.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 106
6 Conclusions 107
BIBLIOGRAPHY 110
INDEX 118

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