給定兩個序列A和B，長度分別為m和n，連續後綴匹配(CSA)問題為計算A序列與所有B序列子字串的最長共同子序列(LCS)，而儲存連續後綴匹配問題的二維矩陣稱為後綴表。給定A(r)和B的S-table與A(r-1)和B計算LCS的結果，我們可以在O(L)的時間解決兩個字串A(r-1)和A(r)相接與另一字串B的LCS，其中L表示LCS的長度。
此論文中，我們針對線性空間後綴表進行研究。Alves等學者首先提出線性空間後綴表的想法，但沒有更多深入的研究與實際的應用。我們提出時間複雜度為O(n)的演算法以解決字串相接與另一字串的LCS問題。我們也提出了時間複雜度為O(nlogn)的演算法來合併兩張線性空間後綴表，改善了原先合併兩張後綴表需要O(nL)的時間。最後，給定A與B的線性空間後綴表，我們提出了一個時間複雜度為O(n)的演算法，以計算Aα與B的線性空間後綴表，其中α為接在A序列後方的新字元。

Given two sequences A and B of lengths m and n, respectively, the consecutive suffix alignment problem is to compute the longest common subsequence (LCS) between A and each suffix of B. The data structure for the consecutive suffix alignment problem is named S-table, which is a two-dimensional matrix. The S-table of A(r) and B can be used to compute the LCS with the concatenation of two substrings (A(r-1) and A(r)) and B in O(L) time, with the alignment result of A(r-1) and B is given, where L denotes the LCS length.
In this thesis, we focus on the linear space S-table. The linear space S-table was fi rst proposed by Alves et al., but without further discussions and practical applications. We propose an algorithm to compute the LCS of two concatenated strings and one string in O(n) time by using the linear space S-table. We also propose an algorithm for merging two linear S-tables in O(n log n) time, instead of merging two S-table in quadratic time previously. At last, we propose an algorithm to compute the linear space S-table of A and B with given the linear space S-table of A and B in O(n) time, where denotes a new character appended to the tail of A.

THESIS VERIFICATION FORM . . . . . . . . . . . . . . . . . . . . . . i
THESIS AUTHORIZATION FORM . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
CHINESE ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 The Longest Common Subsequence Problem . . . . . . . . . . . . . . 3
2.2 The Consecutive Suffix Alignment Problem . . . . . . . . . . . . . . . 6
2.3 Applications of the S-table . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 The Merging of Two S-tables . . . . . . . . . . . . . . . . . . . . . . 11
2.5 The Segment Tree for Range Query and Range Update . . . . . . . . 12
Chapter 3. Our Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 The Alignment with the Linear Space S-table . . . . . . . . . . . . . 18
3.1.1 An O(n log n)-time Sequential Algorithm . . . . . . . . . . . . 21
3.1.2 An Algorithm with O(n) time . . . . . . . . . . . . . . . . . . 23
3.2 Merging two Linear Space S-tables . . . . . . . . . . . . . . . . . . . 27
3.3 An Algorithm for Computing the Linear Space S-table Incrementally 34
Chapter 4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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