python - Linear time v.s. Quadratic time - Stack Overflow
So what the hell is O(x) Time?
Is this an example of Quadratic Time Complexity when it comes to type of big O notation
optimization - Time complexity of quadratic programming - Mathematics Stack Exchange
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A method is linear when the time it takes increases linearly with the number of elements involved. For example, a for loop which prints the elements of an array is roughly linear:
for x in range(10):
print x
because if we print range(100) instead of range(10), the time it will take to run it is 10 times longer. You will see very often that written as O(N), meaning that the time or computational effort to run the algorithm is proportional to N.
Now, let's say we want to print the elements of two for loops:
for x in range(10):
for y in range(10):
print x, y
For every x, I go 10 times looping y. For this reason, the whole thing goes through 10x10=100 prints (you can see them just by running the code). If instead of using 10, I use 100, now the method will do 100x100=10000. In other words, the method goes as O(N*N) or O(N²), because every time you increase the number of elements, the computation effort or time will increase as the square of the number of points.
They must be referring to run-time complexity also known as Big O notation. This is an extremely large topic to tackle. I would start with the article on wikipedia: https://en.wikipedia.org/wiki/Big_O_notation
When I was researching this topic one of the things I learned to do is graph the runtime of my algorithm with different size sets of data. When you graph the results you will notice that the line or curve can be classified into one of several orders of growth.
Understanding how to classify the runtime complexity of an algorithm will give you a framework to understanding how your algorithm will scale in terms of time or memory. It will give you the power to compare and classify algorithms loosely with each other.
I'm no expert but this helped me get started down the rabbit hole.
Here are some typical orders of growth:
- O(1) - constant time
- O(log n) - logarithmic
- O(n) - linear time
- O(n^2) - quadratic
- O(2^n) - exponential
- O(n!) - factorial
If the wikipedia article is difficult to swallow, I highly recommend watching some lectures on the subject on iTunes University and looking into the topics of algorithm analysis, big-O notation, data structures and even operation counting.
Good luck!
I have been learning programming in my own time for years now and I'm coming up on this topic when I've gone back to school. I just can't seem to understand what it asks of me. Can anyone clarify it? I'm a very visual learner, things like a stack, queues, dequeues, etc come easily, but this just slips out of my mind.
Below is the output generated by AI while exploring big O notation:
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O(n2) – Quadratic Time Complexity Definition: The runtime increases quadratically as the input size grows. Doubling the input quadruples the runtime. Characteristics: Typically slower and less efficient, not suitable for large inputs. Examples: Nested loops for comparing each customer to every other customer. Business Case: Small-scale market basket analysis for cross-selling products.
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My query is if it is of the same type as discussed in this MITx Online Differential Calculus course except x3 replaced by n2.
https://www.reddit.com/r/MathHelp/s/XivJaFxhkz
Time complexity of Quadratic Programming.
It was proved by Vavasis at 1991 that the general quadratic program is NP-hard, i.e. it takes more than polynomial time to be solved "exactly" (in reality, its impossible to find an exact solution due to the finite precision arithmetic of the computer). If your QP is convex, there are polynomial time interior point algorithms (e.g. Ye and Tse at 1989 published an extension of Karmarkar's projective algorithm on convex quadratic programs). Also, there are approximation algorithms that return local solutions of nonconvex QPs in polynomial running time (e.g. Ye, 1998).
About quadprog's inner implementation and time-complexity.
Quadprog runs an active-set method (of exponential time-complexity for the worst case input instances) for a general QP. For a really hard QP (indefinite, with a near badly scaled Q matrix) , it may not converge to a solution. In the case that your input's QP is convex, quadprog runs an interior-point method.
I didn't ever check quadprog's time complexity in practice, but you can't lead to an inference by just observing the running times as you scale the dimension of a given QP. It's better to generate a test set, consisting of convex and not-convex QP, manually choose the algorithm you want to be runned by quadprog for each of them, and observe the running times.
I am not familiar with the details of the quadprog function, but I think the issues may be more universal. The cubic time complexity is an asymptotic worst-case bound. It does not mean that growing any problem by 10 times will increase running time by 1000 times. It will often be less, but may be that bad for some problem data.
There is often constant time "overhead" for many algorithms. The overhead may be so large that the running time is essentially independent of the problem size up to a certain point. The asymptotic bound only becomes relevant when comparing large problems with very large problems. In your example this will be hard to observe because quadratic programs with thousands or tens of thousands of variables can take hours or days to solve on a typical personal computer. Memory can become an issue since the problem data alone grows quadratically in the number of variables.
There is a whole industry devoted to these issues. Commercial solvers can cost tens of thousands of dollars. For specific applications, specialized solvers that exploit problem structure are often utilized. Sparsity of the input matrices is the most basic source of running time improvements.
Asymptotic worst-case bounds are often a poor guide for practical problem solving. They are certainly hard to verify using simulations. I suggest you move over to stackexchange for messy, detailed discussions of particular software implementations of algorithms.