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say, i have got 3x'a', 5x'b', 2x'c',4x'd', as char collection.

2 strings are formed, each consists of all the chars given. eg. string A = 'aaabbbbbccdddd' B='abcdabcdabbbdd'

so both strings have length 14

now, we can randomly choose 3 consecutive chars from each string. say, I choose X ='aaa' from A, and Y = 'abc' from B.

define a function good(), which compares the two selected string segments. original flag = false. if the first char from X = the second char from Y then flag = true. if the second char from X = the third char from Y then flag = true. if the second char from X = the second char from Y then flag = true (yes it is not symmetric). good() returns 'good' only if the flag remains false after all the checks.

I know i can write a computer program to figure out the 'good' probability given A and B. But now I would like to find out the max and min of the good probability of different A and B from the char collection. Iterate all possible permutation would be physically infeasible. so I seek an approximate approach.

Any thoughts?

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1 Answer 1

All three comparisons involve one of the two second characters. Only segments that differ at the second character can be good, and we want to minimize/maximize the chance that if they do, the second character also differs from the neighbouring characters on the other segment. That means we should try to minimize/maximize the number of repetitions. Also, the last character in A and the first character in B never get matched; you should put rare/frequent characters there to minimize/maximize the chances of the remaining ones differing from others. I suspect that any strings that obey these two constraints (minimal/maximal number of repetitions, most rare/most frequent characters in the two special slots) will have the same minimal/maximal chance of producing good pairs, or at least to a good approximation.

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