How Can Binary Vectors Be Ordered Through Transformation?

[baxy]

Hi,

This is my first post and I must worn you that I'm not a mathematician. I am just a regular blonk looking for help concerning the following problem.

What i have is a set (T) of binary vectors:A: 100110
B: 000110
C: 010110now what i am trying to figure out is there a transformation, a function to transform given vectors in such a way that

A>B>C

where a relation > on T is given by the fact that to convert

B->A takes 1 step
C->A takes 2 steps
C->B takes 1 step

Can anyone suggest a book or a paper where such things are discussed , or give a few pointers?

Thank you,

baxy

PS

and the relation can ba symmetrical

A<B<C

as long as the order is preserved

 

[chiro]

Hey baxy and welcome to the forums.

I'm not exactly sure what you are trying to do, but it seems that you're given a set T = {T1,T2,T3} and you want to find a transformation that takes T1,T2,T3 in which f(T1) > f(T2) > f(T3) or f(T1) < f(T2) < f(T3).

One easy transformation that comes to mind if we assume that all entries of the set are non-zero is to use a transformation that maps an input vector to an output vector. In other words, using your set with three elements define F to be:

F(T1,T2,T3) = [T1+T2+T3,T2+T3,T3] where the output is a vector with three elements.

Since this is a linear transformation, you can create an appropriate matrix that represents this transformation and define that as your operator.

Is this what you had in mind?

Also I didn't take into account the binary nature of your number, so you would have to compensate for this in your computation.

 

[baxy]

ok, but i didn't quite understand how can i from this output vector figure out that T1>T2>T3 ? so what i need t do is transform my vectors into something and then just by sorting the transforms, figure out that T1_{A} needs 1 transformation step to be converted to T2_{B} and 2 transformation step to be converted to T3_{C}.

This is my problem.

 

[chiro]

ok, but i didn't quite understand how can i from this output vector figure out that T1>T2>T3 ? so what i need t do is transform my vectors into something and then just by sorting the transforms, figure out that T1_{A} needs 1 transformation step to be converted to T2_{B} and 2 transformation step to be converted to T3_{C}.

This is my problem.

Ohh I see what you mean.

Well to sort two elements the first one is min(T1,T2) and the last is max(T1,T2). Can you adapt this for three elements?