# 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.

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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### What does $|V|=O(|E|)$ mean?

I was reading about Dijkstra's algorithm from this Stanford University lecture presentation. On page 18 it says Dijkstra's algorithm is $O(|V|\log|V|+|E|\log|V|)$ and I understand why. But then it ...
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### Arrange in increasing order of asymptotic complexity

I have the following functions that I need to rank in increasing order of Big-O complexity: $$(\log n)^3, 10\sqrt n, n\log n, n\sqrt n, n^4 + n^3, (2.1)^n \cdot n^2, 3^n, 2^n \cdot n^3, n! + n, n^n.$$...
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### Asymptotic Analysis of T(n) = 2T(n/8) + 2T(n/4) + n

Given the recurrence $$T(n) = 2T\bigg(\frac{n}{8}\bigg) + 2T\bigg(\frac{n}{4}\bigg) + n$$ My professor says that $T(n)$ is $O(n\log n)$ but I have calculated a complexity of $O(n)$ as shown below with ...
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### Shifted Big Os. How to say O((n+c)!) = O(n!)?

Suppose an algorithm is $O(n!)$, but we need to run it $n$ times, so the total complexity is $nO(n!) = O(n \cdot n!) = O((n+1)! - n!) = O((n+1)!)$ Strictly, there is no constant factor that would make ...
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### Comparing asymptotic running time of two algorithms $\sqrt n$ and $2^{\sqrt{\log _{2}n}}$

Given two algorithms with their time-complexity $t_a(n)=\sqrt{n}$ and $t_b(n) = 2^{\sqrt{\log _{2}n}}$ and i have to show $t_b(n) = O(t_a(n))$. I´ve made a program to check this statement and it ...
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### Struggling to understand the symbolism around the big oh formal definition

I'm struggling to understand what exactly T(n), and f(n) is in the above text: When we compute the time complexity T(n) of an ...
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### What is the complexity of $i^i$?

What is the complexity of the following algorithm in Big O: for(int i = 2; i < n; i = i^i) { ...do somthing } I'm not sure if there is a valid operator to ...
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### Order notation subtractions in Fibonacci Heap

Can order notation on its own imply: $O(D(n)) + O(t(H)) - t(H) = O(D(n))$ My guess is that you cannot since the constant in the O(t(H)) would still exist after the subtraction if the constant is > 0....
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### Is O(n log n) exponential speedup over O(n^2)?

I would like to know if $O(n \log n)$ is an exponential speedup over $O(n^2)$?
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### Combining Predicate Logic and BigO

I am a beginner to predicate logic and BigO and am having though time understanding the definition of BigO in terms of predicate logic in the picture attached. I particularly am unable to understand ...
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### Is this big O notation format correct? $3^n = 2^{(O(n))}$

I am completing a university exercise deciding whether big notations are true or false. I am stuck on this question : $$3^n = 2^{(O(n))}$$ I want to answer False as the format looks incorrect and ...
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### Big O notation for Average case in Linear search

Average case complexity for linear search is (n+1)/2 i.e, half the size of input n. The average case efficiency of an algorithm can be obtained by finding the average number of comparisons as given ...
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### Does $f(n) \in O(g(n))$ imply $2^{f(n)} \in O(2^{g(n)})$?

Is the following true: $$f(n) \in O(g(n)) \text{ then } 2 ^ {f(n)} \in O(2^{g(n)})$$
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### Is it true that $2^{O(3k)} = 2^{O(k)}$?

Is it true that $2^{O(3k)} = 2^{O(k)}$? But It should be different from $O(2^{k}) = 2^{O(k)}$ ? I will be happy for simple explanation. Thanks.
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### Is $\log(n-1) \in \Omega(\log(n))$?

I saw this question Can I simplify log(n+1) before showing that it is in O(log n)? and wanted to know if a similar situation was also true. Namely, is $\log(n-1) \in \Omega(\log(n))$?
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### Determining Big O of a for loop nested within a while loop

I apologize if this question is a duplicate as i cannot find a similar question in this community forum, please comment the post in which this may be a duplicate of so i can update this post :) Im a ...
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### Finding largest elements

I was asked to find write a pseudocode of an algorithm that extracts the Log(N) largest elements in an array and return them in a sorted list, my attempt is ...
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I have been reading this tutorial on time complexity, and I am a bit puzzled on its explanation of big $O$ notation. It writes: $O(g(n)) =$ { $f(n)$ : there exist positive constants $c$ and $n_0$ ...
### Complexity of $O(\log(n^n))$ vs $O(\log(n!))$
Is $O(\log(n^n)) < O(\log(n!))$? Is there any good/practical algorithm with this kind of complexity? And also, to check my understanding of algorithmic complexity, are these two $> O(n\log(n))$...