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### Re: Challenge: 8 Letters, Most Words

by LanX (Chancellor)
 on Oct 04, 2013 at 17:38 UTC ( #1056904=note: print w/replies, xml ) Need Help??

in reply to Challenge: 8 Letters, Most Words

(Tsts this Limbic~Region sabotaging our work-life balance again! ;)

Some theory:

For a minute I thought this can be trivially solved by counting the normalization of all words (<=8) from the dictionary in a hash... e.g.

```  DB<211> join "",sort split //,"electron"
=> "ceelnort"

DB<212> join "",sort split //,"elector"
=> "ceelort"

As next step successively the count from all smaller words had to be added to covering words, e.g striking the "n" from "ceelnort" leads to "ceelort", so \$count{ceelnort}+=\$count{ceelort}

But than I realized that the best covering word from the dictionary is not necessarily the best solution.

take this counterexample for 3 out of 4 letters, the number showing the coverage-count

```1 a
1   b
3 a b
2 a   c
2   b c
4 a b    d

so the word (a,b,d) is the maximum with a count 4, but the set (a,b,c) would cover 5 words!!!

(yes this also works with repeated letters)

IMHO this problem belongs to the family of Maximum coverage problem and Set_cover_problem, so finding a guarantied best solution shouldn't be trivial w/o brute force.

Cheers Rolf

( addicted to the Perl Programming Language)

##### update

Maybe you can use the above count hash to solve the dual problem:

"which of the n-8 letters cover the minimum of words" (n including repetition)

E.g. "d" is in only one out of 6 words with 4 letters => the remaining 3 letters cover 5 words.

"c" is only in 2 remaining words => (a,b) cover a maximum of 3 words and so on.

Not sure if this leads to the guarantied best solution, sounds to easy... =)

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Re^2: Challenge: 8 Letters, Most Words
by Limbic~Region (Chancellor) on Oct 05, 2013 at 03:31 UTC
LanX,
I couldn't convince myself that anything other than an exhaustive brute force solution would guarantee the correct answer. Now that I have done that, I will think about alternatives.

Cheers - L~R

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