令 A 為一長度 n 的實數數列， L1 和 L2 為兩個整數且 L1 <= L2， R1 和R2 為兩個實數且 R1 <= R2。一段 A 的區間若其長度介於 L1 和 L2 之間且平均介於 R1 和 R2 之間，則該區間為「可行區間」。在本篇論文中，我們探討以下的問題： 找出 A 中全部的可行區間，計算 A 中全部可行區間的個數，找出 A 中的一組個數最多的不相交可行區間，找尋 A 中一個最長的可行區間，以及找尋 A 中的一個最短的可行區間。探討這些問題的動機是出自於找尋 DNA 序列上的 CpG 島 (CpG islands)。在本篇論文中，我們證明所有提出的問題都有 □(nlog n) 的時間下界 (lower bound)，此外，我們也利用幾何方法為所有提出的問題設計出最佳演算法。本篇論文中所有提出的演算法皆為即時演算法 (on-line algorithms)，並都使用 O(n) 的空間。

Let A be a sequence of n real numbers, L1 and L2 be two integers such that L1 <= L2 , and R1 and R2 be two real numbers such that R1 <= R2. An interval of A is feasible if its length is between L1 and L2 and its average is between R1 and R2. In this dissertation, we study the following problems: finding all feasible intervals of A, counting all feasible intervals of A, finding a maximum cardinality set of non-overlapping feasible intervals of A, locating a longest feasible interval of A, and locating a shortest feasible interval of A. The problems are motivated from the problem of locating CpG islands of a DNA sequence. Locating CpG islands is important for gene finding as well as for cancer research. In this dissertation, we firstly show that all the problems have an Ω(n log n)-time lower bound in the comparison model. Then, we use geometric approaches to design optimal algorithms for the problems. All the presented algorithms run in an on-line manner and use O(n) space.

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