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Suppose you have a function quality(x) that returns the quality of a sequence of letters x. Given a string such as "howareyoutoday," what is the most efficient way to determine that the segmentation is "how are you today" (i.e. quality(how)+quality(are)+quality(you)+quality(today) is the maximum quality possible)?

I was thinking that we could have something like the following:

A[0] = h, A[1] = o, ..., A[n] = y 
Q[0] = quality(A[0]), Q[1] = quality(A[0]A[1]), ..., Q[n] = quality(A[0]...A[n])

Now to determine the segmentation, we find max{Q[0], .., Q[n]} which will return some Q[i] (the first space is after this). Then, we find max{Q[i+1], .. Q[n]} which returns another Q[i] (second space is after this), etc. until max returns Q[n].

I have some questions though: is this method even correct, and does it use dynamic programming? It seems to me that it does, since we build the initial Q with subproblems to the original problem. Also, is this an optimal solution? To my understanding, the worst case would be O(n^2), which would be when max returns Q[0], then Q[1], then Q[2], etc.

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    $\begingroup$ Without information on quality it's not clear that such an approach can work. Basically, the "Bellman optimality criterion" requires that there's some operation $\circ$ so that $\operatorname{quality}(s) = \operatorname{quality}(s_1 .. s_k) \circ \operatorname{quality}(s_{k+1})$ for some $k$. $\endgroup$ – Raphael Oct 4 '14 at 12:56
  • $\begingroup$ You should be consider all partitions of the word, not necessary $s_{0,j}$, and next identify the same question, in the subproblems $s_{i_1, i_2}$. Because the string could have characters without sense before appears a real word, and after as well. I think that it's correct vote up for dynamic problem. What's about arxiv.org/pdf/1105.6162.pdf ? $\endgroup$ – jonaprieto Oct 4 '14 at 23:53
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I'm assuming your quality function measures something similar to the likelihood of the string being a word. The problem with your algorithm is that it makes a hard decision about where the first space should go before considering where the other spaces go. If you make a suboptimal decision for where the first space goes it will mess you up later on.

The dynamic programming algorithm for this problem is called the Viterbi algorithm. https://en.wikipedia.org/wiki/Viterbi_algorithm You have to keep a table of the highest quality segmentations seen so far as you process the string from left to right. When the algorithm finishes you will be left with a ranked list of the highest quality segmentations.

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No that doesn't work, without harsh qualifications on 'that' or 'work'. What if the presence vs. absence of space after a letter represents a binary encoding of some hard problem into the string, where quality(x) returns 1 if your segmentation is a solution to the hard problem and 0 if it is not a solution.

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