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Sterten
Posts:
65
Registered:
12/13/04


Re: optimal sorting
Posted:
Dec 11, 2012 11:59 PM


> 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 1dimlist, 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.



