# 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 [tag:landau-notation].

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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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### Show how to get the big-Oh for the following code: [duplicate]

void CountSort (int A[N], int range) { // assume 0 <= A[i] < range for any element A[i] int *pi = new int[range]; for ( int i = 0; i < N; i++ ) pi[A[i]]++; for ( int j = 0; j < range; j+...
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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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### 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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### 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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### 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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### 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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### heap data structure complexity

I'm trying to count running time of build heap in heap sort algorithm ...
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### 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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### 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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### 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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### 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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### $\Omega$-notation for insertion sort [duplicate]

I'm reading the CLRS book and there is a statement for instance, the running time of insertion sort is not $\Omega(n^2)$, since there exists an input for which insertion sort runs in $\Theta(n)$ ...
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### Splay tree amortized cost analysis

I am looking into the amortized analysis of splay trees and seem to be missing something. Pretty much every resource uses the accounting method which I believe I grasp. What confuses me is the part ...
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### BigO time complexity of 3 nested for loops

I'm debating with a friend whether a particular function I wrote is $O(N^3)$ or $O(N \times M \times X)$ I believe it is the latter since all 3 variables differ in size. $N = 100, M = 50, X = 10000$ ...
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### Big oh notation run time [duplicate]

I have this Question , I want the answer and show me how to solve it Please : Analyze the running time of the following algorithm using Big-Oh notation ...
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### Growth rate and runtime [duplicate]

Sorry if this maybe a dumb question, just a little confused But with Big-Oh notation, does it measure the runtime or growth rate of an algo? or both?
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### How can I compare by this two algorithms? [duplicate]

I have two algorithms has a complexity of O (n log n) and the B-complex algorithm (n^2). By imposing NA size the larger issue that a algorithm can solve in a given time and NB size the larger issue ...
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### why is $O(n^2)$ equal to $n^{1.5}$?

I am learning this MIT course, which gives this formula $$O(n^2) = n^{1.5}$$ is there a table to calculate this? like $O(n^{1.5})$, $O(n^{5})$ ? what x takes would have O(x) give $c \cdot n$ where ...
Per wiki |f| is bounded above by g (up to constant factor) asymptotically with this concrete example, $$f(n) = \log n$$ $$g(n) = n^c = n^{0.000001}$$ Does "bounded above (up to constant factor)...