I have some questions to help me understand the graph. If it's something I can read up on, just let me know what to search for. :)
What does the notation on a node mean? (eg. 3/168524)
What is the difference between the dotted lines and the solid lines?
What determines if a node has 1 child vs many? (Maybe this is explained by the notation.)
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Good questions, thanks for asking!There’s a discussion of 2-cycle graphs on Greg Egan’s page: http://www.gregegan.net/SCIENCE/Superpermutations/Superpermutations.html#TCG(He writes 4|123 where I write 4/123. IIRC he and I invented these very similar notations independently and simultaneously; I’ve stuck with the slash because I find it harder to confuse with a 1 when I’m writing by hand.)In short, 4/123 is the 2-cycle that consists of all the permutations that are made from some rotation of 123, with a 4 inserted anywhere. For example, the permutation 2341 belongs to 4/123, because 231 is a rotation of 123.A solid line means that the two 2-cycles share a 1-cycle. The dashed lines mean that the two 2-cycles are neighbours in a 3-cycle. The dotted lines mean they’re neighbours in a 4-cycle.So the long dashed-and-dotted line from 7/123456 at the bottom-right to 7/123564 at the top-left is the principal 4-cycle, i.e. the 4-cycle that begins 1234567.
On Fri, 1 Feb 2019 at 16:44 Justin Hoffman <jstn...@gmail.com> wrote:
I have some questions to help me understand the graph. If it's something I can read up on, just let me know what to search for. :)--
What does the notation on a node mean? (eg. 3/168524)What is the difference between the dotted lines and the solid lines?
What determines if a node has 1 child vs many? (Maybe this is explained by the notation.)
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⇒ so we can only decrease down to 4 remaining chains by just applying extensions
- let's say we can connect these 4 chains through one last partial extension, meaning we remove four 1-links and add three corresponding 2-links // first method
⇒ this is the same as starting from a path (kernel) built from four 1-cycles connected by three 2-links, and adding 29 extensions to it // second method
⇒ the total cost for this construct is
873 = 6*120*[1] + 29*[5] + 6 /*start perm*/ - 4*[1] + 3*[2] // by the first method
= 5*4*[1] + 3*[2] + 6 /*start perm*/ + 29*[29] // by the second method
def back():
if len(diagram.chains) == 0:
=> found a solution
else:
selected_chain = find chain with smallest number of available extensions
if len(selected_chain.available_extensions) == 0:
return
for each available extension: # (will be between 0 and N)
extend the tuple containing this individual extension # (or just this extension if we're done with tuples)
back()
collapse back the tuple extensions
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