Solution for exercise 1.2-3 in Introduction to Algorithms

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本文探讨了如何在一组数字中检测是否存在重复的数字，并提供了两种解决方案：一种是在θ(nlgn)时间内通过排序算法来实现，另一种是如果数字范围较小，则可以使用计数法在θ(n)时间内完成检测。

Consider the problem of determining whether an arbitrary sequence (x₁, x₂, . . . , x_n)of n numbers contains repeated occurrences of some number. Show that this can be done in θ(n lg n) time, where lg n stands for log₂ n.

Solution 1: Any sorting algorithms having log₂ n complexity will be a capable to solve it in θ(n lg n) time.

Solution 2: If the domain of the sequence is not big, this problem can be solved in θ(n) time using the counting method, compute count[X_i] in the first loop, then find if there is any count[X_i] bigger than 1.