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Questions tagged [big-o-notation]

Big O Notation is an informal name of the "O(x)" notation used to describe asymptotic behaviour of functions. It is a special case of Landau notation, where the O is the Greek letter capital omicron. Please consider using the [landau-notation] tag instead if your question is related to small omicron, omega, or theta in Landau notation.

Big O Notation is an informal name of the $\mathcal{O}(x)$ notation used to describe asymptotic behaviour of functions. It is a special case of .

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What is the name the class of functions described by O(n log n)?

In "Big O", common notations have common names (instead of saying, "Oh of some constant factor"): O(1) is "Constant" O(log n) is "Logarithmic" O(n) is "Linear" O(n^2) is "Quadratic" O(n * log n) is ??? Is it just "n log n" or does it have a special…
GlenPeterson
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Understanding of big-O massively improved when I began thinking of orders as sets. How to apply the same approach to big-Theta?

Today I revisited the topic of runtime complexity orders – big-O and big-$\Theta$. I finally fully understood what the formal definition of big-O meant but more importantly I realised that big-O orders can be considered sets. For example, $n^3 + 3n…
mariaprsk
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Time complexity $O(m+n)$ Vs $O(n)$

Consider this algorithm iterating over $2$ arrays $(A$ and $B)$ size of $ A = n$ size of $ B = m$ Please note that $m \leq n$ The algorithm is as follows for every value in A: // code for every value in B: // code The time complexity of…
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Does it make sense to say Big Theta of 1? Or should we just use Big O?

Does saying $f(x) = \Theta(1)$ provide any extra information over saying $f(x) = O(1)$? Intuitively, nothing grows more slowly than a constant, so there should be no extra information in specifying Big Theta over Big O in this case.
MattCochrane
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Is Big-Theta a more accurate description of worst case run time than Big-O?

Question I was asked: Does it make a difference if I say "The worst case run time is $O(n^2)$ vs the worst case run time is $\Theta(n^2)$?" To me, the only difference is that when we say $O(n^2)$, the function may also be $O(n)$, we do not know. But…
Carter Falkenberg
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Can I multiply Big-O time complexities?

Can I multiply Big-O time complexities? For example: $O(n) \cdot O(n) = O(n^2)$? UPDATE: The question came from my observation that different sources analyze their algorithms in different ways. For example, some use the following $O(n \cdot n + n…
illuminato
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Are there any functions with Big O (Busy Beaver(n))?

So, I was reading this article by Scott Aaronson on big numbers, and he mentioned that the Busy Beaver sequence increases faster than all sequences computable by Turing Machines. Faster than exponentials, faster than the Aaronson sequence, and…
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Are there variations of the regular runtimes of the Big-O-Notation?

There are multiple $O$-Notations, like $O(n)$ or $O(n^2)$ and so on. I was wondering, if there are variations of those in reality such as $O(2n^2)$ or $O(\log n^2)$, or if those are mathematically incorrect. Or would it be a right thing to say that…
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Isn't linear time O(n)?

In the question in this video about quicksort luckily picking the median in each recursive call. Tim Roughgarden, the presenter, says at 11:22 Partition needs really linear time, not just $O(n)$ time. What does he mean here? I thought linear time…
heretoinfinity
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Possible Mistake in Skiena's Algorithm Design Manual

In Skiena's book Algorithm Design Manual, 3rd Edition, it is claimed on page 45 that $$ mn - m^2 + m \in \Omega(mn) $$ where $m,n \geq 0$ and $m \leq n$. I claim that this is in fact false, with the following proof. Toward a contradiction, suppose…
Joshua
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What are you allowed to move into the big O notation for it to be still correct?

Can someone tell me what the rules are for moving log or exponents into the $O(n)$ notation so it is still correct? For example: Is this $\log(O(n))= O(\log(n))$ correct? Or is this correct $O(n)^2=O(n^2)$? Or am I not allowed to do this?
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Asymptotics question

Is $\frac {n!} {2!\cdot 4!\cdot 8!\dots (n/2)!}=O(4^n)$? I am really stuck and I tend to believe it's true, but I don't know how to prove it. Any help would be appreciated!
Dudi Frid
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Why is $\sum_{i=1}^n O(i)$ not the same as $O(1)+O(2)+\dots+O(n)$?

The well-known textbook Introduction to Algorithms ("CLRS", 3rd edition, chapter 3.1) claims the following: $$ \sum_{i=1}^n O(i) $$ is not the same as (I'm not using DNE because the book explicitly says "is not the same as") $$ O(1) + O(2) + \dots…
Padaca
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Time Complexity of Linear Search vs Brute Force

I am currently watching the FreeCodeCamp Algorithms and Data Structures Tutorial. In the explanation for exponential time complexity, they explain that using a brute force attack on a combination lock would create O(x^n) time complexity. I am…
jacoboneill2000
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How does $Θ(\log(n!))=Θ(\log(n^n)$?

How does $Θ(\log(n!))=Θ(\log(n^n)$? I understand why $Θ(\log(n!))=Θ(n\log(n))$ and $Θ(\log(n^n))=Θ(n\log(n))$, therefore $Θ(\log(n!))=Θ(\log(n^n)$. But I am having trouble reconciling this with the following: $$\lim_{x\to…
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