Beefy Boxes and Bandwidth Generously Provided by pair Networks
Perl: the Markov chain saw

Re: Algorithm to fit photos into spaces on pages

by blokhead (Monsignor)
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....


Log In?

What's my password?
Create A New User
Node Status?
node history
Node Type: note [id://508728]
and all is quiet...

How do I use this? | Other CB clients
Other Users?
Others cooling their heels in the Monastery: (5)
As of 2017-07-20 22:47 GMT
Find Nodes?
    Voting Booth?
    I came, I saw, I ...

    Results (316 votes). Check out past polls.