new implementation of method is_transitive with linear memory space

[dcoudert]

The previous version of method is_transitive was building the distance matrix of the digraph and so had memory usage in $O(n^2 + m)$. We change that to a version with memory usage in $O(n + m)$. In addition, this new version is faster for non-transitive digraphs.

Before

sage: D = digraphs.Circuit(4)
sage: D.is_transitive()
False
sage: %timeit D.is_transitive()
8.01 µs ± 35.7 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: D = digraphs.RandomDirectedGNP(30,.2)
sage: D.dig6_string()
']AOGb?dA?pG?S?@?OPIGG???_?O?GC??W_H?BCjaJJA?gO@?A??_?VDHG_ACCiBU?O`HaUC?kAAD@_AAJErW_G_ICCGa@?S@Oo?IC????DoCQ?Q?@?_@g?O?C?aAGK??o?cKO_W???A?G?Go?H`??Co'
sage: D.is_transitive()
False
sage: %timeit D.is_transitive()
54.3 µs ± 167 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
sage: D = digraphs.DeBruijn(5, 2)
sage: D.is_transitive()
False
sage: %timeit D.is_transitive()
80.6 µs ± 116 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
sage: D = digraphs.RandomDirectedGNP(20,.2).transitive_closure()
sage: D.dig6_string()
'S^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~w'
sage: D.is_transitive()
True
 sage: %timeit D.is_transitive()
56 µs ± 181 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

With this PR

sage: D = digraphs.Circuit(4)
sage: D.is_transitive()
False
sage: %timeit D.is_transitive()
6.63 µs ± 26.8 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: D =
DiGraph(']AOGb?dA?pG?S?@?OPIGG???_?O?GC??W_H?BCjaJJA?gO@?A??_?VDHG_ACCiBU?O`HaUC?kAAD@_AAJErW_G_ICCGa@?S@Oo?IC????DoCQ?Q?@?_@g?O?C?aAGK??o?cKO_W???A?G?Go?H`??Co')
sage: D.is_transitive()
False
sage: %timeit D.is_transitive()
46.7 µs ± 88.4 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
sage: D = digraphs.DeBruijn(5, 2)
sage: D.is_transitive()
False
sage: %timeit D.is_transitive()
73.8 µs ± 116 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
sage: D = DiGraph('S^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~z~~~^~~w')
sage: D.is_transitive()
True
sage: %timeit D.is_transitive()
58.6 µs ± 118 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

📝 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

[github-actions]

Documentation preview for this PR (built with commit 5fe04f7; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[user202729]

Can you just reuse some existing implementation of dfs/bfs e.g. shortest_path_all_vertices?

[dcoudert]

Can you just reuse some existing implementation of dfs/bfs e.g. shortest_path_all_vertices?

We could do that, but all my trials are way slower. That's why I decided to go this way.
Before, all the complexity of the code was hidden in the call to c_distances_all_pairs.

[user202729]

Of course it's not as much about complexity, but avoiding duplication of code.

Where does the slowness come from? both are O(n ⋅ (n + m)) right? (function call overhead? → cpdef / bfs instead of dfs? / specialized algorithm?)

[dcoudert]

This new version is slightly simpler as it calls simple_BFS for the exploration part. The running time is similar. It uses one extra array of size $n$.