Questions tagged [landau-notation]

Questions about asymptotic notations such as Big-O, Omega, etc.

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Show that if d(n) is O( f (n)) and e(n) is O(g(n)), then d(n)−e(n) is not necessarily O( f (n)−g(n)) [duplicate]

I have this question as an assignment in my Java Algorithms class, and i'm aware that d(n)+e(n) is the same as O(f(n)+g(n)). I dont know why the same doesnt apply to subtracting. Can someone help me? ...
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What does $\log^{O(1)}n$ mean?

What does $\log^{O(1)}n$ mean? I am aware of big-O notation, but this notation makes no sense to me. I can't find anything about it either, because there is no way a search engine interprets this ...
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Adding orders of growth

I am confused as to how this is true: O(n log n) + mO(log n) = O((m + n) log n) I understand that O(n) + O(m) = O(n + m). I'm mostly confused as to how to deal with the coefficient preceding O(log n)....
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What does Big O notation actually specify? [duplicate]

Regarding time complexity I've read conflicting things: 1) That it is worst case. 2) That is average case. For example if I want to know the time complexity for inserting into an arbitrary point in ...
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Big-O Notation Statement True? [duplicate]

Considering functions f and g, is the following true? $(f \in O(g)) \implies (f \in \Theta(g)) \lor (f \in o(g))$ If not, can you please state an example? Despite thinking hard, i could not find ...
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What's the formal definition of Big-O notation for functions of more than one variable?

For functions of a single totally ordered variable, I already know that $f(n)$ is $O(g(n))$ if and only if $\exists m. \exists c. \forall n. (n \ge m) \rightarrow [ f(n) \le c \cdot g(n) ]$. What I ...
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When is the big-O relation preserved under exponentiation?

Suppose that $f, g$ are functions from the positive integers to the positive reals. Under what circumstances will $\log f(n)=O(\log g(n))$ imply $f(n)=O(g(n))$? It's easy to see that this isn't ...
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Why does the Θ-class survive adding a constant only for positive, monotonic, and non-decreasing functions?

I know that for positive monotonically non-decreasing functions, f(n) and g(n), f(n) = O(g(n) + c) entails f (n) = O(g(n)) Why is this always true only for ...
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Is log(n) in complexity class P?

$\log(n)$ is not polynomial; is a problem solvable in $\mathcal{O}(\log n)$ time in P? $n\times \log(n)$ is also not polynomial; is a problem solvable in $\mathcal{O}(n\times \log n)$ time in P? If ...
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Prove transitivity of big-O notation

I'm doing a practice question (not graded HW) to understand mathematical proofs and their application to Big O proofs. So far, however, the very first problem in my text is stumping me wholly. ...
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Big Omega Counterexample?

I am doing homework to practice for my midterm exam and cannot answer this question. I need to decide whether or not this statement is true of false and either give a proof or counter example. For ...
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Which complexity class $3^{n/3}$

Assuming a problem has complexity $O(3^{n/3})$, Which is its class of complexity ? Despite that it is not as $2^{n}$ ,we can say is an exponential ?
I am trying to rank $\log n$, $\log_{10} n$, $n \log n$, $n \log n^2$, $n^{0.8}$, $\sqrt{n}$ in increasing asymptotic complexity. $\log n$ has base 2 unless specified otherwise. The answer I ...