Question

For a homework assignment, I’ve been asked to calculate the runtime of various algorithms. The part that is stumping me is the 2^N, because the size of N is so large.

Assuming a 2^N algorithm with a a data size N=1000 takes 5 seconds to execute, calculate the runtime for data sizes {2000, 3000, 10000}

Now, 2^2000/2^1000 = 2^1000 by the property of exponent division. The result is 5.071509e+301 seconds to execute on a data set of 2000 items.

How can I give a number for the next two sizes? 2^2000 and 2^9000 both return infinity in any calculator that I use. The professors hint is that 2^10 is approximate to 10^3, which means that 1024 is approximate to 1000.

Answer 1

The function of time complexity is f(N) = c * 2^N.

Since f(1000) = 5, we can solve for c = 5 / 2¹⁰⁰⁰.

So f(N) = 5 * 2^{N – 1000}, and f(k * 1000) = 5 * 2^{(k – 1)*1000}.

The blockade seems to be with calculating 2ⁿ, so I will guide you on this.

I will take one of your example to demonstrate the method:

2¹⁰⁰⁰ = 2^{10 * 100} = (2¹⁰)¹⁰⁰ ≈ (10³)¹⁰⁰ = 10^{3 * 100} = 10³⁰⁰ = 1e300

That is using your teacher’s hint, another way to calculate the number more accurately is:

2¹⁰⁰⁰ = 10^{1000 * log₁₀2} = 10^301.03 = 10^0.03 * 10³⁰¹ = 1.0715e301

(As you can see, the more accurate method will be 10.7 times more than the result from the approximate method)