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### Re: Algorithm to fit photos into spaces on pages

 on Nov 15, 2005 at 18:52 UTC ( #508728=note: print w/replies, xml ) Need Help??

in reply to Algorithm to fit photos into spaces on pages

I'm still not sure about an efficient (polynomial-time) algorithm for your original question (minimizingmaximizing the number of distinct layouts used in representing a sequence of photos), but I can address your postscript:
P.S. This does beg the questions of how can one determine if a selection of layouts could deal with every possible selection of photos,
I'm in a formal-languages frame of mind right now, and this question is easily answerable using regular languages & automata. If your layouts are, say, {pll,lp,pp}, then the possible photo sequences you can represent are exactly those sequences which match /^(pll|lp|pp)*\$/.

If you want to know whether this allows you to get all possible photo sequences, then you are asking whether /^(pll|lp|pp)*\$/ is equivalent to /^[pl]*\$/. This is called the "regex universality" problem (does a given regex match all strings?), and it is PSPACE-complete in the general case. But for reasonable sized examples, you can check this by building a DFA from the regex, complementing it, and checking to see if it accepts any strings. I even happen to know of an upcoming Perl project that would make these regular expression & automata operations very easy (which happens to be why I'm in the formal-language frame of mind to begin with) {nudge nudge wink wink}.

and similarly, what is the smallest possible selection of layouts that are similarly "complete"? I don't need to know the answer, but it is intriguing!
Well, the smallest is {p,l} of course ;) To make the problem more interesting, if you are given a set of layouts (say, {pll,lp,pp} again), can you make this set smaller without losing expressivity? What you can do is calculate the minimal generating set of /^(pll|lp|pp)*\$/. This can also be done through some manipulation of the automata for the associated language.

Of course, this only address the question of whether or not you can represent a sequence of photos in a given basis of layouts. It does not address the optimality of the layouts according to your criteria (fewestmaximum number of distinct layouts used). This is where it gets a little harder....

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