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PO(Set of) List of sets (Cartesian product(s)) from graph corresponding to set of lists
 

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    <p>The set of lists (A):</p> <pre><code>{[a,b,d,f], [a,c,d,f], [a,b,e,f], [a,c,e,f]} </code></pre> <p>where a, b, c, d, e and f are items (not necessarily characters in a word), can be factored as a directed acyclic graph (DAG, B, all edges point from left -> to right):</p> <pre><code> b--&gt;d / \ / \ a X f \ / \ / c--&gt;e </code></pre> <p>or as the Cartesian product of 4 sets of items (C, termed axes):</p> <pre><code>{a} * {b,c} * {d, e} * {f} </code></pre> <p>Guava has a nice <a href="http://docs.guava-libraries.googlecode.com/git/javadoc/com/google/common/collect/Sets.html#cartesianProduct%28java.util.List%29" rel="nofollow noreferrer"><em>method</em></a> for generating a set of lists (A) from a list of sets (C).</p> <p>I am trying for an algorithm that accepts a graph like B and returns a list of axes like C (actually one or more, see example below), which can be used with the method above to generate a set of lists like A.</p> <p>However, it is not guaranteed that the set of lists will be a Cartesian product. For example:</p> <pre><code>{[a,b,d,f], -missing- [a,b,e,f], [a,c,e,f]} </code></pre> <p>corresponding to the DAG:</p> <pre><code> b--&gt;d / \ \ a \ f \ \ / c--&gt;e </code></pre> <p>cannot be expressed as <strong>1</strong> Cartesian product but can be expressed as 2:</p> <pre><code>{a}*{b}*{d,e}*{f} and {a}*{c}*{e}*{f} </code></pre> <p>corresponding to the graphs:</p> <pre><code> d / \ a--&gt;b f and a--&gt;c--&gt;e--&gt;f \ / e </code></pre> <p>The lists should have some degree of relatedness (think: a random sample of a very large Cartesian product).</p> <p>Note: lists of different lengths cannot share the same set of axes.</p> <p>Is there an algorithm that does this and I just haven't Googled the right terms? If not, can we create it?</p> <p>Complexity of the algorithm may be an issue as the set could have 10^2 lists and each list could have 10^2 of items, i.e. a fairly large graph. I can guarantee that the input graphs would have the minimal number of nodes possible to represent the set of lists..., and that connected non-branching nodes (a->c->e->f) can be rolled up into single objects (acef).</p> <p>PS. I don't think this is same as the <a href="http://en.wikipedia.org/wiki/Cartesian_product_of_graphs" rel="nofollow noreferrer">Cartesian product of graphs</a>, but there could be some overlap.</p>
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    1. COThere are graph libraries (I am thinking about jgrapht here, but there are probably others), have you tried and poked them to see if they had something approaching?
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    2. CO@fge I am familiar with JGraphT and it does not have such an algorithm.
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    3. COI would consider the lists as words and try to locate longest common suffixes.
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