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blokhead
I'm still not sure about an efficient (polynomial-time) algorithm for your original question (<strike>minimizing</strike>maximizing the number of distinct layouts used in representing a sequence of photos), but I can address your postscript:
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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,
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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)*$/.
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If you want to know whether this allows you to get <i>all</i> possible photo sequences, then you are asking whether /^(pll|lp|pp)*$/ is equivalent to <c>/^[pl]*$/</c>. This is called the "regex universality" problem (does a given regex match all strings?), and it is PSPACE-complete in the general case. But for reasonable sized examples, you can check this by building a DFA from the regex, complementing it, and checking to see if it accepts any strings. I even happen to know of an upcoming Perl project that would make these regular expression & automata operations very easy (which happens to be why I'm in the formal-language frame of mind to begin with) {nudge nudge wink wink}.
<blockquote><i> and similarly, what is the smallest possible selection of layouts that are similarly "complete"? I don't need to know the answer, but it is intriguing!
</i></blockquote>
Well, the smallest is {p,l} of course ;) To make the problem more interesting, if you are given a set of layouts (say, {pll,lp,pp} again), can you make this set smaller without losing expressivity? What you can do is calculate the <i>minimal generating set</i> of /^(pll|lp|pp)*$/. This can also be done through some manipulation of the automata for the associated language.
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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 <i>optimality</i> of the layouts according to your criteria (<strike>fewest</strike>maximum number of distinct layouts used). This is where it gets a little harder....
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