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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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 ...
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Algorithms: Determining Asymptotic Notation from a given execution time

I'm studying for an Algorithms and Data Structure test. There is a type of question that is usually always asked by my professor but I don't know how to answer/solve it. Question 1: An Algorithm with ...
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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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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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19 votes
3 answers
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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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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(...
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4 votes
1 answer
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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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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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Prove $T(n) = T(\left \lceil{\frac{n}{2}}\right \rceil) + 1 = O(\log(n))$

As the title said, prove $T(n) = T(\left\lceil{\frac{n}{2}}\right\rceil) + 1 = O(\log(n))$ My approach is to find $c, n_0 \in \mathbb{R}_+$ such that: $$\forall n \geq n_0, T(n) \leq c\log(n) -d \text{...
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What is the big-$O$ notation of a summation of a log?

For: $$\sum^{n+m}_{i=n} \log(i)$$ I'm wondering what the big O notation is and how to prove it... I believe that we can also write this as $$\log(n) + \log(n+1) + \log(n + 2) + \ldots + \log(n+m)$$ ...
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2 votes
2 answers
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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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2 votes
1 answer
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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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Solving unusual recurrence with two variables

I have the following recurrence relation: $$T(n,k) = T(n-1,k)+T(n-1,k+1)$$ With the following base cases (for some given constant $C$): For all $x \leq C$ and for any $k$: $T(x,k)=1$ For all $y \geq C$...
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Θ, O and Ω, and how they relate to each other as subsets

I am trying to understand how $\Theta(n)$, $O(n)$, and $\Omega(n)$ relate to each other as sets and want to make sure I'm on the right track. I get that $Θ(n) \subseteq O(n)$ since $Θ(n)$ is stronger ...
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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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How does squaring time complexity imply the same time complexity for multiplying different numbers? Isn't it the other way round?

Found this in solutions of a test as being true If you can square an n-bit integer in time $O(n \,log \,n)$, then you can multiply two n-bit integers in time $O(n \, log \,n)$. How does the above ...
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1 answer
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Asymptotic notation for summations

I am struggling to understand why this property of asymptotic notation is true
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2 answers
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Trouble with Big-O notation proof by definition

Let $a,b>0$. Prove $\left(\log\left(n\right)\right)^{a}=O\left(n^{b}\right)$. I'm supposed to find an algorithm to find the log(n) largest elements in an array and return them sorted and explain ...
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Is reduction from Rudrata/Hamiltonian path to Rudrata/Hamiltonian cycle O(1)?

I am reading about P and NP and looking at the reduction of a Rudrata/Hamiltonian path to a Rudrata cycle. I think adding an extra node and 2 edges connecting the start, ...
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1 answer
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Remove range of keys from Binary Search Tree in O(s+h)

I have a binary search tree with integer keys. I have to remove a range (m, n]eZ of keys from the BST in O(s + h) where s is the number of keys to remove and h is the height of the tree. Attempted ...
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0 answers
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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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-4 votes
1 answer
431 views

Is squaring easier than multiplication? [duplicate]

Let $T_1(n)$ be the time complexity of computing the square of an $n$-bit integer, and let $T_2(n)$ be the time complexity of computing the product of two $n$-bit integers. Assuming that addition is ...
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