1. 1 高等演算法 Homework One 暨南大學資訊工程學系 黃光璿 2004/11/11

2. 2 Problem 1

3. 3 Problem 2 Count the number of inversions for the permutation 5 7 1 4 3 2 6

4. 4 Problem 3 Explain when we can say that an algorithm for solving a problem is optimal, in the sense of theoretical analysis.  When the complexity of the algorithm matches the complexity lower bound for the problem that the algorithm intends to solve.

5. 5 Problem 4 Show that to find the largest number in an array of n numbers requires at least n-1 comparisons.  This problem asks to show that n-1 comparisons is the lower bound for any algorithm that guarantees the maximum.  The claim is true under the condition that you want to find the maximum only by comparisons.

6. 6 Let each number be a node in a graph. When two numbers are compared, link them by an edge. If only n-2 comparisons are being used, the graph cannot be connected. There are at least two components, and you cannot make sure which one contains the maximum.

7. 7 However, in the next slide, I will show you an algorithm that compute the maximum without using any comparison.

8. 8 The idea is simple. We can use arithmetic and logical operations to emulate the action of comparisons.  a > b iff a – b > 0  Extract the sign bit of a – b and call it s ( a, b ).  Then returns the maximum of a and b.

9. 9 The computation can be even more complex. In fact, we can compute the index of the maximum without using the temporary variable.

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