Square root filtration values for alpha and delaunay-cech complex

are weights with sqrt=True well-defined (e.g., via $x\mapsto \mathrm{sgn}(x)\sqrt{|x|}$) ?

If the weights make some (squared) values negative, I don't think theory provides anything meaningful for the sqrt, no. There are a number of applications where the weights can only increase filtration values, so they remain non-negative. Adding the same constant to all weights (without sqrt, so possibly negative) preserves a lot of things (the triangulation, the order between filtration values), that's one way to avoid negative values, but it obviously does not preserve filtration values, so it would be confusing. I didn't check if the code replaces only the negative ones with NaN, or if this propagates to random nonsense.

do you think it would be reasonable to put the sqrt to True ? (I'm in favor of it, to make it consistent with other complexes).

I am rather strongly against changing the behavior of old code, that's really not a nice thing to do to users. However

better documentation is always good. The existence of this new option counts as documentation and may bring this issue to users' attention.
for new interfaces (CechPersistence, maybe a function providing the same functionality as AlphaComplex without a class, etc), using sqrt by default sounds good (again with a strong warning in the doc to point out the difference with the old interface), although we need to decide what to do exactly for the weighted case.

Some benchmark to compare std::sqrt inside functions (as done on this PR) and when std::sqrt is performed after construction with for_each_simplex. (cf. src/Alpha_complex/benchmark/Alpha_complex_1d_benchmark.cpp)

master - without sqrt

$ for seed in 40 41 42 43 44; do ./Alpha_complex_1d_benchmark 5000000 $seed; done
... create simplex tree from alpha complex: 7.906s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.398s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 7.837s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.369s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 7.906s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.37s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 7.876s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.38s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 7.887s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.373s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.

sqrt inside functions - cf. branch alpha_sqrt

$ for seed in 40 41 42 43 44; do ./Alpha_complex_1d_benchmark 5000000 $seed; done
... create simplex tree from alpha complex: 7.975s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.43s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 8.021s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.431s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 7.866s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.401s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 7.927s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.395s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 7.87s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.394s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.

for_each_simplex(sqrt) - cf. branch alpha_for_each_sqrt

$ for seed in 40 41 42 43 44; do ./Alpha_complex_1d_benchmark 5000000 $seed; done
... create simplex tree from alpha complex: 8.494s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.499s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 8.099s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.496s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 8.345s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.529s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 8.157s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.468s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... create simplex tree from alpha complex: 8.15s
Alpha complex is of dimension 1 - 9999999 simplices - 5000000 vertices.
... simplex tree assign MEB: 2.391s
Delaunay-Cech complex is of dimension 1 - 9999999 simplices - 5000000 vertices.

GUDHI / gudhi-devel