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Yihan recently learned the asymptotical analysis. The key idea is to evaluate the growth of a function. For example, she now knows that n² grows faster than n. She wants to know whether she really understands the idea, so she has created a little task. First of all, she found a lot of functions here: 91 3n log n² 92: n! log(n!) 93 n²+n 2n+1 94: n³ 96 97 911: √n 916 n 912 917 2n 913 918 nlogn 99 ln ln n 914 √log n 919 5n² 13n+6 log² n 910: 10000 915: log √n 920 n/log n Note: log n = log2 n; ln n = loge n; log²1 n = (log n)2; n! is the factorial of n, i.e., n! = 1 x 2 xxn. 95: 98 log log n (n + 1)! hen let's compare some them pairwise. a) b) c) Compare 912 log log n and 914 √logn. Which one grows faster? Please explain briefly. Compare 92: n! and 913: (n+1)!. Which one grows faster? Please explain briefly. Compare 920 n/logn and 911 √n. Which one grows faster? Please explain briefly. Yihan finds out that 912 log log n and 99: In ln n should be asymptotically equivalent (i.e.. log log n = (In ln n)). Can you prove this? d) :

Yihan recently learned the asymptotical analysis. The key idea is to evaluate the growth of a function. For example, she now knows that n² grows faster than n. She wants to know whether she really understands the idea, so she has created a little task. First of all, she found a lot of functions here: 91 3n log n² 92: n! log(n!) 93 n²+n 2n+1 94: n³ 96 97 911: √n 916 n 912 917 2n 913 918 nlogn 99 ln ln n 914 √log n 919 5n² 13n+6 log² n 910: 10000 915: log √n 920 n/log n Note: log n = log2 n; ln n = loge n; log²1 n = (log n)2; n! is the factorial of n, i.e., n! = 1 x 2 xxn. 95: 98 log log n (n + 1)! hen let's compare some them pairwise. a) b) c) Compare 912 log log n and 914 √logn. Which one grows faster? Please explain briefly. Compare 92: n! and 913: (n+1)!. Which one grows faster? Please explain briefly. Compare 920 n/logn and 911 √n. Which one grows faster? Please explain briefly. Yihan finds out that 912 log log n and 99: In ln n should be asymptotically equivalent (i.e.. log log n = (In ln n)). Can you prove this? d) :

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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**Asymptotical Analysis Exercise** Yihan recently learned about asymptotical analysis. The key idea is to evaluate the *growth* of a function. For example, she now knows that \( n^2 \) grows faster than \( n \). She wants to know whether she really understands the idea, so she has created a little task. First of all, she found a lot of functions here: - \( g_1 : 3^n \) - \( g_2 : n! \) - \( g_3 : n^2 + n \) - \( g_4 : n^3 \) - \( g_5 : \log^2 n \) - \( g_6 : \log n^2 \) - \( g_7 : \log (n!) \) - \( g_8 : 2^{n+1} \) - \( g_9 : \ln \ln n \) - \( g_{10} : 10000 \) - \( g_{11} : \sqrt{n} \) - \( g_{12} : \log \log n \) - \( g_{13} : (n+1)! \) - \( g_{14} : \sqrt{\log n} \) - \( g_{15} : \log \sqrt{n} \) - \( g_{16} : n \) - \( g_{17} : 2^n \) - \( g_{18} : n \log n \) - \( g_{19} : 5n^2 - 13n + 6 \) - \( g_{20} : n / \log n \) **Note:** - \( \log n = \log_2 n \) - \( \ln n = \log_e n \) - \( \log^2 n = (\log n)^2 \) - \( n! \) is the factorial of \( n \), i.e., \( n! = 1 \times 2 \times \cdots \times n \). --- **Task: Compare Some Functions Pairwise** 1. **(a)** Compare \( g_{12} : \log \log n \) and \( g_{14} : \sqrt{\log n} \). Which one grows faster? Please explain briefly. 2. **(b)** Compare \( g_
Transcribed Image Text:**Asymptotical Analysis Exercise** Yihan recently learned about asymptotical analysis. The key idea is to evaluate the *growth* of a function. For example, she now knows that \( n^2 \) grows faster than \( n \). She wants to know whether she really understands the idea, so she has created a little task. First of all, she found a lot of functions here: - \( g_1 : 3^n \) - \( g_2 : n! \) - \( g_3 : n^2 + n \) - \( g_4 : n^3 \) - \( g_5 : \log^2 n \) - \( g_6 : \log n^2 \) - \( g_7 : \log (n!) \) - \( g_8 : 2^{n+1} \) - \( g_9 : \ln \ln n \) - \( g_{10} : 10000 \) - \( g_{11} : \sqrt{n} \) - \( g_{12} : \log \log n \) - \( g_{13} : (n+1)! \) - \( g_{14} : \sqrt{\log n} \) - \( g_{15} : \log \sqrt{n} \) - \( g_{16} : n \) - \( g_{17} : 2^n \) - \( g_{18} : n \log n \) - \( g_{19} : 5n^2 - 13n + 6 \) - \( g_{20} : n / \log n \) **Note:** - \( \log n = \log_2 n \) - \( \ln n = \log_e n \) - \( \log^2 n = (\log n)^2 \) - \( n! \) is the factorial of \( n \), i.e., \( n! = 1 \times 2 \times \cdots \times n \). --- **Task: Compare Some Functions Pairwise** 1. **(a)** Compare \( g_{12} : \log \log n \) and \( g_{14} : \sqrt{\log n} \). Which one grows faster? Please explain briefly. 2. **(b)** Compare \( g_
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