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Topic: optimal sorting
Replies: 3   Last Post: Dec 11, 2012 11:59 PM

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Sterten

Posts: 65
Registered: 12/13/04
Re: optimal sorting
Posted: Dec 11, 2012 11:59 PM
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> given a (symmetrical) finite real square matrix d(n,n)
> find a permutation s in S_n such that
>
> SUM[i=2..n] ( min{d(s(i),s(j))|0<j<i} ) is minimal



think of d(a,b) as the distance of 2 objects, say stars in the multidimensional universe.
Then you want to arrange the stars in a 1-dim-list, when you add a new one to the list
it should be close to a previous one.
What would be the best ordering of the objects so new ones are not so far away from
all the already included ones ?

Or think of objects as (similar) binary numbers, d(a,b) is the number of digits where they differ,
the number of ones in the xor of the numbers.
A compression algorithm lists a new number not by its digits, but rather by all its differences to a
previous number.

One good algorithm is to start with 2 objects of minimal distance. Choose one of them.
Then add new numbers, by the minimal distance to the existing set.
Where distance to a set is the minimum of the individual distances.

Is this always the optimal solution ?
How is the corresponding compression algorithm called ?

I want to apply it to genetical sequences which are very similar with only
occasional mutations.



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