Asymptotics of a function that decreases as n increases

Question

A homework assignment asks me to state the complexity in Big-O notation of the function $$f(n) = 7n – 3n \log n + 100000 $$

I graphed this function and decreases all the way down to zero nearly its entire lifespan. Therefore I concluded that the complexity is bounded by a constant and has the complexity $O(1)$.

Is this correct?

Also out of curiosity, what is the Big-Omega of this function is? The best it could ever run is also O(1). What about Big-Theta? I'm having trouble getting my head around these.

@RanG. negative values is an odd case for Big-O. I don't think it's ever come up... But, carefully check all the definitions, if it's valid, then ok. — Joe, Mar 06 '13 at 05:27
See also our reference question. You have to get your definitions straight! Note also that there is no algorithm anywhere in sight here! Clearly, $f$ can not be the runtime function of any algorithm. Furthermore, see here regardings using plots in this context. — Raphael, Mar 06 '13 at 07:15
@Raphael It can't be the running time of a useful algorithm, but it can be the running time of some algorithm. Take $n=0$. — mrk, Mar 06 '13 at 14:15
@saadtaame In any reasonable computational model, runtime is at least $1$. The given function assumes negative values, so I maintain that it can't be a runtime function (assuming $n$ is to be the input size). — Raphael, Mar 06 '13 at 17:13

score 1 · Answer 1 · answered Mar 06 '13 at 07:20

1

If you use suitable definitions, you'll find that your function is in

$O(2^n)$,
$O(n^4)$,
$o(1)$ and
$\Omega(-n^2)$,

among many others. It's likely that the question (implicitly) asks for sharper bounds; I'll leave that for you to figure out.

answered Mar 06 '13 at 07:20

Raphael

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Asymptotics of a function that decreases as n increases

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