What is the largest n for which one can solve within 10 seconds a problem using an algorithm that requires f(n) bit operations, where each bit operation is carried out in 10-10 seconds, with these functions of n? a. log₂ n b. √n c. n7 d. 10n³
What is the largest n for which one can solve within 10 seconds a problem using an algorithm that requires f(n) bit operations, where each bit operation is carried out in 10-10 seconds, with these functions of n? a. log₂ n b. √n c. n7 d. 10n³
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Complexity and Time Constraints
What is the largest \( n \) for which one can solve within 10 seconds a problem using an algorithm that requires \( f(n) \) bit operations, where each bit operation is carried out in \( 10^{-10} \) seconds, with these functions of \( n \)?
#### Functions of \( n \):
**a.** \( \log_2 n \)
**b.** \( \sqrt{n} \)
**c.** \( n^7 \)
**d.** \( 10n^3 \)
---
To determine the largest \( n \), set up the inequality based on the time constraint:
\[ f(n) \times 10^{-10} \leq 10 \]
Solving for \( n \) will give the maximum size of \( n \) solvable in 10 seconds for each given function.
For each function \( f(n) \), follow these steps:
1. \( \log_2 n \):
\[
\log_2 n \times 10^{-10} \leq 10 \\
\log_2 n \leq 10^{11} \\
n \leq 2^{10^{11}}
\]
2. \( \sqrt{n} \):
\[
\sqrt{n} \times 10^{-10} \leq 10 \\
\sqrt{n} \leq 10^{11} \\
n \leq 10^{22}
\]
3. \( n^7 \):
\[
n^7 \times 10^{-10} \leq 10 \\
n^7 \leq 10^{11} \\
n \leq (10^{11})^{1/7} \\
n \leq 10^{11/7}
\]
4. \( 10n^3 \):
\[
10n^3 \times 10^{-10} \leq 10 \\
n^3 \leq 10^{10} \\
n \leq 10^{10/3} \\
n \leq 10^{3.33}
\]
By calculating these, one can find the largest \( n \) that can be solved within the given constraints for each function](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa06b614-497d-40f0-b077-f4b4fca47baf%2Fc77400db-f6a1-40b7-9cb9-603ccb43ecf5%2Fe1m1fxa_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Complexity and Time Constraints
What is the largest \( n \) for which one can solve within 10 seconds a problem using an algorithm that requires \( f(n) \) bit operations, where each bit operation is carried out in \( 10^{-10} \) seconds, with these functions of \( n \)?
#### Functions of \( n \):
**a.** \( \log_2 n \)
**b.** \( \sqrt{n} \)
**c.** \( n^7 \)
**d.** \( 10n^3 \)
---
To determine the largest \( n \), set up the inequality based on the time constraint:
\[ f(n) \times 10^{-10} \leq 10 \]
Solving for \( n \) will give the maximum size of \( n \) solvable in 10 seconds for each given function.
For each function \( f(n) \), follow these steps:
1. \( \log_2 n \):
\[
\log_2 n \times 10^{-10} \leq 10 \\
\log_2 n \leq 10^{11} \\
n \leq 2^{10^{11}}
\]
2. \( \sqrt{n} \):
\[
\sqrt{n} \times 10^{-10} \leq 10 \\
\sqrt{n} \leq 10^{11} \\
n \leq 10^{22}
\]
3. \( n^7 \):
\[
n^7 \times 10^{-10} \leq 10 \\
n^7 \leq 10^{11} \\
n \leq (10^{11})^{1/7} \\
n \leq 10^{11/7}
\]
4. \( 10n^3 \):
\[
10n^3 \times 10^{-10} \leq 10 \\
n^3 \leq 10^{10} \\
n \leq 10^{10/3} \\
n \leq 10^{3.33}
\]
By calculating these, one can find the largest \( n \) that can be solved within the given constraints for each function
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