New PDF release: Algorithms: Design Techniques and Analysis (Lecture Notes

By M. H. Alsuwaiyel

ISBN-10: 9810237405

ISBN-13: 9789810237400

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Extra resources for Algorithms: Design Techniques and Analysis (Lecture Notes Series on Computing)

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At first glance, one may start by looking a t the while loop, the if statements, etc. in order t o find out how the algorithm works and then compute the number of element assignments. However, it is easy to see that each entry of array B is assigned exactly once. Similarly, each entry of array A is assigned exactly once, when copying B back into A. 2 The number of element assignments performed by Algorithm MERGE to merge two arrays into one sorted array of size n is exactly 2n. n]be an array of n elements.

If there are three elements left, then two (sorted) elements are merged with one element to form a 3-element sorted sequence. Continuing this way, in the j t h iteration, we merge in/2jJ pairs of sorted sequences of size 2 j - I to yield ln/2jJ sorted sequences of size 2 j . If there are k remaining elements, where I 5 k 5 2j-l, Basic Concepts in Algorithmic Analysis 18 then they are passed to the next iteration. If there are k remaining elements, where 2j-l < k < 2 j , then these are merged to form a sorted sequence of size k.

N]in the second iteration, there are two cases to consider according to whether n is even or odd. n]is n/2; otherwise it is ( n - 1)/2. n] is exactly [n/2J. Similarly, the number of remaining elements to be searched in the third iteration is LLn/2]/2] = Ln/41 (see Eq. 3 on page 71). In general, in the j t h pass through the while loop, the number of remaining elements is [n/2j-'J. The iteration is continued until either z is found or the size of the subsequence being searched reaches 1, whichever + + Binary Search 11 occurs first.

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Algorithms: Design Techniques and Analysis (Lecture Notes Series on Computing) by M. H. Alsuwaiyel


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