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By Benoit A., Robert Y., Vivien F.

ISBN-10: 1439898138

ISBN-13: 9781439898130

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Additional info for A guide to algorithm design paradigms, methods, and complexity analysis

Example text

9 n Finally, for n sufficiently large, 2n log n log p 2 2 9 1 log(n) + 2n 2 2n 1 log(n) 2 2. 3. 7 merges two sorted lists. This algorithm stops performing any comparison as soon as all the elements of one of the two input sorted sets have been stored in the resulting array. Furthermore, each time a comparison is done, one element from one of the two input sorted sets is stored in C. Therefore, in the worst case, after (n 1) + (m 1) comparisons, there remains one element in input set A and one in B.

7 merges two sorted lists. This algorithm stops performing any comparison as soon as all the elements of one of the two input sorted sets have been stored in the resulting array. Furthermore, each time a comparison is done, one element from one of the two input sorted sets is stored in C. Therefore, in the worst case, after (n 1) + (m 1) comparisons, there remains one element in input set A and one in B. Then, one final comparison is needed for the algorithm to be able to complete its task. 7 is thus of n + m 1 comparisons.

A ternary tree of height h contains at most 3h leaves (see below). Hence: 3h n! ) n log3 (n) = Θ(n log(n)). We prove by induction that a ternary tree of height h contains at most 3h leaves. When h = 0, the tree is reduced to a single leaf and the result holds. We assume that the result holds for some value h and we consider a tree of height h + 1. Each of the three children of the root define a subtree of the root. These three subtrees of the root are ternary trees of height at most h. Therefore, using the induction hypothesis, each of these three trees contains at most 3h leaves.

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A guide to algorithm design paradigms, methods, and complexity analysis by Benoit A., Robert Y., Vivien F.


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