   Chapter 11.4, Problem 38ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Quantities of the form k 1 n + k 2 n log   n for positive integers k 1 ,   k 2 , and n arise in the analysis of the merge sort algorithm in computer science. Show that for any positive integer k 1 ,   k 2 n   log 2 n is Θ ( n   log 2 n ) .

To determine

To prove:

Show that for any positive integer k, k1n+k2nlog2nis Θ(nlog2n).

Explanation

Given information:

Quantities of the form k1n+k2nlog2n for positive integers k1, k2, and narise in the analysis of the merge sort algorithm in computer science.

Proof:

When n2, then log2nlog22=1

|k1n+k2nlog2n|=|k1n1+k2nlog2n||k1nlog2n+k2nlog2n|=(k1+k2)|nlog2n| whenever n2

Assuming that n0:

|k1n+k2nlog2n||k10+k2nlog2n|=|k2nlog2n|=k2|nlog2n|

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