What does people mean when they say things like:
Big O estimates scalability.
The runtime grows “on the order of the square of the size of the input”, given that the algorithm in the worst case runs in O(n^2).
Does it mean for large n quadruple runtime for the doubled input size (assuming the algorithm runs in O(n^2))?
Is you said YES, then suppose that the number of steps our algorithm takes in the worst case is expressed by the function:
It follows that:
Moreover, we can see that:
But we can't quadruple runtime for the doubled input size n (even for large values of n) in this case, because it would simply be wrong. To see this, check the following plot:
And here is the plot of the ratio f(2n)/f(n):
It doesn't look like this ratio is tending towards 4.
So, what is scalabity, then? What does Big O estimate, if not the ratios like f(2n)/f(n) for large enough n? It doesn't seem to me that f scales like n^2.
So, what is scalabity, then? What does Big O estimate?
Big-O notation gives an upper bound on asymptotic complexity for e.g. classifying algorithms and/or functions.
Does it mean for large n quadruple runtime for the doubled input size
No, you are mixing up asymptotic (time) complexity with actual "runtime count" of your function. When deriving the asymptotic complexity in terms of Big-O notation for, say, a function, we will ignore any constant terms, meaning that for your own example, f(n) and f(2n) are identical in terms of asymptotic complexity. Both for f(n) and for g(n) = f(2n), f and g grows (scales, if you wish) quadratically in they asymptotic region. This does not mean their runtime is exactly quadrupled for a doubled input size (this kind of detailed hypothesis doesn't even make sense in term of asymptotic behavior), it only governs an upper bound on growth behavior for sufficiently large input.
Refer e.g. to my answer to this question for more thorough explanation on the non-relevance on both constant and lower order terms when deriving asymptotic complexity for a function:
Role of lower order terms in big O notation
Consider an algorithm that uses no extra variables except the given input.
How to represent the space complexity in BigO Notation?
O(1)
Where it requires a constant amount of additional space namely 0.
For example, if time complexity of merge sort is O(n log n) then why it is big O not theta or omega. I know the definition of these, but what I do not understand is how to determine the notation based on the definition.
For most algorithms, you are basically concerned on the upper bound on its running time. For example, you have some algorithm to sort an array of numbers. Now you would most likely be concerned that how fast will the algorithm run in the worst possible case.
Hence the complexity of merge sort is mostly written as O(nlogn) even when it will be better to express it as theta(nlogn) because theta notation is a more tighter bound. And merge sort runs in theta(nlogn) time because it will always consume this much time no matter what the input is.
You will not find omega notation again mostly because we are concerned with the upper bounds on running time and not the lower bound.
Question
Hi I am trying to understand what order of complexity in terms of Big O notation is. I have read many articles and am yet to find anything explaining exactly 'order of complexity', even on the useful descriptions of Big O on here.
What I already understand about big O
The part which I already understand. about Big O notation is that we are measuring the time and space complexity of an algorithm in terms of the growth of input size n. I also understand that certain sorting methods have best, worst and average scenarios for Big O such as O(n) ,O(n^2) etc and the n is input size (number of elements to be sorted).
Any simple definitions or examples would be greatly appreciated thanks.
Big-O analysis is a form of runtime analysis that measures the efficiency of an algorithm in terms of the time it takes for the algorithm to run as a function of the input size. It’s not a formal bench- mark, just a simple way to classify algorithms by relative efficiency when dealing with very large input sizes.
Update:
The fastest-possible running time for any runtime analysis is O(1), commonly referred to as constant running time.An algorithm with constant running time always takes the same amount of time
to execute, regardless of the input size.This is the ideal run time for an algorithm, but it’s rarely achievable.
The performance of most algorithms depends on n, the size of the input.The algorithms can be classified as follows from best-to-worse performance:
O(log n) — An algorithm is said to be logarithmic if its running time increases logarithmically in proportion to the input size.
O(n) — A linear algorithm’s running time increases in direct proportion to the input size.
O(n log n) — A superlinear algorithm is midway between a linear algorithm and a polynomial algorithm.
O(n^c) — A polynomial algorithm grows quickly based on the size of the input.
O(c^n) — An exponential algorithm grows even faster than a polynomial algorithm.
O(n!) — A factorial algorithm grows the fastest and becomes quickly unusable for even small values of n.
The run times of different orders of algorithms separate rapidly as n gets larger.Consider the run time for each of these algorithm classes with
n = 10:
log 10 = 1
10 = 10
10 log 10 = 10
10^2 = 100
2^10= 1,024
10! = 3,628,800
Now double it to n = 20:
log 20 = 1.30
20 = 20
20 log 20= 26.02
20^2 = 400
2^20 = 1,048,576
20! = 2.43×1018
Finding an algorithm that works in superlinear time or better can make a huge difference in how well an application performs.
Say, f(n) in O(g(n)) if and only if there exists a C and n0 such that f(n) < C*g(n) for all n greater than n0.
Now that's a rather mathematical approach. So I'll give some examples. The simplest case is O(1). This means "constant". So no matter how large the input (n) of a program, it will take the same time to finish. An example of a constant program is one that takes a list of integers, and returns the first one. No matter how long the list is, you can just take the first and return it right away.
The next is linear, O(n). This means that if the input size of your program doubles, so will your execution time. An example of a linear program is the sum of a list of integers. You'll have to look at each integer once. So if the input is an list of size n, you'll have to look at n integers.
An intuitive definition could define the order of your program as the relation between the input size and the execution time.
Others have explained big O notation well here. I would like to point out that sometimes too much emphasis is given to big O notation.
Consider matrix multplication the naïve algorithm has O(n^3). Using the Strassen algoirthm it can be done as O(n^2.807). Now there are even algorithms that get O(n^2.3727).
One might be tempted to choose the algorithm with the lowest big O but it turns for all pratical purposes that the naïvely O(n^3) method wins out. This is because the constant for the dominating term is much larger for the other methods.
Therefore just looking at the dominating term in the complexity can be misleading. Sometimes one has to consider all terms.
Big O is about finding an upper limit for the growth of some function. See the formal definition on Wikipedia http://en.wikipedia.org/wiki/Big_O_notation
So if you've got an algorithm that sorts an array of size n and it requires only a constant amount of extra space and it takes (for example) 2 n² + n steps to complete, then you would say it's space complexity is O(n) or O(1) (depending on wether you count the size of the input array or not) and it's time complexity is O(n²).
Knowing only those O numbers, you could roughly determine how much more space and time is needed to go from n to n + 100 or 2 n or whatever you are interested in. That is how well an algorithm "scales".
Update
Big O and complexity are really just two terms for the same thing. You can say "linear complexity" instead of O(n), quadratic complexity instead of O(n²), etc...
I see that you are commenting on several answers wanting to know the specific term of order as it relates to Big-O.
Suppose f(n) = O(n^2), we say that the order is n^2.
Be careful here, there are some subtleties. You stated "we are measuring the time and space complexity of an algorithm in terms of the growth of input size n," and that's how people often treat it, but it's not actually correct. Rather, with O(g(n)) we are determining that g(n), scaled suitably, is an upper bound for the time and space complexity of an algorithm for all input of size n bigger than some particular n'. Similarly, with Omega(h(n)) we are determining that h(n), scaled suitably, is a lower bound for the time and space complexity of an algorithm for all input of size n bigger than some particular n'. Finally, if both the lower and upper bound are the same complexity g(n), the complexity is Theta(g(n)). In other words, Theta represents the degree of complexity of the algorithm while big-O and big-Omega bound it above and below.
Constant Growth: O(1)
Linear Growth: O(n)
Quadratic Growth: O(n^2)
Cubic Growth: O(n^3)
Logarithmic Growth: (log(n)) or O(n*log(n))
Big O use Mathematical Definition of complexity .
Order Of use in industrial Definition of complexity .
What is a plain English explanation of Theta notation? With as little formal definition as possible and simple mathematics.
How theta notation is different from the Big O notation ? Could anyone explain in plain English?
In algorithm analysis how there are used? I am confused?
If an algorithm's run time is Big Theta(f(n)), it is asymptotically bounded above and below by f(n). Big O is the same except that the bound is only above.
Intuitively, Big O(f(n)) says "we can be sure that, ignoring constant factors and terms, the run time never exceeds f(n)." In rough words, if you think of run time as "bad", then Big O is a worst case. Big Theta(f(n)) says "we can be sure that, ignoring constant factors and terms, the run time always varies as f(n)." In other words, Big Theta is a known tight bound: it's both worst case and best case.
A final try at intuition: Big O is "one-sided." O(n) run time is also O(n^2) and O(2^n). This is not true with Big Theta. If you have an algorithm run time that's O(n), then you already have a proof that it's not Big Theta(n^2). It may or may not be Big Theta(n)
An example is comparison sorting. Information theory tells us sorting requires at least ceiling(n log n) comparisons, and we have actually invented O(n log n) algorithms (where n is number of comparisons), so sorting comparisons are Big Theta(n log n).
I have always wanted to put this down in Simple words. Here is my try.
If an algorithm's time or space complexity is expressed in
Big O : Ex O(n) - means n is the upper limit. Final Value could be less than or equal to n.
Big Omega : Ex Ω(n) - means n is the lower limit. Final Value could be equal to or more than n.
Theta : Ex Θ(n) - means n is the only possible value. (both upper limit & lower limit)