Big O class for (1/2) ^ n

Question

In which class of Big O does the function (1/2) ^ n fall into?

On a purely mathematical basis, it seems that we would have to put it in O (1), because 1/2 ^ n approaches 0 for any sufficiently large n.

However, when it comes to asymptotic analysis and Big O, we tend to wave our arms a lot and also refer to formulas. 1/2 is technically constant, so it would seem to fall into O (c ^ n).

I lean toward O (c ^ n) because "half the operation" doesn't make sense when it comes to algorithms. Which algorithm takes half time as input increases? In the best case, I see the mathematical formula (1/2) ^ n, related to half of some constant time - say, a minute. So (30 seconds) ^ n becomes a huge number, and the function clearly belongs to O (c ^ n).

Help a little?

Answer 1

The function 0.5 ⁿ is equal to O (1), as well as O (c) for any c> 0 (this is not O (0), since 0.5 ⁿ > 0 for any n).

It is also o (1) (note a little o ).

& Theta; (c) c. = 0, , 0,5 ⁿ > c n. s > 0 lim _{n → & INFIN;} 0,5 ⁿ .

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, , & ​​Theta; (0.5 ⁿ), .

Answer 2

O (1/2 ^ N), N , . "".