字串定址及序列分析的相關問題已被研究了許多年。近年來，學者們更轉而研究這些問題的延伸版本，以符合最新的應用所需。在本論文中，我們致力於探討三個近年來開始受到矚目的問題，分別是(1)縮放字串的定址問題(the indexing problem for scaled strings)，(2)合併
限制下之最長共同子序列問題(the merged LCS problem)及其區塊延伸，以及(3)權重字元限制下的序列對齊問題(the sequence alignment problem with weighted constraints)。

首先，縮放字串定址問題在於如何對一文件字串T進行前置處理，以方便在T中快速搜尋字串P的α倍縮放字串。在本論文中，我們分別針對實數倍放大字串，整數倍放大字串，以及等比例縮放字串提出有效率的定址演算法。我們所提出的演算法大大改進了前人的方法，是目前已知最好的結果。其次，合併限制下之最長共同子序列問題旨在偵測序列之間的交錯關係，可應用於基因及訊號比對。在本論文中，我們亦對merged LCS 問題提出了比前人更有效率的解法。此方法尤其適用於字元種類數多的符號系統。最後，權重字元限制下的序列對齊問題為本論文中所提出的新問題。針對此問題，我們除了提出有效率的解法，亦證明我們的解法可以進一步用來解決其他類似的問題。因此，本論文所提出的演算法在字串定址及序列分析領域上有顯著的貢獻。

Related problems of string indexing and sequence analysis have been widely studied for a long time. Recently, researchers turn to consider extended versions of these problems, which provides more realistic applications. In this dissertation, we focus on three problems of recent interest, which are (1)the
indexing problem for scaled strings, (2)the merged longest common subsequence problem and its variant with blocks, and (3)the sequence alignment
problem with weighted constraints.

The indexing problem for scaled strings asks one to preprocess a text string T, so that the matched positions of a pattern string P in T, with some
scales α applied to P, can be reported efficiently. In this dissertation, we propose efficient algorithms for indexing real scaled strings, discretely scaled
strings, and proportionally scaled strings. Our indexing algorithms achieve either significant improvements to previous results, or the best known results. The merged longest common subsequence (merged LCS) problem aims to detect the interleaving relationship between sequences, which has important applications to genomic and signal comparison. In this dissertation, we propose improved algorithms for finding the merged LCS. Our
algorithms for finding the merged LCS are also more efficient than the previous results, especially for large alphabets. Finally, the sequence alignment problem with weighted constraints is a newly proposed problem in this dissertation. For this new problem, we first propose an efficient solution, and then show that the concept of weighted constraints can be further used to solve many constraint-related problems on sequences. Therefore, our results in this dissertation have significant contributions to the field of string indexing and sequence analysis.

LIST OF FIGURES iv
LIST OF TABLES v
LIST OF SYMBOLS vi
LIST OF ABBREVIATION vii
ABSTRACT viii
1 Introduction 1
1.1 String Matching and Indexing . . . . . . . . . . . . . . . . . . 2
1.2 Sequence Comparison . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Results Summary and Organization . . . . . . . . . . . . . . . 5
2 Prerequisite Knowledge 8
2.1 Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Techniques for String Indexing . . . . . . . . . . . . . . . . . . 9
2.2.1 Suffix Tree . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Suffix Array . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The Least Common Ancestor and the Range Minimum Query 12
2.4 Techniques for Sequence Comparison . . . . . . . . . . . . . . 15
2.4.1 Dynamic Programming for Finding the LCS . . . . . . 15
2.4.2 Hirschberg’s Divide-and-conquer Strategy . . . . . . . 16
3 Indexing Real Scaled Patterns 20
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 The Document Listing Problem . . . . . . . . . . . . . 22
3.2.3 Wang’s Preprocessing for the r-matching Decision Problem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 An Improved Indexing Algorithm . . . . . . . . . . . . . . . . 24
3.3.1 Properties for Indexing . . . . . . . . . . . . . . . . . . 25
3.3.2 New Efficient Indexing with RMQ . . . . . . . . . . . . 26
3.4 Implementation of Other Scaling Functions . . . . . . . . . . . 29
3.5 Summary and Future Work . . . . . . . . . . . . . . . . . . . 30
4 Indexing Discretely Scaled Patterns 32
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Wang’s Indexing Algorithm for Discretely Scaled Patterns . . 34
4.3 Efficient Preprocessing Dependent on Σ . . . . . . . . . . . . 36
4.4 Efficient Preprocessing Independent of Σ . . . . . . . . . . . . 44
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Indexing Proportionally Scaled Patterns 48
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Matching Proportionally Scaled Patterns . . . . . . . . . . . . 49
5.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.2 A Simple Linear Time Matching Algorithm . . . . . . . 51
5.3 Indexing Proportionally-scaled Patterns . . . . . . . . . . . . . 52
5.3.1 Constructing the suffix array of Tふ′ . . . . . . . . . . . 53
5.3.2 Two-phase Searching with RMQ . . . . . . . . . . . . . 55
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Improved Algorithms for Finding the Merged LCS 58
6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 The LCS with Merging Sequences . . . . . . . . . . . . . . . . 61
6.2.1 The Merged LCS Problem . . . . . . . . . . . . . . . . 61
6.2.2 The Block-merged LCS Problem . . . . . . . . . . . . . 62
6.3 The Two-dimensional Minima of Non-negative Integer Points . 63
6.4 Algorithms for the Merged LCS Problem . . . . . . . . . . . . 64
6.4.1 Locating Candidates . . . . . . . . . . . . . . . . . . . 64
6.4.2 The On-line Algorithm for Merged LCS . . . . . . . . . 65
6.4.3 The Off-line Implementation . . . . . . . . . . . . . . . 70
6.5 Extending the Algorithm with Block Constraints . . . . . . . 72
6.5.1 Candidates with Block Constraints . . . . . . . . . . . 72
6.5.2 Extended On-line and Off-line Algorithms . . . . . . . 74
6.5.3 Analysis for Large Alphabets . . . . . . . . . . . . . . 77
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7 Sequence Alignment with Weighted Constraints 80
7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.2 Sequence Alignment with Weighted Constraints . . . . . . . . 82
7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8 Conclusions 90
INDEX 104

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