Questions tagged [landau-notation]

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

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90
votes
3answers
24k views

How does one know which notation of time complexity analysis to use?

In most introductory algorithm classes, notations like $O$ (Big O) and $\Theta$ are introduced, and a student would typically learn to use one of these to find the time complexity. However, there are ...
46
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10answers
11k views

O(·) is not a function, so how can a function be equal to it?

I totally understand what big $O$ notation means. My issue is when we say $T(n)=O(f(n))$ , where $T(n)$ is running time of an algorithm on input of size $n$. I understand semantics of it. But $T(n)$ ...
46
votes
2answers
7k views

Order of growth definition from Reynolds & Tymann

I am reading a book called Principles of Computer Science (2008), by Carl Reynolds and Paul Tymann (published by Schaum's Outlines). The second chapter introduces algorithms with an example of a ...
44
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2answers
3k views

What is the meaning of $O(m+n)$?

This is a basic question, but I'm thinking that $O(m+n)$ is the same as $O(\max(m,n))$, since the larger term should dominate as we go to infinity? Also, that would be different from $O(\min(m,n))$. ...
35
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6answers
9k views

Sorting functions by asymptotic growth

Assume I have a list of functions, for example $\qquad n^{\log \log(n)}, 2^n, n!, n^3, n \ln n, \dots$ How do I sort them asymptotically, i.e. after the relation defined by $\qquad f \leq_O g \...
31
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4answers
42k views

How do O and Ω relate to worst and best case?

Today we discussed in a lecture a very simple algorithm for finding an element in a sorted array using binary search. We were asked to determine its asymptotic complexity for an array of $n$ elements. ...
23
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5answers
10k views

Is O(mn) considered “linear” or “quadratic” growth?

If I have some function whose time complexity is O(mn), where m and n are the sizes of its two inputs, would we call its time complexity "linear" (since it's linear in both m and n) or "quadratic" (...
19
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2answers
7k views

Changing variables in recurrence relations

Currently, I am self-studying Intro to Algorithms (CLRS) and there is one particular method they outline in the book to solve recurrence relations. The following method can be illustrated with this ...
18
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7answers
2k views

Justification for neglecting constants in Big O

Many a times if the complexities are having constants such as 3n, we neglect this constant and say O(n) and not O(3n). I am unable to understand how can we neglect such three fold change? Some thing ...
18
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2answers
3k views

Construct two functions $f$ and $g$ satisfying $f \ne O(g), g \ne O(f)$

Construct two functions $ f,g: R^+ → R^+ $ satisfying: $f, g$ are continuous; $f, g$ are monotonically increasing; $f \ne O(g)$ and $g \ne O(f)$.
15
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4answers
531 views

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 ...
15
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2answers
14k views

Why is there the regularity condition in the master theorem?

I have been reading Introduction to Algorithms by Cormen et al. and I'm reading the statement of the Master theorem starting on page 73. In case 3 there is also a regularity condition that needs to be ...
14
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3answers
635 views

What goes wrong with sums of Landau terms?

I wrote $\qquad \displaystyle \sum\limits_{i=1}^n \frac{1}{i} = \sum\limits_{i=1}^n \cal{O}(1) = \cal{O}(n)$ but my friend says this is wrong. From the TCS cheat sheet I know that the sum is also ...
13
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3answers
4k views

What does Θ(1) memory mean?

I have the definition of an in-situ algorithm from the professor, but I don't understand it. In-situ algorithms refer to algorithms that operate with Θ(1) memory. What does that mean?
12
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2answers
458 views

Infinite chain of big $O's$

First, let me write the definition of big $O$ just to make things explicit. $f(n)\in O(g(n))\iff \exists c, n_0\gt 0$ such that $0\le f(n)\le cg(n), \forall n\ge n_0$ Let's say we have a finite ...
12
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6answers
20k views

n*log n and n/log n against polynomial running time

I understand that $\Theta(n)$ is faster than $\Theta(n\log n)$ and slower than $\Theta(n/\log n)$. What is difficult for me to understand is how to actually compare $\Theta(n \log n)$ and $\Theta(n/\...
11
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2answers
483 views

How to prove that $n(\log_3(n))^5 = O(n^{1.2})$?

This a homework question from Udi Manber's book. Any hint would be nice :) I must show that: $n(\log_3(n))^5 = O(n^{1.2})$ I tried using Theorem 3.1 of book: $f(n)^c = O(a^{f(n)})$ (for $c >...
11
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1answer
7k views

What does tilde mean, in big-O notation?

I'm reading a paper, and it says in its time complexity description that time complexity is $\tilde{O}(2^{2n})$. I have searched the internet and wikipedia, but I can't find what this tilde signifies ...
11
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2answers
262 views

Is $O$ contained in $\Theta$?

So I have this question to prove a statement: $O(n)\subset\Theta(n)$... I don't need to know how to prove it, just that in my mind this makes no sense and I think it should rather be that $\Theta(n)\...
11
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1answer
1k views

Asymptotic Analysis for two variables?

How is asymptotic analysis (big o, little o, big theta, big theta etc.) defined for functions with multiple variables? I know that the Wikipedia article has a section on it, but it uses a lot of ...
10
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3answers
631 views

Error in the use of asymptotic notation

I'm trying to understand what is wrong with the following proof of the following recurrence $$ T(n) = 2\,T\!\left(\left\lfloor\frac{n}{2}\right\rfloor\right)+n $$ $$ T(n) \leq 2\left(c\left\...
10
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2answers
946 views

How to discuss coefficients in big-O notation

What notation is used to discuss the coefficients of functions in big-O notation? I have two functions: $f(x) = 7x^2 + 4x +2$ $g(x) = 3x^2 + 5x +4$ Obviously, both functions are $O(x^2)$, indeed $\...
10
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1answer
5k views

Can a Big-Oh time complexity contain more than one variable?

Let us say for instance I am doing string processing that requires some analysis of two strings. I have no given information about what their lengths might end up being, so they come from two distinct ...
10
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3answers
673 views

Sums of Landau terms revisited

I asked a (seed) question about sums of Landau terms before, trying to gauge the dangers of abusing asymptotics notation in arithmetics, with mixed success. Now, over here our recurrence guru JeffE ...
9
votes
5answers
7k views

What is an Efficient Algorithm?

From the point of view of asymptotic behavior, what is considered an "efficient" algorithm? What is the standard / reason for drawing the line at that point? Personally, I would think that anything ...
9
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3answers
21k views

Big O: Nested For Loop With Dependence

I was given a homework assignment with Big O. I'm stuck with nested for loops that are dependent on the previous loop. Here is a changed up version of my homework question, since I really do want to ...
8
votes
3answers
1k views

Why is $3^n = 2^{O(n)}$ true?

$3^n = 2^{O(n)}$ is apparently true. I thought that it was false though because $3^n$ grows faster than any exponential function with a base of 2. How is $3^n = 2^{O(n)}$ true?
8
votes
4answers
1k views

Nested Big O-notation

Let's say I have a graph $|G|$ with $|E|=O(V^2)$ edges. I want to run BFS on $G$ which has a running time of $O(V+E)$. It feels natural to write that the running time on this graph would be $O(O(V^2)+...
8
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2answers
253 views

Why doesn't $O(1)+O(2)+\cdots+O(n)$ have an interpretation?

In CLRS (on pages 49-50), what is the meaning of the following statement: $\Sigma_{i=1}^{n} O(i)$ is only a single anonymous function (of $i$), but is not the same as $O(1)+O(2)+\cdots+O(n)$, which ...
7
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2answers
278 views

Are functions in O(n) that are nor in o(n) all in Θ(n)?

One of my lectures makes the following statement: $$( f(n)=O(n) \land f(n)\neq o(n) )\implies f(n)=\Theta(n)$$ Maybe I'm missing something in the definitions, but for example bubble sort is $O(n^2)$ ...
7
votes
2answers
153 views

Two functions $g(n)$, $G(n)$ such that $g(n) = o(G(n))$ but $g(n+1) \neq o(G(n))$

The title of the question expresses what I'm looking for - this is to help me better understand the prerequisites for the Non-Deterministic Time Hierarchy Theorem For instance, the Arora-Barak book ...
7
votes
2answers
455 views

$\log^*(n)$ runtime analysis

So I know that $\log^*$ means iterated logarithm, so $\log^*(3)$ = $(\log\log\log\log...)$ until $n \leq 1$. I'm trying to solve the following: is $\log^*(2^{2^n})$ little $o$, little $\omega$, ...
7
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2answers
977 views

Is $n$ times $O(1)$ equivalent to $O(n)$? [duplicate]

I am having a hard time figuring out if $$\sum^n_{i=0} O(1) =O(n)\,.$$ I think it doesn't but I am unable to find a convincing explanation for that, does anyone have an intuitive yet mathematical ...
7
votes
2answers
341 views

Can you operate on and draw conclusions on functions described asymptotically?

This question is homework based (not using actual problem though)! Say you have a function described as: $$f(n) \in O(2n^2) \, .$$ Can you then go on the treat this as: $$f(n) = 2n^2$$ and ...
7
votes
1answer
9k views

Solving $T(n)= 3T(\frac{n}{4}) + n\cdot \lg(n)$ using the master theorem

Introduction to Algorithms, 3rd edition (p.95) has an example of how to solve the recurrence $$\displaystyle T(n)= 3T\left(\frac{n}{4}\right) + n\cdot \log(n)$$ by applying the Master Theorem. I am ...
7
votes
1answer
863 views

Solving the recurrence relation $T(n) = 2T(\lfloor n/2 \rfloor) + n$

Solving the recurrence relation $T(n) = 2T(\lfloor n/2 \rfloor) + n$. The book from which this example is, falsely claims that $T(n) = O(n)$ by guessing $T(n) \leq cn$ and then arguing $\qquad \...
7
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2answers
2k views

Variations of Omega and Omega infinity

Some authors define $\Omega$ in a slightly different way: let’s use $ \overset{\infty}{\Omega}$ (read “omega infinity”) for this alternative definition. We say that $f(n) = \overset{\infty}{\Omega}(g(...
6
votes
3answers
1k views

How did they cancel out O-terms in this fraction?

While reading a book about algorithms, I came across this derivation: $$ \frac{2a_0(2N) \ln(2N) + O(2N)}{2a_0N\ln N+O(N)} = \frac{2\ln(2N) + O(1)}{\ln N+O(1)} = 2 + O\left(\frac{1}{\log N}\right). $$ ...
6
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3answers
3k views

What do f(x) and g(x) represent in Big O notation?

I have been reading about Big O notation. People writing about Big O often use the terms $f(x)$ and $g(x)$. For instance, I often see people write things like $f(x) = O(g(x))$ or $f(x) \in O(g(x))$. ...
6
votes
2answers
344 views

Are $\log_{10}(x)$ and $\log_2(x)$ in the same big-O class of functions?

Are $\log_{10}(x)$ and $\log_{2}(x)$ in the same big-O class of functions? In other words, can one say that $\log_{10}(x)=O(\log x)$ and $\log_{2}(x)=O(\log x)$?
6
votes
2answers
105 views

Why does $2^{O(\log n)} = n^{O(1)}$ hold?

I was recently looking at the article on the P complexity class on Wikipedia. In the section on relationships to other classes, it mentions that P is known to be at least as large as L, giving this ...
6
votes
2answers
315 views

Landau Notation, Definitions: Limits vs. Corman's

When dealing with Landau notation, $\Theta, O,\Omega,o,\omega$, why do some texts choose the Corman style definitions, i.e.: $$o(g(n))=\{ f(n): \forall c>0:\exists n_0>0:\; 0\leq f(n) < cg(n)...
6
votes
2answers
581 views

Why do Θ-bounds not survive taking differences?

$f_1$, $f_2$, $g_1$, and $g_2$ are functions such that: $$f_1 = \Theta(f_2)$$ $$g_1 = \Theta(g_2)$$ I was able to prove that: $$\frac{f_1}{g_1} = \Theta\biggl(\frac{f_2}{g_2}\biggr)$$ But I can't ...
6
votes
2answers
5k views

Why does heapsort run in $\Theta(n \log n)$ instead of $\Theta(n^2 \log n)$ time?

I am reading section 6.4 on Heapsort algorithm in CLRS, page 160. ...
6
votes
1answer
332 views

What does the “big O complexity” of a function mean?

What do people mean when they refer to the "big O complexity" of a function? What is the big O complexity of $9n^2 + 10n$, for example?
6
votes
2answers
98 views

Is $\{\Theta(f)|f:\mathbb{N}\rightarrow\mathbb{N}\}$ Dedekind-complete?

Let $\Theta$ and $o$ be defined as usual (Landau-notation). For two equivalence classes defined by $\Theta$ we define $$\Theta(f) <_o \Theta(g) :\Leftrightarrow f \in o(g)\qquad.$$ Let $$\mathbb{F}:...
6
votes
2answers
896 views

Is Big-Oh notation preserved under monotonic functions?

I was just looking at the big-Oh notation. I wanted to know if the following is true in general $$f(n)=O(g(n)) \implies \log (f(n)) = O(\log (g(n)))$$ I can prove that this is true if $g$ is ...
6
votes
1answer
128 views

Use of Big O Notation in a recent paper by Khot et al

I'm reading a paper about Constraint Satisfaction Problems, specifically "A Characterization of Strong Approximation Resistance", Subhash Khot, Madhur Tulsiani, Pratik Worah (ECCC TR13-075). The ...
6
votes
1answer
160 views

What is the name for the complexity class $O(n^{1+\epsilon})$

Sometimes problems can be solved in $O(n^c)$ time for any $c > 1$, but not for $c=1$. Typically this is written as $O(n^{1 + \epsilon})$, since $\epsilon$ is understood to be some small positive ...
5
votes
3answers
511 views

How to prove $(n+1)! = O(2^{(2^n)})$

I am trying to prove $(n+1)! = O(2^{(2^n)})$. I am trying to use L'Hospital rule but I am stuck with infinite derivatives. Can anyone tell me how i can prove this?