BigO always checks the upper bound. So we can measure ..the way we write the code, so that there will be less time complexity and thus increase our code performance. But why do we use the lowerbound (omega) ? I did not understand the use of omega in real time. Can anybody please suggest me on this
It's a precision feature. It happens that it is usually easier to prove that an algorithm will take, say, O(n) operations to complete than proving that it will take at least O(n) operations (BTW, in this context operation means elementary computations such as the logical and arithmetic ones.)
By providing a lower bound, you are also giving an estimate of the best case scenario, as the big-O notation only provides an upper bound.
From a practical viewpoint, this has the benefit of telling that no matter what, any algorithm will require so many (elementary) steps (or more).
Note also, that it is also useful to have estimates of the average, the worst and the best cases, because these will shed more light on the complexity of the algorithm.
There are problems whose inherent complexity is known to be at least of some order (meaning there is a mathematical theorem proving the fact). So, no matter the algorithm, these problems cannot be solved with less that a certain number of calculations. This is also useful because lets you know whether a given algorithm is sub-optimal or matches the inherent complexity of the problem.
Related
I understand that Big Omega defines the lower bound of s function (or best-case runtime).
Considering that almost every search algorithm could "luck out" and find the target element on the first iteration, would it be fair to say that its Big-Omega time complexity is O(1)?
I also understand that defining O(1) as the big Omega may not be useful -other lower bounds may be tighter, or closer to the evaluated function-, but the question is, is it correct?
I've found multiple sources claiming the linear search is Big-Omega O(n), even if some cases could complete in a single step, which is different from the best-case scenario as I understand it.
The lower bound (𝛺) is not the fastest answer a given algorithm can give.
The lower bound of a given problem is equal to the worst case scenario of the best algorithm that solves the problem. When doing complexity analysis, you should never forget that "luck" is always in the hands of the input (the instance the algorithm is trying to solve).
When trying to find a lower bound, you will imagine the "perfect algorithm" and you will try to "trap" it in a very hard case. Usually the algorithm is not defined and is only described based on its (hypotetical) performances. You would use arguments such as "If the ideal algorithm is that fast, it will not have this particular knowledge and will therefore fail on this particular instance, ie. the ideal algorithm doesn't exist". Replace ideal with the lower bound you are trying to prove.
For example, if we search the lower bound for the min-search problem in an unsorted array is 𝛺(n). The proof for this is quite trivial, and like most of the time, is made by contradiction. Basically, an algorithm A in o(n) will not see at least one item from the input array, if that item it did not saw was the minimum, A will fail. The contradiction proves that the problem is in 𝛺(n).
Maybe you can have a look at that answer I gave on a similar question.
The notations O, o, Θ, Ω, and ω are used in characterizing mathematical functions; for example, f(n) = n3 log n is in O(n4) and in Ω(n3).
So, the question is what mathematical functions we apply them to.
The mathematical functions that we tend to be interested in are things like "the worst-case time complexity of such-and-such algorithm, as a function of the size of its input", or "the average-case space complexity of such-and-such procedure, as a function of the largest element in its input". (Note: when we just say "the complexity of such-and-such algorithm", that's usually shorthand for its worst-case time complexity, as a function of some characteristic of its input that's hopefully obvious in context. Either way, it's still a mathematical function.)
We can use any of these notations in characterizing those functions. In particular, it's fine to use Ω in characterizing the worst case or average case.
We can also use any of these functions in characterizing things like "the best-case […]" — that's unusual, but there are times when it may be relevant. But, notably, we're not limited to Ω for that; just as we can use Ω in characterizing the worst case, we can also use O in characterizing the best case. It's all about what characterizations we're interested in.
You are confusing two different topics: Lower/upper bound, and worst-case/best-case time complexity.
The short answer to your question is: Yes, all search algorithms have a lower bound of Ω(1). Linear search (in the worst case, and on average) also has a lower bound of Ω(n), which is a stronger and more useful claim. The analogy is that 1 < π but also 3 < π, the latter being more useful. So in this sense, you are right.
However, your confusion seems to be between the notations for complexity classes (big-O, big-Ω, big-θ etc), and the concepts of best-case, worst-case, average case. The point is that the best case and the worst case time complexities of an algorithm are completely different functions, and you can use any of the notations above to describe any of them. (NB: Some claim that big-Ω automatically and exclusively describes best case time complexity and that big-O describes worst case, but this is a common misconception. They just describe complexity classes and you can use them with any mathematical functions.)
It is correct to say that the average time complexity linear search is Ω(n), because we are just talking about the function that describes its average time complexity. Its worst-case complexity is a different function, which happens not to be Ω(n), because as you say it can be constant-time.
https://www.freecodecamp.org/news/big-o-notation-why-it-matters-and-why-it-doesnt-1674cfa8a23c/
Exponentials have greater complexity than polynomials as long as the coefficients are positive multiples of n
O(2ⁿ) is more complex than O(n⁹⁹), but O(2ⁿ) is actually less complex
than O(1). We generally take 2 as base for exponentials and logarithms
because things tends to be binary in Computer Science, but exponents
can be changed by changing the coefficients. If not specified, the
base for logarithms is assumed to be 2.
I thought O(1) is the simplest in complexity. Could anyone help me explain why O(2ⁿ) is less complex than O(1) ?
Errata. The author made an obvious mistake and you caught it. It's not the only mistake in the article. For example, I would expect O(n*log(n)) to be the more appropriate complexity for sorting algorithms than the one they claim (quoted below). Otherwise, you'd be able to sort a set without even seeing all of the data.
"As complexity is often related to divide and conquer algorithms, O(log(n)) is generally a good complexity you can reach for sorting algorithms."
It might be worthwhile to try to contact the author and give him a heads up so he can correct it and avoid confusing anyone else with misinformation.
I was asked this question in an interview recently and was curious as to what others thought.
"When should you calculate Big O?"
Most sites/books talk about HOW to calc Big O but not actually when you should do it. I'm an entry level developer and I have minimal experience so I'm not sure if I'm thinking on the right track. My thinking is you would have a target Big O to work towards, develop the algorithm then calculate the Big O. Then try to refactor the algorithm for efficiency.
My question then becomes is this what actually happens in industry or am I far off?
"When should you calculate Big O?"
When you care about the Time Complexity of the algorithm.
When do I care?
When you need to make your algorithm to be able to scale, meaning that it's expected to have big datasets as input to your algorithm (e.g. number of points and number of dimensions in a nearest neighbor algorithm).
Most notably, when you want to compare algorithms!
You are asked to do a task, for which several algorithms can be applied to. Which one do you choose? You compare the Space, Time and development/maintenance complexities of them, and choose the one that best fits your needs.
Big O or asymptotic notations allow us to analyze an algorithm's running time by identifying its behavior as the input size for the algorithm increases.
So whenever you need to analyse your algorithm's behavior with respect to growth of the input, you will calculate this. Let me give you an example -
Suppose you need to query over 1 billion data. So you wrote a linear search algorithm. So is it okay? How would you know? You will calculate Big-o. It's O(n) for linear search. So in worst case it would execute 1 billion instruction to query. If your machine executes 10^7 instruction per second(let's assume), then it would take 100 seconds. So you see - you are getting an runtime analysis in terms of growth of the input.
When we are solving an algorithmic problem we want to test the algorithm irrespective of hardware where we are running the algorithm. So we have certain asymptotic notation using which we can define the time and space complexities of our algorithm.
Theta-Notation: Used for defining average case complexity as it bounds the function from top and bottom
Omega-Notation: Bounds the function from below. It is used for best-time complexity
Big-O Notation: This is important as it tells about worst-case complexity and it bounds the function from top.
Now I think the answer to Why BIG-O is calculated is that using it we can get a fair idea that how bad our algorithm can perform as the size of input increases. And If we can optimize our algorithm for worst case then average and best case will take care for themselves.
I assume that you want to ask "when should I calculate time complexity?", just to avoid technicalities about Theta, Omega and Big-O.
Right attitude is to guess it almost always. Notable exceptions include piece of code you want to run just once and you are sure that it will never receive bigger input.
The emphasis on guess is because it does not matter that much whether complexity is constant or logarithmic. There is also a little difference between O(n^2) and O(n^2 log n) or between O(n^3) and O(n^4). But there is a big difference between constant and linear.
The main goal of the guess, is the answer to the question: "What happens if I get 10 times larger input?". If complexity is constant, nothing happens (in theory at least). If complexity is linear, you will get 10 times larger running time. If complexity is quadratic or bigger, you start to have problems.
Secondary goal of the guess is the answer to question: 'What is the biggest input I can handle?". Again quadratic will get you up to 10000 at most. O(2^n) ends around 25.
This might sound scary and time consuming, but in practice, getting time complexity of the code is rather trivial, since most of the things are either constant, logarithmic or linear.
It represents the upper bound.
Big-oh is the most useful because represents the worst-case behavior. So, it guarantees that the program will terminate within a certain time period, it may stop earlier, but never later.
It gives the worst time complexity or maximum time require to execute the algorithm
I understand O(NlgN) is linearithmic. But what is O(N^m(lgN))? Would is just be considered polynomial running time since the polynomial part grows faster?
If you know that the asymptotic behaviour of a function or algorithm can be described by O(n^m log N), you should probably stick with just that. However, you could naturally say that one upper bound on the time complexity of that same function/algorithm is one of polynomial time, i.e.:
This is acceptable as the Big-O upper bound on the asymptotic behaviour mustn't necessarily be a tight one.
Now, let's say you have some algorithm and have found an upper asymptotic bound on it as O(N^(m+1)), but know that you used quite coarse tools when deriving this bound; i.e., possibly there exists tighter asymptotic bounds. However, before proceeding on a crusade in calculus and analysis, you could ask yourself: is this bound good enough for my purposes? (E.g. making sure the algorithm doesn't run in exponential time). If so, just use the so-so but acceptable bound you've derived.
In case you've already derived a tighter better bound, however, it is probably most favourable to stick with that bound when presenting the asymptotic behaviour of your function or algorithm.
What is the use of Big-O notation in computer science if it doesn't give all the information needed?
For example, if one algorithm runs at 1000n and one at n, it is true that they are both O(n). But I still may make a foolish choice based on this information, since one algorithm takes 1000 times as long as the other for any given input.
I still need to know all the parts of the equation, including the constant, to make an informed choice, so what is the importance of this "intermediate" comparison? I end up loosing important information when it gets reduced to this form, and what do I gain?
What does that constant factor represent? You can't say with certainty, for example, that an algorithm that is O(1000n) will be slower than an algorithm that's O(5n). It might be that the 1000n algorithm loads all data into memory and makes 1,000 passes over that data, and the 5n algorithm makes five passes over a file that's stored on a slow I/O device. The 1000n algorithm will run faster even though its "constant" is much larger.
In addition, some computers perform some operations more quickly than other computers do. It's quite common, given two O(n) algorithms (call them A and B), for A to execute faster on one computer and B to execute faster on the other computer. Or two different implementations of the same algorithm can have widely varying runtimes on the same computer.
Asymptotic analysis, as others have said, gives you an indication of how an algorithm's running time varies with the size of the input. It's useful for giving you a good starting place in algorithm selection. Quick reference will tell you that a particular algorithm is O(n) or O(n log n) or whatever, but it's very easy to find more detailed information on most common algorithms. Still, that more detailed analysis will only give you a constant number without saying how that number relates to real running time.
In the end, the only way you can determine which algorithm is right for you is to study it yourself and then test it against your expected data.
In short, I think you're expecting too much from asymptotic analysis. It's a useful "first line" filter. But when you get beyond that you have to look for more information.
As you correctly noted, it does not give you information on the exact running time of an algorithm. It is mainly used to indicate the complexity of an algorithm, to indicate if it is linear in the input size, quadratic, exponential, etc. This is important when choosing between algorithms if you know that your input size is large, since even a 1000n algorithm well beat a 1.23 exp(n) algorithm for large enough n.
In real world algorithms, the hidden 'scaling factor' is of course important. It is therefore not uncommon to use an algorithm with a 'worse' complexity if it has a lower scaling factor. Many practical implementations of sorting algorithms are for example 'hybrid' and will resort to some 'bad' algorithm like insertion sort (which is O(n^2) but very simple to implement) for n < 10, while changing to quicksort (which is O(n log(n)) but more complex) for n >= 10.
Big-O tells you how the runtime or memory consumption of a process changes as the size of its input changes. O(n) and O(1000n) are both still O(n) -- if you double the size of the input, then for all practical purposes the runtime doubles too.
Now, we can have an O(n) algorithm and an O(n2) algorithm where the coefficient of n is 1000000 and the coefficient of n2 is 1, in which case the O(n2) algorithm would outperform the O(n) for smaller n values. This doesn't change the fact, however, that the second algorithm's runtime grows more rapidly than the first's, and this is the information that big-O tells us. There will be some input size at which the O(n) algorithm begins to outperform the O(n2) algorithm.
In addition to the hidden impact of the constant term, complexity notation also only considers the worst case instance of a problem.
Case in point, the simplex method (linear programming) has exponential complexity for all known implementations. However, the simplex method works much faster in practice than the provably polynomial-time interior point methods.
Complexity notation has much value for theoretical problem classification. If you want some more information on practical consequences check out "Smoothed Analysis" by Spielman: http://www.cs.yale.edu/homes/spielman
This is what you are looking for.
It's main purpose is for rough comparisons of logic. The difference of O(n) and O(1000n) is large for n ~ 1000 (n roughly equal to 1000) and n < 1000, but when you compare it to values where n >> 1000 (n much larger than 1000) the difference is miniscule.
You are right in saying they both scale linearly and knowing the coefficient helps in a detailed analysis but generally in computing the difference between linear (O(cn)) and exponential (O(cn^x)) performance is more important to note than the difference between two linear times. There is a larger value in the comparisons of runtime of higher orders such as and Where the performance difference scales exponentially.
The overall purpose of Big O notation is to give a sense of relative performance time in order to compare and further optimize algorithms.
You're right that it doesn't give you all information, but there's no single metric in any field that does that.
Big-O notation tells you how quickly the performance gets worse, as your dataset gets larger. In other words, it describes the type of performance curve, but not the absolute performance.
Generally, Big-O notation is useful to express an algorithm's scaling performance as it falls into one of three basic categories:
Linear
Logarithmic (or "linearithmic")
Exponential
It is possible to do deep analysis of an algorithm for very accurate performance measurements, but it is time consuming and not really necessary to get a broad indication of performance.