Loopy Lattices Redux by Marko A. Rodriguez.
Comparison of Titan and Faunus counting the number of paths in a 20 x 20 lattice.
Interesting from a graph-theoretic perspective but since the count can be determined analytically, I am not sure of the utility of being about to count the paths?
In some ways this reminds me of Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions by Marc Timme, Frank van Bussel, Denny Fliegner and Sebastian Stolzenberg, New Journal of Physics 11 (2009) 023001.
The question Timme and colleagues were investigating was the coloring of nodes in a graph which depended upon the coloring of other nodes. For a chess board sized graph, the calculation is estimated to take billions of years. The technique developed here takes less than seven (7) seconds for a chess board sized graph.
Traditionally, assigning a color to a vertex required knowledge of the entire graph. Here, instead of assigning a color, the color that should be assigned is represented by a formula stating the unknowns. Once all the nodes have such a formula:
The computation of the chromatic polynomial has been reduced to a process of alternating expansion of expressions and symbolically replacing terms in an appropriate order. In the language of computer science, these operations are represented as the expanding, matching and sorting of patterns, making the algorithm suitable for computer algebra programs optimized for pattern matching.
What isn’t clear is whether a similar technique could be applied to merging conditions where the merging state of a proxy depends upon, potentially, all other proxies.