Speed up OnDigraphs for a digraph and a permutation

[ChrisJefferson]

This came up from some code which was heavily using digraphs. this almost doubled the speed of the algorithm, as it turns out these GAP-level checks were dominating the run-time of OnDigraphs.

• Instead of checking if the permutation fixes the vertices, check
  if it is the identity (which is much cheaper, although catches
  less cases)

• Do not check explictly if the permutation maps the vertices
  of the graph to itself, do the mapping then check afterwards.

[wilfwilson]

Thanks for noticing this @ChrisJefferson. I made a couple of comments - I'll give @james-d-mitchell a chance to say what he thinks, and see how attached he is to the error message.

[wilfwilson]

Another solution is that we could create some NC versions of functions that are particularly performance-critical, including OnDigraphsNC.

[wilfwilson]

@ChrisJefferson I know this is a bit late (sorry) but I intend to come up with a resolution for this soon that keeps the performance improvement.

[james-d-mitchell]

I'd propose improving the performance of OnDigraphs at gap days in Brussels in 2025, if we have the time.

[reiniscirpons]

I tried my best to incorporate the improved methods, but I just do not see the timing improvements claimed. I settled on implementing a separate OnDigraphsNC method which skips the more expensive ForAll and ForEach checks and only does the p = () optimization. I then replaced internal usage of the OnDigraphs method with OnDigraphsNC. Here are some runtime stats:

Test	Old method time, ms	Proposed new OnDigraphsNC method time, ms	Proposed new OnDigraphs method time, ms
1000 random digraphs with 100 vertices, IsIsomorphicDigraph	703	689	-
829 random topologically sortable digraphs with 100 vertices, DigraphTransitiveReduction	65	52	-
100 random transitive digraphs with 1000 vertices, IsIsomorphicDigraph	5476	5374	-
100 random transitive digraphs with 1000 vertices, DigraphTransitiveReduction	24020	23947	-
100 random digraphs with 1000 vertices, OnDigraphs or OnDigraphsNC respectively	1415	1403	1487

There seem to be some minor improvement (within margin of error in some cases) with the proposed OnDigraphsNC method, but it slows down OnDigraphs by a bit.

[james-d-mitchell]

Thanks @reiniscirpons looks good