nicekikah

2021-08-15

What is the largest n for which one can solve within a day using an algorithm that requires f(n) bit operations, where each bit operation is carried out in ${10}^{-11}$ seconds, with these functions f(n)?
a) $\mathrm{log}n$
b) $1000n$
c) ${n}^{2}$

AGRFTr

Step 1
Consider the provided question,
Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you that means 16. (a), (b), (c). If you want remaining sub-parts to be solved, then please resubmit the whole question and specify those sub-parts you want us to solve.
a) Each bit operation is carried out in ${10}^{-11}seconds.T={10}^{-11}$ seconds.
The algorithm can take at most 1 day which contains $24×60×60=86400$ seconds, while there are $\frac{t}{T}=\frac{86400}{{10}^{-11}}=8.64×{10}^{-15}$ possible bit operation in 86400 seconds. Step 2
Algorithm requires $f\left(n\right)=\mathrm{log}n$ bit operations:
$\mathrm{log}n=8.64×{10}^{15}$
Since, the algorithm has base 2, because bits only have 2 possible values.
${\mathrm{log}}_{2}n=8.64×{10}^{15}$
Now, use the property ${\mathrm{log}}_{a}x=b⇒x={a}^{b}$
So, $n={2}^{8.64×{10}^{15}}$
Thus, $n={2}^{8.64×{10}^{15}}$
Step 3
b) Each bit operation is carried out in ${10}^{-11}$ seconds. $T={10}^{-11}$ seconds.
The algorithm can take at most 1 day which contains $24×60×60=86400$ seconds, while there are $\frac{t}{t}=\frac{86400}{{10}^{-11}}=8.64×{10}^{15}$ possible bit operation in 86400 seconds.
Algorithm requires $f\left(n\right)=1000n$ bit operations:
$1000n=8.64×{10}^{15}$
$⇒n=\frac{8.64×{10}^{15}}{1000}$
$⇒n=8.64×{10}^{12}$
Thus, $n=8.64×{10}^{12}$
Step 4
c) Each bit operation is carried out in ${10}^{-11}$ seconds. $T={10}^{-11}$ seconds.
The algorithm can take at most 1 day which contains $24×60×60=86400$ seconds, while there are $\frac{t}{T}=\frac{86400}{{10}^{-11}}=8.64×{10}^{15}$ possible bit operation in 86400 seconds.
Algorithm requires $f\left(n\right)={n}^{2}$ bit operations:
${n}^{2}=8.64×{10}^{15}$
$⇒n=\sqrt{8.64×{10}^{15}}$
$⇒n\approx 9.295×{10}^{7}$
Thus, $n\approx 9.295×{10}^{7}$

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