摘 要
編輯距離問題已經被研究數十年。給予兩個序列(字串)A 和B，編輯距離問
題即是求得將A 轉換成B 的最小花費。根據可以被使用的編輯操作，編輯操作
的花費，以及輸入的字串格式，編輯距離問題可以分為多種的變形問題。變動長
度編碼字串以及循環的字串的編輯距離問題是輸入序列方面的變型問題。區塊編
輯問題是運算方面的變型問題。考慮連續刪除以及新增編輯操作的編輯距離問題
是編輯操作花費方面的變型問題。除此之外，基因重組問題也可以視為是編輯距
離問題的一種變型問題。在本論文中，我們回顧許多編輯距離問題相關的演算法，
變型問題以及基因重組問題。我們也透過實作不同的演算法來進行許多實驗，進
而說明不同演算法的實際執行效率。

關鍵詞：編輯距離、區塊編輯、基因重組、最長共同子序列、動態規劃、相似度

Abstract
The edit distance problem has been studied for several decades. Given sequences
(strings) A and B with length m and n, respectively, m ≤ n, the edit distance
problem is to find the minimum cost of operations required to transform A to B.
According to different models of cost functions, operations and input sequences, the
problem has several variants. The edit distance on run-length encoding sequences and
cyclic sequences are the variants on the input aspect. The block edit problem is a
variant on the operation aspect. The edit distance considering consecutive insertions
and deletions is another variant on the cost function. Besides, the genome
rearrangement problem can also be viewed as a variant, whose operations include
inversions, reversals and transpositions. In this thesis, we survey some algorithms for
the edit distance problem, its variants and the genome rearrangement problem. We
also perform some experiments to illustrate the execution efficiency of various
algorithms.
Keywords: Edit Distance, Block Edit, Genome Rearrangement, Longest Common
Subsequence, Dynamic Programing, Similarity

VERIFICATION FORM i
THANKS iii
CHINESE ABSTRACT iv
ENGLISH ABSTRACT v
LIST OF FIGURES ix
LIST OF TABLES xiv
LIST OF SYMBOLS xvi
1 Introduction 1
2 Preliminaries 3
2.1 Edit Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Longest Common Subsequence . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Longest Increasing Subsequence . . . . . . . . . . . . . . . . . . . . . 7
2.5 Run-length Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Edit Distance and Similarity 10
3.1 Simple Version by Wagner and Fischer . . . . . . . . . . . . . . . . . 10
3.2 Algorithm by Lowrance and Wagner . . . . . . . . . . . . . . . . . . 11
3.3 Edit Distance of Considering Consecutive Insertions and Deletions by
Waterman et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Time Bounds by Wong and Chandra . . . . . . . . . . . . . . . . . . 14
3.5 Algorithm by Masek and Paterson . . . . . . . . . . . . . . . . . . . . 14
3.6 Relation between Edit Distance and Similarity by Smith et al. . . . . 17
3.7 Algorithm for Edit Distance Considering Consecutive Insertions and
Deletions with Linear Cost Function by Gotoh . . . . . . . . . . . . . 19
3.8 Algorithm for Edit Distance Considering Consecutive Insertions and
Deletions with Concave Cost Function by Waterman . . . . . . . . . 21
3.9 Distance Function by Smith et al. . . . . . . . . . . . . . . . . . . . . 23
3.10 Algorithm by Myers . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.11 Algorithm for Edit Distance with Concave Cost Function by Miller
and Myers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.12 Algorithm for Edit Distance of Considering Consecutive Insertions
and Deletions with Mixed Convex and Concave Cost Function by
Eppstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.13 Algorithm by Wu et al. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.14 Cyclic String-to-String Correction Problem by Maes . . . . . . . . . . 41
3.15 Edit Distance on Run-length Encoding Sequences by Bunke and Csirik 42
3.16 Edit Distance of Cyclic Sequences by Marzal and Barrachina . . . . . 44
3.17 Edit Distance of Run-length Encoding Sequences by Arbell et al. . . . 45
3.18 Edit Distance of RNA Structures by Jiang et al. . . . . . . . . . . . . 50
3.19 Edit Distance between Run-length Encoding Sequence and Uncompressed
Sequence by Liu et al. . . . . . . . . . . . . . . . . . . . . . . 53
3.20 Block Edit Problem by Ann et al. . . . . . . . . . . . . . . . . . . . . 59
4 Genome Rearrangement 63
4.1 Sequence Alignment with Non-overlapping Inversions by Schoniger
and Waterman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Exact and Approximation Algorithms for Reversal Distance on Permutations
by Keceioglu and Sanko . . . . . . . . . . . . . . . . . . . 66
4.3 Lower Bounds for Reversal Distance on Permutations by Bafna et al. 69
4.4 Approximation Algorithm for Transposition Distance on Permutations
by Walter et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Upper Bounds and Lower Bounds for Reversal Distance and Transposition
Distance on Binary Sequences by Christie and Irving . . . . 75
4.6 An Ecient Algorithm for Computing Non-overlapping Inversion and
Transposition Distance by Ta et al. . . . . . . . . . . . . . . . . . . . 77
5 Experimental Results 80
5.1 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6 Conclusions and Future Work 103
BIBLIOGRAPHY 109
A Miscellaneous Experimental Results 115

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