## Function Growthi. **Organize the following functions into six columns.** Items in the same column should have the same asymptotic growth rates (big-O). If a column is to the left of another column, all its growth rates should be slower than those of the column to its right.- Functions: \( n^2, \, n!, \, \log_2 n, \, n \log_2 n, \, 3n, \, 5n^2 + 3, \, 2^n, \, 10000, \, n \log_3 n, \, 100n, \, 3\log_3 n \)ii. **Using the definition of big-O, show \( 100n + 5 = O(n) \).** Give a particular \( c \) and \( n_0 \).iii. **Using the definition of big-O, show that \( n = O(2^n) \).** Give a particular \( c \) and \( n_0 \).iv. **Identify a function \( f(n) \) which has \( 3n + 4n^2 + 3n! = O(f(n)) \).** You may use the facts that \( n = O(n!) \) and \( n^2 = O(n!) \).

i Organize the following functions into six columns. Items in the same column should have the same asymptotic growth rates (big-0). If a column is to the left of another column, all its growth rates should be slower than those of the column to its right. n², n!, log2 n, n log2 n, 3n, 5n² + 3, 2", 10000, n log3 n, 100n, 3log3n ii Using the definition of big-O, show 100n + 5 = 0(n). Give a particular e and no. %3D iii Using the definition of big-O, show that n = 0(2"). Give a particular e and no- iv Identify a function f(n) which has 3n + 4n? + 3n! = n= 0(n!) and n² = 0(n!). O(f(n)). You may use the facts that

i Organize the following functions into six columns. Items in the same column should have the same asymptotic growth rates (big-0). If a column is to the left of another column, all its growth rates should be slower than those of the column to its right. n², n!, log2 n, n log2 n, 3n, 5n² + 3, 2", 10000, n log3 n, 100n, 3log3n ii Using the definition of big-O, show 100n + 5 = 0(n). Give a particular e and no. %3D iii Using the definition of big-O, show that n = 0(2"). Give a particular e and no- iv Identify a function f(n) which has 3n + 4n? + 3n! = n= 0(n!) and n² = 0(n!). O(f(n)). You may use the facts that

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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Question

## Function Growth

i. **Organize the following functions into six columns.**
Items in the same column should have the same asymptotic growth rates (big-O). If a column is to the left of another column, all its growth rates should be slower than those of the column to its right.

- Functions: \( n^2, \, n!, \, \log_2 n, \, n \log_2 n, \, 3n, \, 5n^2 + 3, \, 2^n, \, 10000, \, n \log_3 n, \, 100n, \, 3\log_3 n \)

ii. **Using the definition of big-O, show \( 100n + 5 = O(n) \).**
Give a particular \( c \) and \( n_0 \).

iii. **Using the definition of big-O, show that \( n = O(2^n) \).**
Give a particular \( c \) and \( n_0 \).

iv. **Identify a function \( f(n) \) which has \( 3n + 4n^2 + 3n! = O(f(n)) \).**
You may use the facts that \( n = O(n!) \) and \( n^2 = O(n!) \).

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