improvement of method vertex_separation_BAB

[dcoudert]

Method vertex_separation_BAB implements a branch and bound algorithm for the vertex separation of graphs. it uses the concept of memoization to cut some branches of the exploration tree of the branch and bound. Roughly, it stores some prefixes (sets of vertices) to avoid exploring multiple times branches starting with prefixes on the same set of vertices.
Previously, this was implemented by using a set of frozensets, each containing the set of vertices of a prefix.
This PR stores instead the bitset representation of a prefix as a tuple of integers. Hence, it uses less memory per stored prefix. In addition, the running time is slightly improved.

Before:

sage: for i in range(2, 7):
....:     G = graphs.MycielskiGraph(i)
....:     print(f"{G}: {G.order()} vertices; vs = {vertex_separation_BAB(G)[0]}")
....:     timeit("vs = vertex_separation_BAB(G)[0]")
....: 
Mycielski Graph 2: 2 vertices; vs = 1
625 loops, best of 3: 2.79 μs per loop
Mycielski Graph 3: 5 vertices; vs = 2
625 loops, best of 3: 4.58 μs per loop
Mycielski Graph 4: 11 vertices; vs = 5
625 loops, best of 3: 15.4 μs per loop
Mycielski Graph 5: 23 vertices; vs = 10
625 loops, best of 3: 353 μs per loop
Mycielski Graph 6: 47 vertices; vs = 20
5 loops, best of 3: 85.2 ms per loop

After:

Mycielski Graph 2: 2 vertices; vs = 1
625 loops, best of 3: 2.57 μs per loop
Mycielski Graph 3: 5 vertices; vs = 2
625 loops, best of 3: 4.23 μs per loop
Mycielski Graph 4: 11 vertices; vs = 5
625 loops, best of 3: 13.5 μs per loop
Mycielski Graph 5: 23 vertices; vs = 10
625 loops, best of 3: 277 μs per loop
Mycielski Graph 6: 47 vertices; vs = 20
5 loops, best of 3: 56.8 ms per loop

Before

sage: for i in range(5, 11):
....:     G = graphs.Grid2dGraph(i, i)
....:     print(f"{G}: {G.order()} vertices; vs = {vertex_separation_BAB(G)[0]}")
....:     timeit("vs = vertex_separation_BAB(G)[0]")
....: 
2D Grid Graph for [5, 5]: 25 vertices; vs = 5
625 loops, best of 3: 99.7 μs per loop
2D Grid Graph for [6, 6]: 36 vertices; vs = 6
625 loops, best of 3: 629 μs per loop
2D Grid Graph for [7, 7]: 49 vertices; vs = 7
125 loops, best of 3: 3.9 ms per loop
2D Grid Graph for [8, 8]: 64 vertices; vs = 8
25 loops, best of 3: 24.4 ms per loop
2D Grid Graph for [9, 9]: 81 vertices; vs = 9
5 loops, best of 3: 182 ms per loop
2D Grid Graph for [10, 10]: 100 vertices; vs = 10
5 loops, best of 3: 67.7 s per loop

After:

2D Grid Graph for [5, 5]: 25 vertices; vs = 5
625 loops, best of 3: 85.2 μs per loop
2D Grid Graph for [6, 6]: 36 vertices; vs = 6
625 loops, best of 3: 489 μs per loop
2D Grid Graph for [7, 7]: 49 vertices; vs = 7
125 loops, best of 3: 2.88 ms per loop
2D Grid Graph for [8, 8]: 64 vertices; vs = 8
25 loops, best of 3: 17.6 ms per loop
2D Grid Graph for [9, 9]: 81 vertices; vs = 9
5 loops, best of 3: 142 ms per loop
2D Grid Graph for [10, 10]: 100 vertices; vs = 10
5 loops, best of 3: 64.4 s per loop

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

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[fchapoton]

nice. Looks so simple..

[dcoudert]

Indeed. I don't know why I didn't thought about it before. I found it while working on a different code.