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Re^7: Challenge: 8 Letters, Most Words

by Limbic~Region (Chancellor)
on Oct 05, 2013 at 21:08 UTC ( #1057084=note: print w/replies, xml ) Need Help??

in reply to Re^6: Challenge: 8 Letters, Most Words
in thread Challenge: 8 Letters, Most Words

The C code I wrote to calculate it is below. It finished nearly instantly. I was trying to determine how to write the rest of the code (could it fit in memory).
#include <stdio.h> // Program to count how many 8 character combinations are necessary fo +r brute force // Constants to remove magic numbers #define Z 25 int main (void) { static const short int max[26] = {4, 3, 3, 4, 4, 4, 4, 3, 3, 2, 3, + 4, 3, 4, 4, 3, 1, 4, 5, 3, 4, 2, 3, 2, 3, 4}; // from STDERR of filt short int have[26] = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, + 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}; int combo = 0; int i, j, k, l, m, n, o, p; for (i = 0; i <= Z; i++) { if (++have[i] > max[i]) { --have[i]; continue; } for (j = i; j <= Z; j++) { if (++have[j] > max[j]) { --have[j]; continue; } for (k = j; k <= Z; k++) { if (++have[k] > max[k]) { --have[k]; continue; } for (l = k; l <= Z; l++) { if (++have[l] > max[l]) { --have[l]; continue; } for (m = l; m <= Z; m++) { if (++have[m] > max[m]) { --have[m]; continue; } for (n = m; n <= Z; n++) { if (++have[n] > max[n]) { --have[n]; continue; } for (o = n; o <= Z; o++) { if (++have[o] > max[o]) { --have[o]; continue; } for (p = o; p <= Z; p++) { if (++have[p] > max[p]) { --have[p]; continue; } ++combo; --have[p]; } --have[o]; } --have[n]; } --have[m]; } --have[l]; } --have[k]; } --have[j]; } --have[i]; } printf("%i\n", combo); return 0; }
It also ended up being part of my final solution since I realized it was unnecessary to keep them in memory using a high water mark algorithm.

Cheers - L~R

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Re^8: Challenge: 8 Letters, Most Words
by LanX (Bishop) on Oct 05, 2013 at 21:28 UTC
    isn't the goal to find a clever Perl solution for the whole problem which does it within less than an hour? :)

    I have two good ideas but no time to hack them. :(

    Cheers Rolf

    ( addicted to the Perl Programming Language)

      Absolutely - but I am not that smart. I was hoping someone would provide a solution that was guaranteed to be correct that didn't require exhaustive searching. So far, all of the heuristic methods have flaws. That isn't to say I am not smart enough to come up with a heuristic solution but I needed to know for certain.

      Cheers - L~R

        hi Limbic~Region

        IMHO all "heuristic" methods must have flows... I'm quite experienced with these kind of problems and people tend to underestimate them (I'm not smart just trained :).

        And shouldn't be to difficult to show this belongs to NP-class.

        Anyway since I was able to calculate all possible 8-letter-combinations within 3 minutes with an non-optimized recursion it should be possible to find an approach to rapidly calculate each covering-number and to choose the maximum.

        Let me explain:

        First of all a good branch-and-bound could avoid useless branches which can't possibly beat the current maximum (my recursion just needs more criteria to bound)

        (this doesn't contradict NP, NP is about run-time and not correctness)

        In order to speed up calculation, one should cache sub-solutions which can be rapidly added.

        Let l be a letter-combination to be checked and L-x all derived combinations by striking x letters and n(l) the number of dictionary-words which can be formed with exactly all letters.

        its easy to see that the covering

        C(l)= n(l)+ sum { C($_) } L-1 - sum { C($_) } L-2

        e.g.  C('abc') = n('abc') + C('ab')+C('ac')+C('bc') - C(a) -C(b)-C(c)

        and with the table from this post

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

        we see C('abc') = 0+3+2+2-(1+1+0) = 5 and indeed the first 5 entries in the table can be constructed out of a,b and c!

        So to calculate the covering of an 8-letter tuple we just need to look up the covering of (at most) 8 7-letter sub-tuples and 28 6-letter sub-tuples and add them.

        Starting with all 1-letter tuples one can successively calculate the covering of all 2-letter tuples and so on.

        -max: 1 26 -max: 2 350 -max: 3 3247 -max: 4 23312 -max: 5 137909 -max: 6 698996 -max: 7 3116882 -max: 8 12461993 sum 16442715

        Holding 16 million hash-entries shouldn't be a problem. A worst case of 591_937_740 (=16442715 *(28+8)) additions neither, which are already 1000 times less operations than in your C program.

        This should be fast enough for 8 letters and I doubt one can do it faster...

        Implementation left as an exercise! :)

        Cheers Rolf

        ( addicted to the Perl Programming Language)

        ) for simplification this table only holds ordered words, extending it to count different permutations of the same letters ("cab" and "abc" => n("abc") =2 ) wouldn't change the math!

        ) and normally one can always construct an edge case where the recursion never bounds.

        ) please note the complexity rises exponentially for more letters, it's still NP!

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