基因序列的比對是目前在計算生物領域非常重要的問題。許多文獻指出，如果有一組基因在多個基因序列上的距離相互靠的很近，那麼這一組基因往往具有功能上或歷史上的關聯性。若有一組基因在不同物種的染色體上兩兩相鄰的距離都不超過一個給定的臨介值 δ，便稱這組基因為一個「基因組」。「基因組樹」則是一種簡練的方法，可以用來表示出所有不同臨介值 δ 下可以產生出來的所有基因組。這一篇論文提出了一些新演算法能更有效率造出兩條染色體的基因組樹。對於這個問題，Zhang 和 Leong 曾經提出一個時間為O(n lg^2 n) 的演算法，其中 n 為基因的總數。這篇論文提出兩個改進演算法，時間分別為O(n lg n lglg n) 以及 O(n lg n α(n))，其中 α(n) 為 Ackermann's function 的反函數。和 Zhang 及 Leong 提出的演算法一樣，這兩個演算法都可以被推廣來造出 k > 2 條染色體的基因組樹，其時間都只會增加 k 倍。在實際的應用中，所有基因在染色體上的位置皆為整數。除了上述兩個改進的演算法外，在假設所有基因在染色體上的位置皆為整數的情況下，這篇論文另外提出了一個時間為 O(n lg n + z) 的演算法來造基因組樹，其中 z 為兩兩相鄰基因中最長的距離。在實際應用中，z 通常會小於 n，在這個狀況下，第三個演算法會比前兩個演算法更有效率。同樣的，這個演算法也可以被推廣來造出 k 條染色體的基因組樹，其所需的時間為 O(kn lg n + z)。

Comparing multiple genome sequences is an important method to discover new biological insights. If a group of genes remain physically close to each other in multiple genomes, often called a conserved gene cluster, then the genes may be either historically or functionally related. A gene team is a set of genes that appear in two or more species, possibly in a different order yet with the distance between adjacent genes in the team for each chromosome always no more than a certain threshold δ. A gene team tree is a succinct way to represent all gene teams for every possible value of δ. In this dissertation, new efficient algorithms are presented for the problem of constructing a gene team tree of two chromosomes. For this problem, Zhang and Leong had an O(n lg^2 n)-time algorithm, where n is the number of genes. In this dissertation, two improved algorithms are presented, which require, respectively, O(n lg n lglg n) and O(n lg n α(n)) time, where α(n) is the inverse of Ackermann's function. Similar to Zhang and Leong's gene-team-tree algorithm, the presented algorithms can be extended to k chromosomes with the time complexities increased only by a factor of k, where k > 2 is an integer. In practice, the distance between two genes is integer. In addition to the two improved algorithms, assuming that the positions of genes are integers, this dissertation presents an O(n lg n + z)-time algorithm, where z is the maximum distance between any two adjacent genes. For real-world applications, z is usually smaller than n and thus the third algorithm is more efficient than the other two algorithms. Similarly, it can be extended to k chromosomes. The extended algorithm requires O(kn lg n + z) time.

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