在資訊科學中，字串比對是一個被充分研究的領域。它常常被應用在生物資訊當中，像是DNA序列的比對等等。由於DNA序列的長度可以是非常大的，所以在這個領域中，如何能壓縮空間是一個非常重要的課題。字串比對的問題有很多種變形，在這篇論文中，我們將探討其中有通配符的參數化字串比對問題，並且提出在時間空間上都有效率的解法。

我們將從只有一個通佩符的問題著手。我們首先介紹用通佩符樹的解法，它使用了O(nlogn)個字的空間，並且可以在O(p+occ)的時間內處理一個詢問字串（query pattern），其中n是文本（text）的長度、p是詢問字串的長度、occ是詢問字串在文本裡出現的次數。我們接著給出了空間壓縮的解法，它使用了O(nlog(sigma))個字的空間，並且可以在O(p(loglogn+log(sigma))+occlog(sigma))的時間內處理一個詢問，其中sigma是字母種類的數量。

接著，這個問題被延伸成有k個通佩符。我們首先建造出有k層的通佩符樹來解決這個問題，第一個解法用了O(nlog^k(n))個字的空間，並可以在O((2^k)(p+k^2)+occ)的時間內處理一個詢問。第二種解法壓縮了最後一層的通配符樹，使用O(nlog^(k-1)(n)log(sigma))個字的空間，並能在O((2^k)p(k+log(sigma))+occlog(sigma))的時間內處理一個詢問。

Pattern matching is a well-studied problem in Computer Science. It is frequently applied in bioinformatics, like the searching in DNA sequences whose lengths can be large, so compression of space becomes an important issue in this area. There are many variations of this problem. In this thesis, we focus on the problem of parameterized pattern matching with wildcards in pattern, and provide efficient indexing solutions to this problem.

We start with the 1-wildcard case. A solution using wildcard tree is introduced. It takes O(nlogn)-word indexing space and O(p+occ) query time, where n is the length of text, p is the length of pattern, and occ is the number of times that the pattern occurs in the text. We also give a second solution with compressed space, which uses O(nlog(sigma))-word indexing space with O(p(loglogn+log(sigma))+occlog(sigma)) query time, where sigma is the size of alphabet set.

Next, the problem is further extended to k-wildcards. Wildcard tree of k-layers is built to solve the problem. The first solution uses O(nlog^k(n))-word indexing space and can solve the problem with O((2^k)(p+k^2)+occ) query time. Similarly, we also provide a second solution with compressed space, which is obtained by compressing the last layer of the wildcard trees. It takes O(nlog^(k-1)(n)log(sigma))-word indexing space and each query takes O((2^k)p(k+log(sigma))+occlog(sigma)) time.

1 Introduction.........................................1
1.1 Problem Definition................................2
1.2 Baker’s Encoding and Parameterized Suffix Tree....4
2 AnO(nlogn)-word Index................................6
2.1 Handling Case 1...................................7
2.2 Handling Case 2...................................8
3 Compressing to anO(nlogσ)-word Index................10
3.1 Mapping..........................................10
3.1.1 Level 1 Mapping................................12
3.1.2 Level 2 Mapping................................12
3.2 Matching.........................................17
3.2.1 Proof of Lemma 4...............................19
4 Extension tok-PIWC..................................20
4.1 Naive Method.....................................21
4.2 Blind Search.....................................22
4.3 Compressing thek-th Layer........................25
5 Conclusion and Open Problems........................27

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