算法：T(N) =2^N

Question

Suppose the number of operations required by a particular algorithm is exactly T(n) = 2^n and our 1.6 Ghz computer performs exactly 1.6 billion operations per second. What is the largest problem, in terms of n, that can be solved in under a second? In under a day?

I tired 2^1.6 for a second and 2^(1.6*60*24), but I think I misunderstood the problem.

Answer 1

What we know:

• For being under 1 sec you need to perform less than 1.6*10^9 operations
• The number of operations needed is T(n)=2^n with n the size of the problem

We are looking for the maximum n (the maximum size of the problem under 1 sec). So we can write:

2^n <= 1.6*10^9 
n <= ln(1.6*10^9) / ln(2)
n <= 30

So in one sec you can compute a problem of size 30

Now 1 day is 24 * 60 * 60 seconds so:

2^n <= 86400 * 1.6*10^9 
n <= ln(86400 * 1.6*10^9)/ln(2)
n <= 46

So in one day you can compute a problem of size 46

Imagine the time needed for a problem of size 64...