對於兩序列 A = a1a2a3 … am 以及 B = b1b2b3 … cn， 一個多重集區間為 ∆(A, i, j) = [ax | i ≤ x ≤ j]， 以及一個同時出現在兩序列的多重集，共同多重集區間 （common multisets interval，CMI）為∆(A, iA, jA) = ∆(B, iB, jB) 對於 $iA, jA, iB, jB. 1 ≤ iA ≤ jA ≤ m 且 1 ≤ iB ≤ jB ≤ n。
先前，研究者推出了用以找到兩個排列（permutation）以及序列（sequence）的共同區間演算法。 在這篇碩士論文中，我們推出兩個用來在兩序列中找到共同多重集區間的演算法。第一個演算法是 occurrence counting 演算法，它計算出現元素在兩輸入序列中所有區間的次數並且計算元素出現次數的差值。它的時間複雜度是O(n3)，n 代表輸入序列的長度。第二個演算法是 hash key 演算法，使用質數乘積以及模運算來建立哈希表以便加快搜尋。第二個演算法的時間複雜度是 O(n2 + Gn + qn) 或 O(n2|Σ| + G|Σ|+ q|Σ|)， G 代表答案的數量而 q 代表錯誤碰撞的數量。在我們的實驗中，我們使用CPE (Collegiate Programming Examination of Taiwan) 中的C/C++程式碼作為我們分類用的資料集。實驗結果顯示BKS (behavior knowledge space) 混合 LCS (longest common subsequence) 和 CMI 可以得到比兩個方法單獨使用還要高的準確度。

For two sequences A = a1a2a3 … am and B = b1b2b3 … cn, a multiset interval ∆(A, i, j) = [ax | i ≤ x ≤ j], and a common multisets interval (CMI) is ∆(A, iA, jA) = ∆(B, iB, jB) for some $iA, jA, iB, jB. 1 ≤ iA ≤ jA ≤ m and 1 ≤ iB ≤ jB ≤ n, which is a multiset that appears in both sequences.
Previously, researchers have proposed algorithms for finding the common set interval of permutations and sequences. In this thesis, we propose two algorithms to find common multiset intervals of two sequences. The first is the occurrence counting algorithm, which counts the occurrences of the characters in all intervals of the two input sequences and calculate the difference of character occurrences. Its time complexity is O(n3) time, where n denotes the length of the input sequences. The second is the hash key algorithm, which use the product of prime numbers and the modulo operation to build a hash table for quick search. The time complexity of the second algorithm is O(n2 + Gn + qn) or O(n2|Σ| + G|Σ|+ q|Σ|), where G denotes the number of answers and q denotes the number of error collisions. In our experiments, we use C/C++ source codes in CPE (Collegiate Programming Examination of Taiwan) as the data set for classification. The experimental results show that the BKS (behavior knowledge space) method with the combination of the LCS (longest common subsequence) and CMI classifiers can obtain better accuracy than the two methods alone.

中文論文審定書 i
英文論文審定書 ii
謝辭 iii
中文摘要 iv
英文摘要 v
TABLE OF CONTENTS vii
LIST OF FIGURES viii
LIST OF TABLES ix
Chapter 1. Introduction 1
1.1 Definitions 1
Chapter 2. Previous Works 3
2.1 The Common Set Intervals of Two Permutations 3
2.1.1 Algorithm 1 of Uno and Yagiura 4
2.1.2 Algorithm 2 of Uno and Yagiura 4
2.1.3 Algorithm 3 of Uno and Yagiura 6
2.1.4 Algorithm 4 of Uno and Yagiura 7
2.2 The Common Set Intervals of k Permutations 7
2.3 The Common Set Intervals of Two Sequences 8
2.4 The Common Set Intervals of k Sequences 10
2.5 The Longest Common Subsequence Problem 11
2.6 The Behavior Knowledge Space Method 11
Chapter 3. Algorithms for the Common Multiset Interval Problem 13
3.1 The Occurrence Counting Algorithm 13
3.2 The Hash Key Algorithm 14
Chapter 4. Experimental Results 26
4.1 Assembly Process 26
4.2 BKS with LCS and CMI 28
Chapter 5. Conclusions 31
BIBLIOGRAPHY 32

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