What is the difference between Θ(n) and O(n)?

f(n) = o(g(n examples))
big theta problems
big o notation - discrete math
asymptotic notation questions
big omega
f(n) = o(f(n/2))
big theta of n factorial
what is the difference between big oh, big omega and big theta notations?

Sometimes I see Θ(n) with the strange Θ symbol with something in the middle of it, and sometimes just O(n). Is it just laziness of typing because nobody knows how to type this symbol, or does it mean something different?

Short explanation:

If an algorithm is of Θ(g(n)), it means that the running time of the algorithm as n (input size) gets larger is proportional to g(n).

If an algorithm is of O(g(n)), it means that the running time of the algorithm as n gets larger is at most proportional to g(n).

Normally, even when people talk about O(g(n)) they actually mean Θ(g(n)) but technically, there is a difference.


More technically:

O(n) represents upper bound. Θ(n) means tight bound. Ω(n) represents lower bound.

f(x) = Θ(g(x)) iff f(x) = O(g(x)) and f(x) = Ω(g(x))

Basically when we say an algorithm is of O(n), it's also O(n2), O(n1000000), O(2n), ... but a Θ(n) algorithm is not Θ(n2).

In fact, since f(n) = Θ(g(n)) means for sufficiently large values of n, f(n) can be bound within c1g(n) and c2g(n) for some values of c1 and c2, i.e. the growth rate of f is asymptotically equal to g: g can be a lower bound and and an upper bound of f. This directly implies f can be a lower bound and an upper bound of g as well. Consequently,

f(x) = Θ(g(x)) iff g(x) = Θ(f(x))

Similarly, to show f(n) = Θ(g(n)), it's enough to show g is an upper bound of f (i.e. f(n) = O(g(n))) and f is a lower bound of g (i.e. f(n) = Ω(g(n)) which is the exact same thing as g(n) = O(f(n))). Concisely,

f(x) = Θ(g(x)) iff f(x) = O(g(x)) and g(x) = O(f(x))


There are also little-oh and little-omega (ω) notations representing loose upper and loose lower bounds of a function.

To summarize:

f(x) = O(g(x)) (big-oh) means that the growth rate of f(x) is asymptotically less than or equal to to the growth rate of g(x).

f(x) = Ω(g(x)) (big-omega) means that the growth rate of f(x) is asymptotically greater than or equal to the growth rate of g(x)

f(x) = o(g(x)) (little-oh) means that the growth rate of f(x) is asymptotically less than the growth rate of g(x).

f(x) = ω(g(x)) (little-omega) means that the growth rate of f(x) is asymptotically greater than the growth rate of g(x)

f(x) = Θ(g(x)) (theta) means that the growth rate of f(x) is asymptotically equal to the growth rate of g(x)

For a more detailed discussion, you can read the definition on Wikipedia or consult a classic textbook like Introduction to Algorithms by Cormen et al.

Difference between O(N^2) and Θ(N^2), Read and learn for free about the following article: Big-θ (Big-Theta) notation. The measurements of Big-O, Big-Theta, and Big-Omega would often be different Big-O tells you which functions grow at a rate >= than f(N), for large N θ on the other hand is both a lower and an upper bound. So a linear runtime algorithm is in θ(n) but not in θ(n^2). Mathematically θ(n^2) is a set of functions which don't grow faster that c*n^2 for some c but also doesn't grow slower than c*n^2 for another smaller c. For example this set contains c * x^2, c1 * x^2 + c2 * x,

There's a simple way (a trick, I guess) to remember which notation means what.

All of the Big-O notations can be considered to have a bar.

When looking at a Ω, the bar is at the bottom, so it is an (asymptotic) lower bound.

When looking at a Θ, the bar is obviously in the middle. So it is an (asymptotic) tight bound.

When handwriting O, you usually finish at the top, and draw a squiggle. Therefore O(n) is the upper bound of the function. To be fair, this one doesn't work with most fonts, but it is the original justification of the names.

Big-θ (Big-Theta) notation (article), Before, we used big-Theta notation to describe the worst case running time of binary search, which is Θ(lg n). The best case running time is a completely different  If an algorithm is of θ(g(n)), it means that the running time of the algorithm as n (input size) gets larger is proportional to g(n). If an algorithm is of O(g(n)), it means that the running time of the algorithm as n gets larger is at most proportional to g(n).

one is Big "O"

one is Big Theta

http://en.wikipedia.org/wiki/Big_O_notation

Big O means your algorithm will execute in no more steps than in given expression(n^2)

Big Omega means your algorithm will execute in no fewer steps than in the given expression(n^2)

When both condition are true for the same expression, you can use the big theta notation....

Big-O notation (article) | Algorithms, Dropping lower order terms is always fine because there will always be a n0 after which Θ(n3) has higher values than Θn2) irrespective of the constants involved  The upper bound on f(n) is n^3. So our function f(n) is clearly O(n^3). But is it Θ(n^3)? For f(n) to be in Θ(n^3) it has to be sandwiched between two functions one forming the lower bound, and the other the upper bound, both of which grown at n^3. While the upper bound is obvious, the lower bound can not be n^3.

Rather than provide a theoretical definition, which are beautifully summarized here already, I'll give a simple example:

Assume the run time of f(i) is O(1). Below is a code fragment whose asymptotic runtime is Θ(n). It always calls the function f(...) n times. Both the lower and the upper bound is n.

for(int i=0; i<n; i++){
    f(i);
}

The second code fragment below has the asymptotic runtime of O(n). It calls the function f(...) at most n times. The upper bound is n, but the lower bound could be Ω(1) or Ω(log(n)), depending on what happens inside f2(i).

for(int i=0; i<n; i++){
    if( f2(i) ) break;
    f(i);
}

Analysis of Algorithms, O(f(n)) means that the curve described by f ( n ) f(n) f(n) is an upper bound for the resource needs of a function. This means that if we were to draw a graph of the  What’s the difference between \n (newline) and \r (carriage return)? In particular, are there any practical differences between \n and \r? Are there places where one should be used instead of the other? All the answers are fairly predictable, but I'd be interested to know if there are any PRACTICAL differences between \n and \r.

Theta is a shorthand way of referring to a special situtation where the big O and Omega are the same.

Thus, if one claims The Theta is expression q, then they are also necessarily claiming that Big O is expression q and Omega is expression q.


Rough analogy:

If: Theta claims, "That animal has 5 legs." then it follows that: Big O is true ("That animal has less than or equal to 5 legs.") and Omega is true("That animal has more than or equal to 5 legs.")

It's only a rough analogy because the expressions aren't necessarily specific numbers, but instead functions of varying orders of magnitude such as log(n), n, n^2, (etc.).

Big-O, Little-o, Theta, Omega · Data Structures and Algorithms, For example, we may write T(n) = n - 1 ∊ O(n2). This is indeed true, but not very useful. Ω and Θ notation. Big Omega is used to  “N” and “KN” editions of Windows aren’t prevented from using these media playback features. Instead, they’re just not installed by default. If you want to enable these disabled multimedia features on a N or KN edition of Windows, download the free Media Feature Pack from Microsoft.

Big O notation: definition and examples · YourBasic, Unlike Big Ω (omega) and Big θ (theta), the 'O' in Big O is not greek. caveat is that Big O notation often refers to something different academically In plain english it says: our runtime, which we call f(n), is Big O of g(n) if and  Windows 10 N editions are specifically designed for Europe and Switzerland to comply with the European law. The N stands for Not with Media Player and does not come with Windows Media Player pre-installed. KN is specially designed for Korean market and does not include Windows Media Player (WMP) and an instant messenger.

Total n00b's guide to Big O, Big Ω, & Big θ - Russell Tepper, Let be two arbitrary functions. Definition of Big O. if there exists integer and constant , such that for . Example: We claim that Then, we pick and Therefore, for n  An MIT graduate who brings years of technical experience to articles on SEO, computers, and wireless networking. Home and business owners looking to buy networking gear face an array of choices. Many products conform to the 802.11a, 802.11b/g/n, and/or 802.11ac wireless standards collectively known as Wi-Fi technologies.

What is the difference between big oh, big omega and big theta , Big O notation is a mathematical notation that describes the limiting behavior of a function when with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, There are two formally close, but noticeably different, usages of this notation: and say that the algorithm has order of n2 time complexity. If a < b, then a & b can never be equal in magnitude, and a is always strictly smaller than b. With Big Oh notation, if we say , then the function g (n) forms an asymptotic bound for f (n), but f (n) can come within a constant factor of g (n) in the limit (we sometimes call this an asymptotically tight bound).

Comments
  • It's not obvious, but this question is a duplicate of this one stackoverflow.com/questions/464078/… from yesterday.
  • Possible duplicate of Difference between lower bound and tight bound?
  • If "If an algorithm is of O(g(n)), it means that the running time of the algorithm as n gets larger is at most proportional to g(n)." Then how do you say that "Basically when we say an algorithm is of O(n), it's also O(n2), O(n1000000), O(2n)," ??
  • @Andy897 It follows from the definition of "proportional". From Wikipedia: "In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier. The constant is called the coefficient of proportionality or proportionality constant."
  • What does >= \Omega(...) mean? I understand if we say it's a member of \Omega(...), but if it's greater than it? What sense does it make?
  • I usually never go below 3-4 answers on any questions. This was worth the ride. Thanks for sharing the trick. :D
  • But is wrong! The number of steps is bounded above by n^2 as n gets very large. However, an algorithm that runs in n^2 + c steps takes more than n^2 steps, but is still O(n^2). Big-O notation only describes asymptotic beahviour.
  • This is not a end all be all definition. It's just a launching point.... Since we are talking about asymptotic notations as n approaches infinity. The constant C becomes a non factor.
  • While I like the simplicity of this answer, it should be noted that an O(n^2) algorithm could very well take 1,000,000,000*n^2 steps to execute, which is certainly much larger than n^2. An algorithm being O(n^2) just means that it will take no more than k*n^2 steps to execute, where k is some positive real number.
  • What do you mean by "asymptotic runtime"?
  • Asymptotic in this context means "for large enough n". The runtime of code fragment whose asymptotic runtime is Θ(n) will grow linearly as n increases, e.g. runtime T can be expressed as T(n) = a*n+b. For small values of n (e.g. n=1 or 2) this may not be the best way of describing the behaviour - perhaps you have some initialization code that takes a lot longer than f(i).
  • You have messed up the words and graphs.
  • @kushalvm, thanks for your honesty. Could you kindly explain what do you mean specifically? For the sake of my learning and others that may get confused with this answer. :-)