### Analyzing Asymptotic Growth RatesAlice has asked you to help her classify these functions based on their asymptotic growth rate. That is, she wants you to prove a tight big-Theta bound for each function by following these steps:1. **Algebraic Simplification:** Use algebra to simplify the function for ease of analysis. Remember, the goal of simplification is to isolate terms and identify the dominant term. For instance, we might want to expand \((n + 1)^2\) to \(n^2 + 2n + 1\) to see that \(n^2\) is the dominant term.2. **Guess the Tight Bound:** Use your intuition, the limit test, or algebra to come up with a good guess of a tight bound. For instance, after isolating \(n^2\) as the dominant term, we might guess our polynomial is in \(\Theta(n^2)\).3. **Proof of Guess:** Use the limit test or the definitions to prove that your guess is correct. Both methods can provide correct proofs, and it’s a good idea to practice both methods. Sometimes our guess is incorrect, and we need to return to the previous step.#### Example FunctionGiven the function:\[f_5(n) = \log_2 \left( n^{n^3} \right)\]

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Alice has asked you to help her classify these functions based on their asymptotic growth rate. That is, she wants you to prove a tight big-Theta bound for each function by following these steps. i. Use algebra to simply the function for ease of analysis. Remember the goal of simplification is to isolate terms and identify the dominant term. For instance, we might want to expand (n+1)² to n² +2n+1 to see that n² is the dominant term. ii. Use your intuition, the limit test, or algebra to come up with a good guess of a tight bound. For instance, after isolating n² as the dominant term we might guess our polynomial is in (n²). iii. Use the limit test or the definitions to prove that your guess is correct. Both methods can provide correct proofs, and its a good idea to practice both methods. Sometimes our guess is incorrect and we need to return to the previous step.

Alice has asked you to help her classify these functions based on their asymptotic growth rate. That is, she wants you to prove a tight big-Theta bound for each function by following these steps. i. Use algebra to simply the function for ease of analysis. Remember the goal of simplification is to isolate terms and identify the dominant term. For instance, we might want to expand (n+1)² to n² +2n+1 to see that n² is the dominant term. ii. Use your intuition, the limit test, or algebra to come up with a good guess of a tight bound. For instance, after isolating n² as the dominant term we might guess our polynomial is in (n²). iii. Use the limit test or the definitions to prove that your guess is correct. Both methods can provide correct proofs, and its a good idea to practice both methods. Sometimes our guess is incorrect and we need to return to the previous step.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Compare the functions according to their order of growth. Prove the accuracy of your ordering. Note: Your analysis must be rigorous and precise. Merely stating the ordering without providing any mathematical analysis will not be accepted. a) (n2)! b) (n!)n
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Alice has asked you to help her classify these functions based on their asymptotic growth rate. That is, she wants you to prove a tight big-Theta bound for each function by following these steps. i. Use algebra to simply the function for ease of analysis. Remember the goal of simplification is to isolate terms and identify the dominant term. For instance, we might want to expand (n + 1)² to n² +2n+1 to see that n² is the dominant term. ii. Use your intuition, the limit test, or algebra to come up with a good guess of a tight bound. For instance, after isolating n² as the dominant term we might guess our polynomial is in (n²). iii. Use the limit test or the definitions to prove that your guess is correct. Both methods can provide correct proofs, and its a good idea to practice both methods. Sometimes our guess is incorrect and we need to return to the previous step. Transcribed Image Text: Alice has asked you to help her classify these functions based on their asymptotic growth rate.…
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