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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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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Prove or disprove that $log^{k}(n) \in O(\sqrt{n}) \forall k > 0$

I'm trying to solve the problem described in the title. By using the free version of wolfram and testing some increasing values of $k$ I get that: $$\lim_{n \rightarrow \infty} \frac{log^{k}(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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1answer
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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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2answers
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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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1answer
70 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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3answers
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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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1answer
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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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1answer
31 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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1answer
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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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1answer
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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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3answers
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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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1answer
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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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1answer
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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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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
579 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 ...
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2answers
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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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1answer
50 views

Time complexity, Big-O for this function?

def f(n): if n < 100000: return 0; for(int i = 0; i < n*n; i++){ return f(n-1) } What is the time complexity? My answer is $O((...
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1answer
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Asymptotic notation and random variables

I have two random variables $X$ and $Y$ and I want to bound the value of one in terms of the other (for now, I don't care about the actual distribution of their values). Suppose that the two ...
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2answers
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How to find i-th root of n whose remainder is the smallest?

Given a number n, what is the most assymptotically fast algorithm to express it in terms of base^exponent + rem such that rem is the smallest possible and base is limited from 2 to some relatively ...
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1answer
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How to calculate Big O of $T(n) = aT(n^b) + f(n)$?

I'm a student studying Big O. I know that we can solve $T(n) = aT(\frac{n}{b}) + f(n)$ by compering $n^{\log_b{a}}$ to $f(n)$ or $O(n^{\log_b{a}} + f(n))$ Today I was faced with $T(n) = T(\sqrt n)...
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$\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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2answers
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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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1answer
52 views

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 ...
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2answers
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What does “bounded above” mean in Family of Bachmann–Landau notations?

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)...
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how to compute $O(n^{0.000001})$ [duplicate]

this MIT course gives a formula about Big O $$n^{0.999999} \log n = O(n^{0.999999} \cdot n^{0.000001})$$ going through wiki, i cannot find a similar Big O properties or usages. how to compute $O(n^{...
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Are “of the order of n” and “Big O” the same thing?

I am learning from the MIT course Introduction to Algorithms. The professor says: Now, remember $\Theta(n)$ is essentially something that says "of the order of $n$". What does "of the order ...
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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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3answers
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What should I call algorithms with non-linear non-constant time?

I am writing a paper in which I want to refer to a group of algorithms. Some of these algorithms are of complexity O(NlogN), and some of the are more complex (e.g ...
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1answer
27 views

Comparing different asymptotic notations

Suppose we have 3 algorithms complexity times at the worst case: A = $O(nlogn)$ B = $O(n\sqrt{n})$ C = $\Theta(n)$ In my opinion, it is not possible to define the best solution, since we don't know ...
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1answer
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Asymptotics question

Is $\frac {n!} {2!\cdot 4!\cdot 8!\dots (n/2)!}=O(4^n)$? I am really stuck and I tend to believe it's true, but I don't know how to prove it. Any help would be appreciated!
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2answers
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Traveling Salesman Problem: Big O Complexity of Algorithm

I'm trying to figure out how to do this problem in my intro algorithm class, but I'm a little confused. The Traveling Salesman problem (TSP) is famous. Given a list of cities and the distances ...
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3answers
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How can i prove this asymptotic comparison? [duplicate]

This is an exercise that's part of my assignment, but it is optional and flagged as a "challenge". I would like to discuss its solution: Prove that: $$ 27\log{n} + \sqrt{n} = \theta(\sqrt{n})$$ ...
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2answers
173 views

How to find the big o running time if the recursion function have different cases of recursion with different fraction of n?

How to find the big o running time if the recursion function have different cases of recursion with different fraction of n? If I have a recursive function like this for example (This is just an ...
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0answers
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Probability in 1-universal hash function

I am trying to prepare for an exam and I am not sure how to solve this task: Given is a hash function with m buckets, which uses a 1-universal hash function h: U -> H and handles collisions with ...