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[HELP] combinatorics on strings
By baddestguy, history, 4 years ago, In English

I want to find the number of strings that are lexicographically smaller than A and has substring B NOTE: the length of the strings should be <= |A|.

e.g. A = "da" and B = "c" then answer = 29: {'c', 'ac', 'bc', cX}, where X is any alphabet. also, constraints are pretty low, 1 <= |B|,|A| <= 1000 Please help thanks.

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4 years ago, # |
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Does "contains" mean "subsequence" or "substring"? Is "smaller" lexicographically smaller or anything else?

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    4 years ago, # ^ |
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    it means substring, and smaller means lexicographically smaller

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      4 years ago, # ^ |
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      Is the number of such strings not infinite in your case? "c", "ca", "caa", "caaa", "caaaa" etc all have "c" as a substring and are lexicographically smaller than "da".

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        4 years ago, # ^ |
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        Oh, then the length of the strings should be <= |A|.

        is there a name for this type of ordering cause I think I'm not phrasing it correctly?

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          4 years ago, # ^ |
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          I don't think it has a standard name, you just had to be more explicit.

          Anyway, I think your problem can be solved with dynamic programming, states like this: $$$\mathrm{dp}[i][j][a][b]$$$ counts the number of strings lexicographically smaller than $$$A$$$, where:

          • $$$i$$$ is the length of the string;
          • $$$j$$$ is the length of the longest suffix of the string that is also a prefix of $$$B$$$;
          • $$$a$$$ is 0 if the string is a prefix of $$$A$$$, 1 otherwise;
          • $$$b$$$ is 0 if the string does not contain $$$B$$$ as a substring, 1 otherwise.

          I think you should be able to do updates in a KMP-like way.

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