## Big theta problems

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While trying to understand the difference between Theta and O notation I came across the following statement :. But I do not understand this. The book explains it mathematically, but it's too complex and gets really boring to read when I am sram road groupset not understanding.

Everything that is Theta f n is also O f nbut not the other way around. For this reason big-Theta is more informative than big-O notation, so if we can say something is big-Theta, it's usually preferred.

However, it is harder to prove something is big Theta, than to prove it is big-O. Omega n is asymptotic lower bound. Note that this notation is not related to the best, worst and average cases analysis of algorithms.

Each one of these can be applied to each analysis. I recommend reading pages section 1. People often confuse O-notation by assuming that it gives an exact order of Growth; they use it as if it specifies a lower bound as well as an upper bound. Big-Oh is for approximations. However, we do not know at what values of N in that sense we know approximately. If the running time is expressed in big-O notation, you know that the running time will not be slower than the given expression.

It expresses the worst-case scenario. But with Theta notation you also known that it will not be faster.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. In that case, g is both an upper bound and a lower bound on the growth of f.

We wish to evaluate the cost function c n where n is the size of the input list. Since it grows neither slower nor faster, it must grow at the same speed. This is the interesting property of big-theta notation: it's both an upper bound and a lower bound on the complexity. In other words Theta f n 'describes' a function T nif both O [big O] and Omega, 'describe' the same T, with the same f. Learn more. How to calculate big-theta Ask Question. Asked 8 years, 7 months ago.

**Theta Notation - Definition & Example**

Active 6 years, 10 months ago. Viewed 32k times. Can some one provide me a real time example for how to calculate big theta. Bill the Lizard k gold badges silver badges bronze badges. Navin Leon Navin Leon 1, 2 2 gold badges 19 19 silver badges 41 41 bronze badges. Have you read e. Active Oldest Votes.

Victor Nicollet Victor Nicollet 23k 2 2 gold badges 51 51 silver badges 87 87 bronze badges. For the example algorithm above: Should min be initially negative infinity? Jan 28 '18 at Oliver Charlesworth Oliver Charlesworth k 25 25 gold badges silver badges bronze badges.

Navin: Big-O, etc.Big-O, Little-o, Omega, and Theta are formal notational methods for stating the growth of resource needs efficiency and storage of an algorithm. There are four basic notations used when describing resource needs.

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The math in big-O analysis can often be intimidates students. One of the simplest ways to think about big-O analysis is that it is basically a way to apply a rating system for your algorithms like movie ratings. It tells you the kind of resource needs you can expect the algorithm to exhibit as your data gets bigger and bigger.

Think about the graphs in the grow rate section. The way each curve looks. That is the most important thing to understand about algorithms analysis. This means that if we were to draw a graph of the resource needs of a particular algorithm, it would fall under the curve described by f n f n f n. What's more, it doesn't need to be under the exact curve described by f n f n f n. It could be under a constant scaled curve for f n f n f n In fact it can be any constant, as long as it is a constant.

A constant is simply a number that does not change with n. So as n n n gets bigger, you cannot change what the constant is. The actual value of the constant does not matter though. In summary, when we are looking at big-O, we are in essence looking for a description of the growth rate of the resource increase. The exact numbers do not actually matter in the end.

Big-O, Little-o, Theta, Omega. O f n O f n 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.

No results matching " ".Using Big-Theta notation to classify the traditional grade school algorithms for addition and multiplication. That is, if asked to add two numbers each having n-digits, how many individual additions must be performed?

If asked to multiply two n-digit numbers, how many individual multiplication are required? First, do you understand theta notation? Do you understand it well enough to tell me, using theta notation, how long this algorithm takes to run?

How many times do you add two one-digit numbers? It depends on how many times you have to carry, but there's an upper limit and a lower limit. So, my question for you is, and this is possibly a hard question, but one you need to be good at answering in order to be good at thinking about hard problems: What, precisely, do you not understand?

Thanks for the reply Rashakil Fol, The problem is that I do not know how to apply the big-O to this problem, perhaps if you could give me an example, I will appreciate it. Well, I know that.

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Why can't you solve this problem? Where in the solution process do you go, I don't know how to do this? You can't just draw a blank. That's not how problems get solved. Do you understand what the problem's asking you to do? If yes, then do you know how to calculate the minimum and maximum possible number of addition operations when adding two numbers? If yes, then can you express those bounds using theta notation? Here, "No" might mean that it's just impossible.

I think Rashakil Fol, it is the part where expressing those bounds using theta notation?

I need help in here, this area. Well, what are the bounds? If the numbers have length n, then the number of addition operations you have to make is between n and 2n-1, right? Anyway, looking at large enough values of n, we know that the number of addition operations is greater than 0.

This means that Theta n would describe the number of addition operations, where n is the length of the two numbers.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. My guess is that f n is BigTheta g nsince both functions are constants wich means the functions are proportionalbut a my teacher insists that I am wrong.

Am I right? Is there any way I could rest my case? Sorry if this sounds like a noob question! Yes, you are right. You are correct. Assuming you quoted the problem correct and there's no misunderstanding, your teacher is wrong if they said they are not theta-of-each-other. Without knowing the actual text of the exam problem though, and the exact text you wrote or circledthere is no way we on the internet can know there isn't some sort of misreading of the problem.

You should also note that you could still be wrong in your justification even though your answer is right. For the record, not only is f x in the set BigTheta g xbut g x is in the set BigTheta f x. It would also imply that BigTheta is a symmetric relation.

You now have the suitable tools to ask "why do you think I am wrong? Learn more. Big Theta problem Ask Question. Asked 8 years, 10 months ago. Active 5 years, 10 months ago. Viewed times.

Greg Hewgill k gold badges silver badges bronze badges. It is not a homework, it's a part of my exam. On the site here, the homework tag tends to encompass actual homework, exams, etc Aside from that being a central way to unify a type of question, it's also pragmatic: if you look at the tags, you'll see that homework has almost followers and almost questions, whereas exam has only one follower and about questions. Active Oldest Votes. Ishtar Ishtar Thank you very much! That is the exact text, only translated from my language into English.

It is not the justification that bothers him, but the answer. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann[1] Edmund Landau[2] and others, collectively called Bachmann—Landau notation or asymptotic notation.

In computer sciencebig O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. Big O notation characterizes functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation. The letter O is used because the growth rate of a function is also referred to as the order of the function.

A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Let f be a real or complex valued function and g a real valued function. Let both functions be defined on some unbounded subset of the real positive numbersand g x be strictly positive for all large enough values of x.

In many contexts, the assumption that we are interested in the growth rate as the variable x goes to infinity is left unstated, and one writes more simply that. As g x is chosen to be non-zero for values of x sufficiently close to aboth of these definitions can be unified using the limit superior :.

In typical usage the O notation is asymptotical, that is, it refers to very large x. In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant.

As a result, the following simplification rules can be applied:. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of xnamely 6 x 4. Now one may apply the second rule: 6 x 4 is a product of 6 and x 4 in which the first factor does not depend on x.

Omitting this factor results in the simplified form x 4. Thus, we say that f x is a "big-oh" of x 4. In both applications, the function g x appearing within the O This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

Big O notation is useful when analyzing algorithms for efficiency. Ignoring the latter would have negligible effect on the expression's value for most purposes.

### Big-Theta Problems!! HOW?

Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term n 3 or n 4. Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either.

Big O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. Consider, for example, the exponential series and two expressions of it that are valid when x is small:.

If the function f can be written as a finite sum of other functions, then the fastest growing one determines the order of f n.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It only takes a minute to sign up. This is not homework. I have the solution but it's not what I'm getting.

I know there are multiple solutions to the problem but I want to make sure that I'm not missing anything. Divide the inequality by the largest order n-term. This is the only way I know how to solve these equations.

We choose C 1 to be 1. Am I wrong somewhere? Your solution is fine. So your approach is valid, and it's possible that the solution from your class is also valid: that's not a contradiction. Let me give you one piece of advice for structuring your proof in the future, though. Experience has proven that that kind of reasoning is error-prone, though: it's easy for you to make a mistake along the way.

It is also hard for others to check. When you write up your proof, I suggest you do the reasoning in the opposite direction.

This better matches how people think and makes it easier for the reader to understand what's going on. And the purpose of a proof is to communicate an idea to the reader, so the proof should be structured to make the reader's life easier: to make it as easy as possible for the reader to verify that the proof's reasoning is correct and valid.

It's just like a good book; good writing is chosen to make life good for the reader. Proofs are the same way. And, ultimately, this will benefit you too: in my experience, if you follow this approach, then you're less likely to make a subtle mistake in your proof.

But you know what? That's OK. This style of proof will be better for you, and better for the reader. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

Big Theta Proof on polynomial function Ask Question. Asked 6 years, 7 months ago. Active 6 years, 7 months ago.

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