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**Problem:** **Given:** Arrange the following functions in increasing order according to their growth: 1. \(\sqrt{2}\) 2. \(\log n\) 3. \(n^2\) 4. \(n!\) 5. \(\log (\log (n^5))\) 6. \(n^n\) 7. \(2^n\) **Solution:** To solve this problem, compare the growth rates of each function as \(n\) approaches infinity. Generally, functions grow in this order: \(\log n < \sqrt{2} < \log (\log (n^5)) < n^2 < 2^n < n! < n^n\) This arrangement helps in understanding the relative scalability of different algorithms and mathematical expressions. **Explanation:** - **\(\log n\)**: Logarithmic function grows very slowly. - **\(\sqrt{2}\)**: A constant, hence does not grow with \(n\). - **\(\log (\log (n^5))\)**: A double logarithm function grows very slowly, marginally faster than a constant. - **\(n^2\)**: Polynomial function, grows faster than logarithmic functions. - **\(2^n\)**: Exponential growth, very fast increase with \(n\). - **\(n!\)**: Factorial growth, faster than exponential as \(n\) increases. - **\(n^n\)**: This is the most rapid growth function of all listed here. Understanding these concepts is essential for analyzing algorithm efficiency and other computational problems.

**Problem:** **Given:** Arrange the following functions in increasing order according to their growth: 1. \(\sqrt{2}\) 2. \(\log n\) 3. \(n^2\) 4. \(n!\) 5. \(\log (\log (n^5))\) 6. \(n^n\) 7. \(2^n\) **Solution:** To solve this problem, compare the growth rates of each function as \(n\) approaches infinity. Generally, functions grow in this order: \(\log n < \sqrt{2} < \log (\log (n^5)) < n^2 < 2^n < n! < n^n\) This arrangement helps in understanding the relative scalability of different algorithms and mathematical expressions. **Explanation:** - **\(\log n\)**: Logarithmic function grows very slowly. - **\(\sqrt{2}\)**: A constant, hence does not grow with \(n\). - **\(\log (\log (n^5))\)**: A double logarithm function grows very slowly, marginally faster than a constant. - **\(n^2\)**: Polynomial function, grows faster than logarithmic functions. - **\(2^n\)**: Exponential growth, very fast increase with \(n\). - **\(n!\)**: Factorial growth, faster than exponential as \(n\) increases. - **\(n^n\)**: This is the most rapid growth function of all listed here. Understanding these concepts is essential for analyzing algorithm efficiency and other computational problems.

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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**Problem:** **Given:** Arrange the following functions in increasing order according to their growth: 1. \(\sqrt{2}\) 2. \(\log n\) 3. \(n^2\) 4. \(n!\) 5. \(\log (\log (n^5))\) 6. \(n^n\) 7. \(2^n\) **Solution:** To solve this problem, compare the growth rates of each function as \(n\) approaches infinity. Generally, functions grow in this order: \(\log n < \sqrt{2} < \log (\log (n^5)) < n^2 < 2^n < n! < n^n\) This arrangement helps in understanding the relative scalability of different algorithms and mathematical expressions. **Explanation:** - **\(\log n\)**: Logarithmic function grows very slowly. - **\(\sqrt{2}\)**: A constant, hence does not grow with \(n\). - **\(\log (\log (n^5))\)**: A double logarithm function grows very slowly, marginally faster than a constant. - **\(n^2\)**: Polynomial function, grows faster than logarithmic functions. - **\(2^n\)**: Exponential growth, very fast increase with \(n\). - **\(n!\)**: Factorial growth, faster than exponential as \(n\) increases. - **\(n^n\)**: This is the most rapid growth function of all listed here. Understanding these concepts is essential for analyzing algorithm efficiency and other computational problems.
Transcribed Image Text:**Problem:** **Given:** Arrange the following functions in increasing order according to their growth: 1. \(\sqrt{2}\) 2. \(\log n\) 3. \(n^2\) 4. \(n!\) 5. \(\log (\log (n^5))\) 6. \(n^n\) 7. \(2^n\) **Solution:** To solve this problem, compare the growth rates of each function as \(n\) approaches infinity. Generally, functions grow in this order: \(\log n < \sqrt{2} < \log (\log (n^5)) < n^2 < 2^n < n! < n^n\) This arrangement helps in understanding the relative scalability of different algorithms and mathematical expressions. **Explanation:** - **\(\log n\)**: Logarithmic function grows very slowly. - **\(\sqrt{2}\)**: A constant, hence does not grow with \(n\). - **\(\log (\log (n^5))\)**: A double logarithm function grows very slowly, marginally faster than a constant. - **\(n^2\)**: Polynomial function, grows faster than logarithmic functions. - **\(2^n\)**: Exponential growth, very fast increase with \(n\). - **\(n!\)**: Factorial growth, faster than exponential as \(n\) increases. - **\(n^n\)**: This is the most rapid growth function of all listed here. Understanding these concepts is essential for analyzing algorithm efficiency and other computational problems.
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