最長共同遞增子序列問題 (LCIS)是為了找出兩條序列的最大長度的共同遞增子序列。在這篇論文中，我們提出了一個O((n+L(m-L))loglog |Σ|)時間複雜度的演算法來解決LCIS的問題，其中m和n分別為A和B的長度，m≤n，L是LCIS的長度，Σ是指字母集。我們的演算算法主要思想是擴展一些以前可行的支配操作來解決方案的答案。我們利用van Emde Boas tree的資料結構來完成延伸和支配的操作。從我們的時間複雜度可以看出，當L非常小或是非常大的時候會很有效率，最後我們的實驗數據可以顯示我們演算法的效率。

The longest common increasing subsequences　(LCIS) problem is to find out a common increasing subsequence with the maximal length of two given sequences. In this thesis, we propose an algorithm for solving the LCIS problem in O((n+L(m-L))loglog |Σ|) time, where m and n denote the lengths of A and B, respectively, m≤n, L denotes the LCIS length, andΣdenotes the alphabet set. The main idea of our algorithm is to extend the answer from some previously feasible solutions, in which the domination operation is invoked. To accomplish the extension and domination operations, the data structure of van Emde Boas tree is utilized. From the time complexity, it is obvious that our algorithm is extremely efficient when L is very small or very large. Some experiments are executed to show the efficiency of our algorithm.

VERIFICATION FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

THESIS AUTHORIZATION FORM . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

CHINESE ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ENGLISH ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 The Diagonal Algorithm for the Longest Common Subsequence Problem 4
2.2 The Longest Increasing Subsequence Problem . . . . . . . . . . . . . 6
2.3 The Longest Common Increasing Subsequence Problem . . . . . . . . 7

Chapter 3. The Algorithm for the Longest Common Increasing
Subsequence Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 The Diagonal Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Efficient Implementation with van Emde Boas Trees . . . . . . . . . . 16

Chapter 4. Experimental Results . . . . . . . . . . . . . . . . . . . . . . 19

4.1 The Longest Common Increasing Subsequence . . . . . . . . . . . . . 19
4.2 The Longest Common Weakly Increasing Subsequence . . . . . . . . 24

Chapter 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Appendixes

A. The Complete LCIS Results . . . . . . . . . . . . . . . . . . . . . . . 47

B. The Complete LCWIS Results . . . . . . . . . . . . . . . . . . . . . . 55

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