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.

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Time complexity for concatenating strings

I was going through this piece of code from an algorithms books and something doesn't look clear Please ignore the spelling errors, How does 0(x + 2x + nx) reduce to o(xn^2) ? My analogy, assuming ...
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Asymptotic growth of a function containing a sum

How to compare the asymptotic growth of a function containing a sum with another function? I'm not sure how I'm supposed to dissolve the sum. Usually I just take the limis of f(x)/g(x). If that fails ...
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Time complexity of $O(n)$ loop which has a multiplication ($O(n^2)$) in it

Assume we know that the implementation for the multiplication operator for a language is known to be $O(n^2)$. Given this pseudocode: ...
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Is this program O(n^2logn) or O(nlog^2(n))?

I was wondering whether this program (I'm using a C syntax, hope it's not an issue) is to be considered $O(n^2 \log(n))$ or $O(n\log^2(n))$ or something else entirely. ...
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Big-O Notation: Runtime Analysis

I have a problem with an exercise, I have to analyze the following For-Loops Then I have to write down the explicit notation, my problem is that I don't know how to get the right m. I tried this but ...
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Is $(n^5 + n^7)\in \Omega(n^7)$? Shouldn't it be in $\Omega(n^5)$?

Is $(n^5 + n^7)\in \Omega(n^7)$? Shouldn't it be in $\Omega(n^5)$? I understand Omega to be a "lower bound" on a function. Shouldn't the largest lower bound on the function $n^5 + n^7$ be $n^5$? (...
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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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1answer
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How to compare n number of m-dimensional points among one another with minimum time complexity?

Suppose there are four points (n = 4) which are four dimensional (m = 4) . Lets say these points are : A(4,1,1,1) , B(3,2,1,1) , C(2,3,3,3) , D(1,4,4,4). What is the best data structure to compare all ...
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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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Why is n log n dominated by n log^2 n?

Does the rule of $n ^ a$ dominate $n ^ b$ if $a > b$ apply here as well? My understanding is that $n \log n$ will be dominated by $n \log ^2 n$ because of $\log$ being raised to the power of $2$.
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About Big O properties

Suppose I have something like the following: $f(x) = g(x) + O(x^n)$ And I apply a power $m$ to both sides $f(x)^m = g(x)^m + \cdots + O(x^n)^m$ My question is whether the following is well ...
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Which function grows faster: N Log N or N^(1+ε/√(log N)) [duplicate]

How would you go about solving this problem? I thought about using a limit infinity approach, but got confused and Wolfram Alpha didn't provide any explanation.
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Improving time complexity from O(log n/loglog n) to O((log ((nloglog n)/log n))/loglog ((nloglog n)/log n))

Suppose I have an algorithm whose running time is $O(f(n))$ where $f(n) = O\left(\frac{\log n}{\log\log n}\right)$ And suppose I can change this running time in $O(1)$ steps into $O\left(f\left(\...
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How do I simplify $O\left({n^2}/{\log{\frac{n(n+1)}{2}}}\right)$

I'm not very certain about how to deal with asymptotics when they are in the denominator. For $$O\left(\frac{n^2}{\log{\frac{n(n+1)}{2}}}\right)$$, my intuition tells me that it should be treated in a ...
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Summing big-O-notation

prove or disprove $$\text{If } f(n)=g(n)+h(n), \text{ then } O(f(n)) = O(g(n))+O(h(n)).$$ I have no idea about where to begin. what are the theories which should be used here?
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How does $n^c \lg n, 0<c<1$ compare to other common time complexities

Between what two common time complexities would you place $n^c lg n, 0<c<1$? The following table illustrates the common time complexities. Source: wikipedia
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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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Big-O notation for the given function whose runtime complexity grows faster than the input

I struggle to determine the runtime complexity of a function I thought of while trying to solve this quiz. The quiz itself goes like this: Write a program to find the n-th ugly number. Ugly numbers ...
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Summation of asymptotic notation

How can we solve summation of asymptotic notations like given below: $$ \sum_{k=1}^{n-1} O(n). $$
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Asymptotics of $\frac{1}{\log(\frac{2^n}{2^n-1})}$

I am trying to understand the asymptotics of \begin{equation} f(n) = \frac{1}{\log(\frac{2^n}{2^n-1})} \end{equation} In particular, is there some $c \geq 1$ such that $f(n) = O(n^c)$?
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How to determine time complexity with a simple way?

I'm learning about time complexity but all the cases we did in class were rather simple. Now I'm working on my home work and the cases our teacher let us have was: $$f(n) = 4n(n + 2 \log^2 n^2) + e^{−...
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Show that the union of Θ and o is not O

Show that: $\Theta(n\log n)\cup o(n\log n)\neq O(n\log n)$ I tried to start this in many ways but I don't really know how... intuitively isn't $\Theta \cup o = o$? So that would mean that I would ...
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Difference Between $n^{\Omega{(1)}}$ and $\Omega{(n)}$ [closed]

I am not sure about the difference between $n^{\Omega(1)}$ and $\Omega(n)$. It seems to me that the only difference is that $n^{\Omega(1)}$ can contain some sublinear functions, i.e., $n^{\frac{1}{2}}$...
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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 log(n) equivalent to (log(n))^x for big-O analysis?

My professor noted that we could treat any logarithmic function with an exponent as equivalent to log(n) for the purposes of big-O analysis. ie. $(n log(n) + 1)^2 + (log(n) + 1)(n^2 + 1)$ From the ...
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Simplifying this expression with big O when several variables are involved

I have an algorithm which depends on three variables an where the running time is in $\mathcal{O}(m+2 m\cdot n\cdot p+p\cdot(n+m))$ and I would like to simplified it. I proceeded as follows : \begin{...
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Why is building a heap $\mathcal O(n)$ and not $\theta(n)$?

From what I see online, all seem to suggest that heapifying takes $\mathcal O (n)$ time, but it seems like it should always takes $\theta(n)$ time, even in the best case. Is something wrong with my ...
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Find the error in the following “proof” that $O(n) = O(n^2)$

Let $f(n) = n^2 , g(n) = n$, and $h(n) = g(n)−f(n)$. It is clear that $h(n) ≤ g(n) ≤ f(n)$ for all $n ≥ 0$. Therefore, $f (n) = \max(f (n), h(n))$. Thus, $O(n) = O(g(n)) = O(f(n) + h(n)) = O(\max(f(n),...
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Compare log^k(n) with n^(1/2)

I'm trying to prove or disprove that $\log^{k}(n) \in O(\sqrt{n}), \ \forall k > 0$. By using the free version of wolfram and testing some increasing values of $k$ I get that: $$\lim_{n \...
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What is the difference between $\ Ω $ , $\ Θ $ , and O , [duplicate]

What are these difference $\ Ω $ , $\ Θ $ and O ? Why $\ n^3 + 4n^2 = Ω(n^2) $ But, $\ n^3 + 4n^2 = O(n^5) $
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Big Oh Notation Complexity [duplicate]

How it can possible $\ 1/2n^2 -3n = Θ(n^2) $ Why we dont care $\ 3n $ and $\ 1/2 $
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Derive a while loop (which seemingly have some logarithmic traits) runs in $\Theta(n)$

I know for a fact that algorithm A runs in $\Theta(n)$, but how does one derive that? Algorithm A ...
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116 views

Time-complexity for basic multiplication and division algorithms

The algorithms below has been taken from Discrete Mathematics and it's Applications 7th edition book by Rosen. p.253 says that "number of shifts required is $O(n^2)$" and "a total of $O(n^2)$ ...
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1answer
440 views

Solving the recurrence relation T(n) = 2T(n/2) + nlog n via summation

I have seen a few examples of using the master theorem on this to obtain O(n*log^2(n)) as an answer. I am trying to solve this by unrolling and solving the summation, but I can't seem to get the same ...
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How can $\Theta$ and $O$ complexities be different?

From the definition of the $\Theta$-notation, $$f(n)=\Theta(g(n))\\\implies \exists n_0, \exists c_1,c_2\gt 0, \forall n\gt n_0, c_1\cdot g(n)\le f(n)\le c_2\cdot g(n)$$ We can see that the ...
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confused with Time Complexity [duplicate]

I was reading book related to Time Complexity, and came up with 4 lines of equations that I could not understand properly, could you please explain why are those true? 1) $n = o(n\log\log n)$ 2) $...
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Runtime complexity of a brute force factoring algorithm? (in terms of bits)

Let N be an n bit number. A brute force algorithm factors N by trying to divide N by all of the numbers between 2 and sqrt(N). Given that dividng two n bit integers takes O(n^2) time, what is the ...
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How to prove that ($56n^2+106n+48)(\log(264n^2+200)) = Θ(𝑛^2\log n)$

I understand that essentially we have to prove that $$c_1(n^2\log n)\le (56n^2+106n+48)(\log(264n^2+200)) \le c_2(n^2\log n)\,.$$ I am confused on how to simplify this further? And ...
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59 views

How to prove the performance, Big Omega ,of building a binary heap using recursive method is Ω(nlog(n))

We can learn the big-O of building a binary heap using recursive method is O(n log n) from wiki "This approach, called Williams’ method after the inventor of binary heaps, is easily seen to run in O(n ...
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Complexity analysis using big - O, Omega and Theta notation

I was reading a research paper and there I read the following: $t=O\left(d^{2} \log _{d}^{2} n\right)$ matches the lower bound $\Omega\left(d^{2} \log _{d} n\right)$ in the regime where $d=\Theta\...
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How to find running time complexity of divide and conquer method without Master Theorem

I understand that Master Theorem can be used to solve divide-and-conquer run times if they're in the form of $T(n) = aT(\frac{n}{b}) + n^clog^k(n)$ The reason behind it has to do with drawing a tree ...
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Asymptotics and logarithms/exponents

We have four categories: additive constants, multiplicative constants, polynomials, and exponentials When determining the growth order of functions, we only care about polynomials and ...
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Exact meaning of $2^{\mathcal{O}(f(n))}$

In Sipser's Introduction to the Theory of Computation he uses the notation $2^{\mathcal{O}(f(n))}$ to denote some asymptotic running time. For example he says that the running time of a single-tape ...
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Why is heap insert O(logN) instead O(n) when you use an array?

I am studying about the arrays vs heap for make a priority queue For check the heap implementation I am reviewing this code: Heap , but I have the following question. Heap is based on array, and ...
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1answer
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Asymptotic Relationship from Limit

F(n) = n-100 G(n) = n-200 I am trying to show the asymptotic relationship between these two functions using limits. I take the limit n->∞ f(n) / g(n) and I get the result 1 which is constant c. ...
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Help with Big-O homework [duplicate]

"er" is the Danish equivalent of "is" in English. I need some help with the square root one. Additionally, it would be nice to know if the other ones are correct.
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145 views

heap data structure complexity

I'm trying to count running time of build heap in heap sort algorithm ...
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1answer
45 views

Helping prove this notation for Big-O(n!)

Hello, I was wondering if anyone can help me prove the right part of this double equation. I know the left one is possible due tolog(n!) = Θ(nlogn). Any help is ...
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2answers
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Big-O Notation and Calculus?

I was wondering if there are any calculus relationships implicit in Big-O notation. For example, an algorithm linear according to Big-O notation reduces the size of the problem by a constant amount ...
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1answer
592 views

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 ...