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.

  • $\begingroup$ a tricky one. did you think about $O(-n\log n)$? $\endgroup$ – Ran G. Mar 6 '13 at 5:07
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    $\begingroup$ It does not just decrease to zero, it turns negative. $\endgroup$ – Karolis Juodelė Mar 6 '13 at 5:07
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    $\begingroup$ Are you talking about time complexity? $\endgroup$ – Sid Mar 6 '13 at 5:18
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    $\begingroup$ @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. $\endgroup$ – Joe Mar 6 '13 at 5:27
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    $\begingroup$ 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. $\endgroup$ – Raphael Mar 6 '13 at 7:15

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.

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