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I make a program in C++, that factorizes natural numbers. The only problem is to create a function that does the following:
input: it receives a matrix vector< vector< int> > M.
output: it gives a vector v so that a result of multiplying v and M is a vector that all its coordinates are equal to 0.
Everything must be modulo 2, so coefficients of M and v consists only 0s and 1s
Schould I use a Gauss elimination method? If so, how do this? The problem that implementations I saw on the net don't use vectors and the vectors are necessary in my main program
I would be grateful if someone helped me.
Regards
This is an interesting problem. The steps to be taken can be found in this exercise. link.
The tricky part is to understand that there are always only a finite number of solutions, i.e, only the trivial solution exists or non-trivial solutions exist.
Once you finish the row reduction steps and if non-trivial solutions exist, there is always going to be at least one independent variable (it can take any value 0/1) and the rest of the variables depend on the independent variables.
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Guys I what is the fastest algorithm to solve a modular equation in the format of a^b = c mod p where p is a really big prime and b is unknown.
e.g.:
2^k = 15 mod 30903154482632612361920641803533
I already tried trial and error using boost library in C++ but it would take very long time to reach the answer.
You're trying to solve what is called a discrete logarithm. If there was an efficient solution to this, I imagine whoever discovered it would wreak chaos on cryptographic systems long before it would be posted here.
You will find quite a couple of algorithms on Wikipedia with varying time complexity. Some of these are quite easy to implement. See The computational complexity of discrete log for the best space complexity.
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I just wanted to know if the above question is trivial or not. More importantly, how can you recognize if an algorithm is trivial?
It depends what do you mean by trivial. If you talk about complexity, it is O(n*M(N)) where M(N) is the complexity of the underlying multiplication algorithm with N maximum of the array's values and n is the length of the array.
If you talk about implementation, it is one loop with one check that the sum of the neighbors is a perfect square. If the elements fit into int, double etc. you have sqrt function in the standard library. If your elements are arbitrary length integers or float point numbers, you either need to use an appropriate library or implement the handling of these numbers on your own, which might be not trivial.
This understanding should help you to answer your last question
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Some days ago I wanted to use C++ sort() function to sort an array of strings, but I had a problem!
What algorithm does it use to sort the array? Is it a deterministic one or may it use different algorithms based on the type of the array?
Also, is there a clear time complexity analysis about it?
Does this function use the same algorithm for sorting numbers array and strings array?
It might or it might not. That is not specified by the standard.
And if we use it to sort an array of strings which the total size of them is less than 100,000 characters, would it work in less than 1 second(in the worst case)?
It might or it might not. It depends on the machine you're running the program on. Even if it will work in less than 1 second in worst case on a particular machine, it would be difficult to prove. But you can get a decent estimation by measuring. A measurement only applies to the machine it was performed, of course.
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This is a part programming, part statistical math question.
I have a dataset where I want to get the most frequent number (mode), the problem is that I am dealing with values with slight variation.
So normally {1,2,50,50,90} the most frequent number would be 50
But in my case the numbers look like this:
{1,2,49,50,51,90} but the result is still 50
So my question is how can I efficiently calculate this number and is there a statistical term for this number?
Some pseudo code:
Float items.val[] = {1,2,49,50,51,90};
Float threshold = 4;
For (item in items) {
For (subitem in items){
Float dist=Distance(time,subitem)
If (dist < threshold){
item.dist += dist
}
}
}
Output=Sort(item.dist)[0]
There are various ways to go about this.
(1) the most careful, exact way is to assume a probabilistic model for the observed values, and look for the mode (as the expected value or most probable or some other criterion) of the inferred values. I am going to guess this is far too much work in this case, although given unlimited time I would certainly want to approach it that way.
(2) construct a histogram, and look for the bin which has the greatest density (with density = (#items in bin)/(width of bin)). This doesn't necessarily yield a single value.
(3) fit a parametric distribution to the observed values, and report the mode of the fitted distribution.
You might get more traction for this question at stats.stackexchange.com. Good luck and have fun.
EDIT: After looking at your example code, I see it is not too different from (2) above. It seems like a reasonable and workable approach.
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I'm developing an estimation algorithm in C++ and performance is key. Basically there is loop where in each iteration, a decision is made on whether to add a column vector to a matrix or to remove one.
I have implemented my own matrix and vector classes and used Intel MKL for matrix operations. However after the first version I'm now looking into using Armadillo.
I would like to know what the best strategy is for dynamic growing matrices inside loops. I know the maximum size of the matrix, so I could preallocate.
First of all, is there another matrix library you would recommend other than Armadillo for small matrices (50 X 50)?
Secondly, what would be the best way to tackle this problem using Armadillo?