Question

How much time does an algorithm using ${\mathbf{2}}^{\mathbf{50}}$operations need if each operation takes these amount of time?

role="math" localid="1668588341959" $$\begin{gathered} \mathit{a}\mathbf{)}\mathbf{}{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{s} \\ \mathbf{b}\mathbf{)}\mathbf{}{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathbf{s} \\ \mathbf{c}\mathbf{)}\mathbf{}{\mathbf{10}}^{\mathbf{-}\mathbf{12}}\mathbf{s} \end{gathered}$$

Short answer

a) Time taken is 36 years.

b) Time taken is13 days.

c) Time taken is 19 minutes.

Step-by-step solution

(a)Step 1: Find the time T needed by the algorithm when t=10-6

$T=N\times t$

Where,

T = Time taken

$N=2^{50}$(Number of operations)

$t={10}^{-6}$(time per operation)

On solving,

$\begin{array}{r}T=2^{50}\times {10}^{-6}\\ \approx 1.126\times {10}^9\text{seconds}\\ \approx 1.876\times {10}^7\min \\ \approx 312749\text{hours}\\ \approx 13031\text{days}\\ \approx 36\mathrm{yrs}\end{array}$

Time taken is $\approx 36years$.

(b)Step 2: Find the time T needed by the algorithm when t=10-9

$T=N\times t$

Where,

T = Time taken

$N=2^{50}$(Number of operations)

$t={10}^{-9}$(time per operation)

On solving,

$\begin{array}{r}T=2^{50}\times {10}^{-9}\\ \approx 1.126\times {10}^6\text{seconds}\\ \approx 18765\min \\ \approx 13\text{days}\end{array}$

Time taken is 13 days.

(c) Step 3: Find the time T needed by the algorithm when t=10-12

$T=N\times t$

Where,

T = Time taken

$N=2^{50}$(Number of operations)

$t={10}^{-12}$(time per operation)

On solving,

$\begin{array}{r}T=2^{50}\times {10}^{-12}\\ \approx 1125\mathrm{seconds}\\ \approx 19\min \end{array}$

Time taken $\approx 19$minutes.