birkhoff | Birkhoff decomposition for doubly stochastic matrices

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Birkhoff decomposition for doubly stochastic matrices.

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birkhoff Examples and Code Snippets

Birkhoff's algorithm for doubly stochastic matrices,Basic usage

Python

Lines of Code : 15

License : Strong Copyleft (GPL-3.0)

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import numpy
from birkhoff import birkhoff_von_neumann_decomposition

# Create a doubly stochastic matrix.
#
# D = numpy.array(...)

# The decomposition is given as a list of pairs in which the right element
# is a permutation matrix and the left ele

Birkhoff's algorithm for doubly stochastic matrices,Testing

Python

Lines of Code : 2

License : Strong Copyleft (GPL-3.0)

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pip install -r requirements-test.txt
nosetests

Birkhoff's algorithm for doubly stochastic matrices,Installation requirements

Python

Lines of Code : 1

License : Strong Copyleft (GPL-3.0)

Copy

pip install -r requirements.txt

Community Discussions

Trending Discussions on Math

Checking if two strings are equal after removing a subset of characters from both

Special Number Count

How do I calculate square root in Python?

Why does this numeric equation using a cosine produce a different result between a console application and windows application?

How to write a portable constexpr std::copysign()?

Linearize nested for loops

Create a sequence of sequences of numbers

split geometric progression efficiently in Python (Pythonic way)

How to create Polynomial Ring which has Float coefficients Julia

Convert GPS Coordinates to Match Custom 2d outdoor layout Image

QUESTION

Checking if two strings are equal after removing a subset of characters from both

Asked 2022-Mar-29 at 22:42

I recently came across this problem:

You are given two strings, s1 and s2, comprised entirely of lowercase letters 'a' through 'r', and need to process a series of queries. Each query provides a subset of lowercase English letters from 'a' through 'r'. For each query, determine whether s1 and s2, when restricted only to the letters in the query, are equal. s1 and s2 can contain up to 10^5 characters, and there are up to 10^5 queries.

For instance, if s1 is "aabcd" and s2 is "caabd", and you are asked to process a query with the subset "ac", then s1 becomes "aac" while s2 becomes "caa". These don't match, so the query would return false.

I was able to solve this in O(N^2) time by doing the following: For each query, I checked if s1 and s2 would be equal by iterating through both strings, one character at a time, skipping the characters that do not lie within the subset of allowed characters, and checking to see if the "allowed" characters from both s1 and s2 match. If at some point, the characters don't match, then the strings are not equal. Otherwise, the s1 and s2 are equal when restricted only to letters in the query. Each query takes O(N) time to process, and there are N queries, for a total of O(N^2) time.

However, I was told that there was a way to solve this faster in O(N). Does anyone know how this might be done?

...

ANSWER

Answered 2022-Mar-28 at 11:30

The first obvious speedup is to ensure your set membership test is O(1). To do that, there's a couple of options:

Represent every letter as a single bit -- now every character is an 18-bit value with only one bit set. The set of allowed characters is now a mask with these bits ORed together and you can test membership of a character with a bitwise-AND;
Alternatively, you can have an 18-value array and index it by character (c - 'a' would give a value between 0 and 17). The test for membership is then basically the cost of an array lookup (and you can save operations by not doing the subtraction -- instead just make the array larger and index directly by character.

Thought experiment

The next potential speedup is to recognize that any character which does not appear exactly the same number of times in both strings will instantly be a failed match. You can count all character frequencies in both strings with a histogram which can be done in O(N) time. In this way, you can prune the search space if such a character were to appear in the query, and you can test for this in constant time.

Of course, that won't help for a real stress-test which will guarantee that all possible letters have a frequency matched in both strings. So, what do you do then?

Well, you extend the above premise by recognizing that for any position of character x in string 1 and some position of that character in string 2 that would be a valid match (i.e the same number of character x appears in both strings up to their respective positions), then the total count of any other character up to those positions must also be equal. For any character where that is not true, it cannot possibly be compatible with character x.

Concept

Let's start by thinking about this in terms of a technique known as memoization where you can leverage precomputed or partially-computed information and get a whole lot out of it. So consider two strings like this:

Source https://stackoverflow.com/questions/71642925

QUESTION

Special Number Count

Asked 2022-Mar-09 at 04:56

It is a number whose gcd of (sum of quartic power of its digits, the product of its digits) is more than 1. eg. 123 is a special number because hcf of(1+16+81, 6) is more than 1.

I have to find the count of all these numbers that are below input n. eg. for n=120 their are 57 special numbers between (1 and 120)

I have done a code but its very slow can you please tell me to do it in some good and fast way. Is there is any way to do it using some maths.

...

ANSWER

Answered 2022-Mar-06 at 18:14

The critical observation is that the decimal representations of special numbers constitute a regular language. Below is a finite-state recognizer in Python. Essentially we track the prime factors of the product (gcd > 1 being equivalent to having a prime factor in common) and the residue of the sum of powers mod 2×3×5×7, as well as a little bit of state to handle edge cases involving zeros.

From there, we can construct an explicit automaton and then count the number of accepting strings whose value is less than n using dynamic programming.

Source https://stackoverflow.com/questions/71370656

QUESTION

How do I calculate square root in Python?

Asked 2022-Feb-17 at 03:40

I need to calculate the square root of some numbers, for example √9 = 3 and √2 = 1.4142. How can I do it in Python?

The inputs will probably be all positive integers, and relatively small (say less than a billion), but just in case they're not, is there anything that might break?

Related

_{Note: This is an attempt at a canonical question after a discussion on Meta about an existing question with the same title.}

...

ANSWER

Answered 2022-Feb-04 at 19:44

Option 1: math.sqrt()

The math module from the standard library has a sqrt function to calculate the square root of a number. It takes any type that can be converted to float (which includes int) as an argument and returns a float.

Source https://stackoverflow.com/questions/70793490

QUESTION

Why does this numeric equation using a cosine produce a different result between a console application and windows application?

Asked 2022-Feb-12 at 13:17

I write a mathematical function to be benchmark function in my optimization algorithm.

...

ANSWER

Answered 2022-Feb-12 at 13:14

In the platform that produces “-4,09139395927863E+154”, the Math.Cos routine is broken. It apparently uses a processor instruction that does not support operands outside [−2⁻⁶³, +2⁻⁶³].

Since I do not use C#, here is a C program that reproduces the correct behavior:

Source https://stackoverflow.com/questions/71090136

QUESTION

How to write a portable constexpr std::copysign()?

Asked 2022-Feb-08 at 10:13

In particular, it must work with NaNs as std::copysign does. Similarly, I need a constexpr std::signbit.

...

ANSWER

Answered 2021-Sep-20 at 19:54

If you can use std::bit_cast, you can manipulate floating point types cast to integer types. The portability is limited to the representation of double, but if you can assume the IEEE 754 double-precision binary floating-point format, cast to uint64_t and using sign bit should work.

Source https://stackoverflow.com/questions/69259995

QUESTION

Linearize nested for loops

Asked 2022-Feb-01 at 08:27

I'm working on some heavy algorithm, and now I'm trying to make it multithreaded. It has a loop with 2 nested loops:

...

ANSWER

Answered 2021-Dec-20 at 09:25

A third attempt:

I've taken your code, and at last got it to run properly (in python):

Source https://stackoverflow.com/questions/70413446

QUESTION

Create a sequence of sequences of numbers

Asked 2022-Jan-27 at 22:54

I would like to make the following sequence in R, by using rep or any other function.

...

ANSWER

Answered 2022-Jan-04 at 15:43

Use sequence.

Source https://stackoverflow.com/questions/70579403

QUESTION

split geometric progression efficiently in Python (Pythonic way)

Asked 2022-Jan-22 at 10:09

I am trying to achieve a calculation involving geometric progression (split). Is there any effective/efficient way of doing it. The data set has millions of rows. I need the column "Traded_quantity"

Marker Action Traded_quantity 2019-11-05 09:25 0 0 09:35 2 BUY 3 09:45 0 0 09:55 1 BUY 4 10:05 0 0 10:15 3 BUY 56 10:24 6 BUY 8128

turtle = 2 (User defined)

base_quantity = 1 (User defined)

...

ANSWER

Answered 2022-Jan-22 at 10:09

This should work

Source https://stackoverflow.com/questions/70810857

QUESTION

How to create Polynomial Ring which has Float coefficients Julia

Asked 2022-Jan-18 at 23:30

I want to create a polynomial ring which has float Coefficients like this. I can create with integers but, Floats does not work.

...

ANSWER

Answered 2022-Jan-18 at 23:30

While I do not have previous experience with this particular (from appearances, rather sophisticated) package Oscar.jl, parsing this error message tells me that the function you are trying to call is being given a BigFloat as input, but simply does not have a method for that type.

At first this was a bit surprising given that there are no BigFloats in your input, but after a bit of investigation, it appears that the culprit is the following

Source https://stackoverflow.com/questions/70763117

QUESTION

Convert GPS Coordinates to Match Custom 2d outdoor layout Image

Asked 2022-Jan-17 at 04:19

I don't know if this is possible, but I am trying to take the image of a custom outdoor football field layout and have the players' GPS coordinates correspond to the image xand y position. This way, it can be viewed via the app to show the players' current location on the field as a sort of live tracking.

I have also looked into this Convert GPS coordinates to coordinate plane. The problem is that I don't know if this would work and wanted to confirm beforehand. The image provided in the post was for indoor location, and it was from 11 years ago.

I used Location and Google Maps packages for flutter. The player's latitude and longitude correspond to the actual latitude and longitude that the simulator in the android studio shows when tested.

The layout in question and a close comparison to the result I am looking for.

Any help on this matter would be appreciated highly, and thanks in advance for all the help.

Edit:

After looking more at the matter I tried the answer of this post GPS Conversion - pixel coords to GPS coords, but it wasn't working as intended. I took some points on the image and the correspond coordinates, and followed the same logic that the answer used, but reversed it to give me the actual image X, Ypositions.

The formula that was given in the post above:

...

ANSWER

Answered 2022-Jan-12 at 08:20

First of All, Yes you can do this with high accuracy if the GPS coordinates are accurate.

Second, the main problem is rotation if the field are straight with lat lng lines this would be easy and straightforward (no bun intended).

The easy way is to convert coordinate to rotated image similar to the real field then rotated every X,Y point to the new straight image. (see the image below)

Here is how to rotate x,y knowing the angel:

Source https://stackoverflow.com/questions/70603285

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