Trac #33645: Another improvement for face iterator of simple/simplici…

…al polytopes

Follow up on #30040:

If we are in a lattice, such that all intervals `[F, G]` are boolean, if
`F` is not the lower bound, then any face except the lower bound, has a
unique representation as meet of coatoms.

This means that we do not need to mark a face as visited, when removing
it from the coatoms.

Before (benchmarks without parallelization):

{{{
sage: P = polytopes.cyclic_polytope(10, 22)
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
43.2 ms ± 40 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
sage: P = polytopes.associahedron(['A', 8])
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
20.1 ms ± 69.3 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
sage: P = polytopes.hypercube(12, backend='field')
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
38.6 ms ± 464 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
sage: P = polytopes.Birkhoff_polytope(5)  # neither simple nor
simplicial
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
275 ms ± 531 µs per loop (mean ± std. dev. of 7 runs, 1 loop each)
}}}

After (benchmarks without parallelization):
{{{
sage: P = polytopes.cyclic_polytope(10, 22)
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
38.6 ms ± 1.07 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
sage: P = polytopes.associahedron(['A', 8])
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
18.3 ms ± 23.7 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: P = polytopes.hypercube(12, backend='field')
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
30.9 ms ± 40 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
sage: P = polytopes.Birkhoff_polytope(5)  # neither simple nor
simplicial
sage: %timeit C = CombinatorialPolyhedron(P); C.f_vector(1, 0)
274 ms ± 410 µs per loop (mean ± std. dev. of 7 runs, 1 loop each)
}}}

URL: https://trac.sagemath.org/33645
Reported by: gh-kliem
Ticket author(s): Jonathan Kliem
Reviewer(s): Travis Scrimshaw

src/sage/geometry/polyhedron/combinatorial_polyhedron/face_iterator.pyx

@@ -1970,7 +1970,13 @@ cdef inline int next_face_loop(iter_t structure) nogil except -1:
         # In this case there exists ``faces[0].faces[n_faces]``, of which we
         # have visited all faces, but which was not added to
         # ``visited_all`` yet.
-        add_face_shallow(visited_all[0], faces[0].faces[n_faces])
+
+        if not faces[0].polyhedron_is_simple:
+            # In case of a simple lattice, this step needs not to be applied:
+            # Every element, except the lower bound, has a unique representation of coatoms in this case.
+            # Hence, as the face is already removed from faces[0], any subfaces will not be visited.
+            # (If we manually ignore subfaces, faces will still be added to visited_all).
+            add_face_shallow(visited_all[0], faces[0].faces[n_faces])
     else:
         # Once we have visited all faces of ``faces[n_faces]``, we want
         # to add it to ``visited_all``.
@@ -2193,7 +2199,14 @@ cdef inline int prepare_face_iterator_for_partial_job(
 
             for i in range(job_id_c):
                 # Fast forwarding the jobs.
-                add_face_shallow(structure.visited_all[d], structure.new_faces[d].faces[structure.new_faces[d].n_faces -1])
+
+                if not structure.new_faces[d].polyhedron_is_simple:
+                    # In case of a simple lattice, this step needs not to be applied:
+                    # Every element, except the lower bound, has a unique representation of coatoms in this case.
+                    # Hence, as the face is already removed from faces[0], any subfaces will not be visited.
+                    # (If we manually ignore subfaces, faces will still be added to visited_all).
+                    add_face_shallow(structure.visited_all[d], structure.new_faces[d].faces[structure.new_faces[d].n_faces -1])
+
                 structure.new_faces[d].n_faces -= 1
 
             parallel_struct.current_job_id[current_depth -1] = job_id_c