Hello monks

I am turning to you after a long quest. I have been searching for the answer to a problem that does not allow me to sleep, eat or think. In past few months I have been struggling with a problem that has now turned to my obsession. Although many have tried to convince me that this is an NP problem ergo, has no practical solution, I have this gut feeling that if i somehow break the problem into small fragments i'll be able to solve it.

Now before i discourage you from even attempting to read the post i'll explain the problem

I am given a long string over the 3 character alphabet.

S= acccabacbbacabcabcacabc....
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This string is usually 5-10 bilion characters long. What I am trying to figure out is how long is the shortest unique substring of that string on some position S(x) given that i allow k number of mismatches.
So let say that I am looking at the position S(10)

S[10..13] = baca
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now this is the shortest unique substring on S(10), however the shortest unique substring on the same position allowing a single mismatch is :

S[10..16] = bacabca




S[10..16] = bacabca
                  x
S[19..25] = cacabc.
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The way you get those results traditionally is by choping your initial string into n-mers and comparing them to every position in your string. and you repeat that by increasing the n-mer length until the only hit left is a self hit. Then you know that you found your answer

My approach was based on "divide and conquer" strategy. I chopped my strings into n-mers (n is a small number) to reduce the search space and hashed them. then i compared all-vs-all n-mers to get their relative distances, therefore producing a complex but still manageable graph. and now im stuck. there is no way to reconstruct the alignments from the info i produced in any reasonable amount of time.

Does anyone has any pointers or suggestions. I'm not looking for a complexity reduction solution but anything that can make these kind of queries, implementation vise fast. (and memory efficient )

Thank you

baxy