Prufer coding for trees

[SimonCoste]

Hi,
I added elementary functionalities related to trees. There are 3 exported functions,

• is_tree checks if a simple graph is a tree
• prufer_encode creates the Prüfer sequence associated to a tree
• prufer_decode creates the tree associated with a Prüfer sequence

This is my first PR; I did my best at following the style guide and provide a doc, please do not hesitate to explain how things should be done.

I also modified the docs but I did not succeed in having it correctly deployed locally; I get a new trees pages but none of the docstrings I wrote seem to be parsed by the Documenter. I also get the following possibly related warning:

Warning: Documenter could not auto-detect the building environment Skipping deployment.
└ @ Documenter ~/.julia/packages/Documenter/bFHi4/src/deployconfig.jl:75

Related: https://github.com/JuliaGraphs/Graphs.jl/pull/194. I'll add a random Cayley tree generator very soon.

[codecov]

Codecov Report

Merging #206 (426d438) into master (897e183) will increase coverage by 0.02%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #206      +/-   ##
==========================================
+ Coverage   97.40%   97.42%   +0.02%     
==========================================
  Files         109      113       +4     
  Lines        6470     6531      +61     
==========================================
+ Hits         6302     6363      +61     
  Misses        168      168              

[simonschoelly]

Hi, I already added a few comments so far - I will add more later, but I need to figure out how these Prüfer sequences work first.

[simonschoelly]

A few more minor things (and one major) - appart from that I haven't worked with the documentation for a long time, so I have not found out what is wrong here, maybe @gdalle knows more about that, otherwise I will try too look at it in the next fews days.

[SimonCoste]

Hi, I adapted to all your remarks.

Importantly, I also added the uniform_tree(n) function. It generates a uniform tree over the $n^{n-2}$ such trees by decoding a random uniform Prüfer sequence.

Edit: I also corrected the docs, you were right about the space after ```: if you forget it, the function isnt parsed by Documenter. The doc looks fine now.

[SimonCoste]

I fixed a small bug (must specify seed=nothing in rng_from_rng_or_seed arguments). Everything should be fine now !

[simonschoelly]

The documentation is also working now, right? Then it does indeed seem ready to be merged.

[SimonCoste]

Yes, the doc is properly working!

[simonschoelly]

Great, thank you - for your next PR, the tests should also run automatically, its just for someones first PR, that we have to to trigger the tests automatically

[SimonCoste]

In the near future (~2 months) I'll add more random tree generators, possibly: 

• random binary trees
• random recursive trees
• random uniform trees with a fixed degree sequence
• random uniform trees with a fixed number of leaves.

Thanks Simon for the support !

[simonschoelly]

In the near future (~2 months) I'll add more random tree generators, possibly: 

* random binary trees

* random recursive trees

* random uniform trees with a fixed degree sequence

* random uniform trees with a fixed number of edges.

Thanks Simon for the support !

That are definitely welcome additions. When taking inspiration from existing code for random graphs here, you just have to be a bit careful, that code sometimes quite old and buggy, and might not even calculate the expected distribution (If I remember correctly, for random_regular_ graph).

[SimonCoste]

What is this potential bias in random_regular_graph ? I'll have a look.

[simonschoelly]

I don't know where exactly the error is, but lets look at an example, the 2-regular graphs on 6-vertices, assuming that the vertices are labeled:

There 60 connected graphs (a single cycle) given by 5!/2 = 60.
There are 10 disconnected graphs (two cycles of 3 vertices each) given by {5 choose 2} = 10.

Therefore the probability of getting a connected graph should be 60 / (10 + 60) = 6/7 ≈ 0.857.

We can also estimate this probability by sampling repeatedly with random_regular_graph:

julia> using Graphs

julia> p_connected(n) = count(is_connected(random_regular_graph(6, 2)) for _ in 1:n) / n
p_connected (generic function with 1 method)

julia> p_connected(10^6)
0.6911

so there is something off with that algorithm And it can also not be for the (probably much more difficult) problem of random regular graphs on unlabeled vertices, because then the probability would be 1/2.