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Re^10: Challenge: 8 Letters, Most Wordsby LanX (Saint) |
on Oct 06, 2013 at 12:26 UTC ( [id://1057142]=note: print w/replies, xml ) | Need Help?? |
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)² 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
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.
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...³ Implementation left as an exercise! :) Cheers Rolf ( addicted to the Perl Programming Language) ¹) 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! ²) and normally one can always construct an edge case where the recursion never bounds. ³) please note the complexity rises exponentially for more letters, it's still NP!
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