#1769 - Faster box approximation of VPolytope

[schillic]

Closes #1769.

In 2D this implementation is less efficient for a moderate number of vertices due to the additional conversion from low/high in the Hyperrectangle constructor. I guess in higher dimensions it will be more useful. Should I remove the dispatch for VPolygon or add an invoke of the old method based on the number of vertices?

julia> P1 = VPolytope([[0.0, 0.0], [2.0, 1.0], [1.0, 2.0]]);
julia> P2 = VPolytope([rand(2) for _ in 1:100]);
julia> P3 = VPolytope([rand(2) for _ in 1:1000]);

# this branch

julia> @time box_approximation(P1)
[0.0, 0.0][2.0, 2.0]
  0.000008 seconds (5 allocations: 416 bytes)
Hyperrectangle{Float64,Array{Float64,1},Array{Float64,1}}([1.0, 1.0], [1.0, 1.0])
julia> @time box_approximation(P2)
[0.001830275380493962, 0.005165832293896333][0.9791389830947983, 0.9914106144943142]
    0.000009 seconds (5 allocations: 416 bytes)
Hyperrectangle{Float64,Array{Float64,1},Array{Float64,1}}([0.49048462923764613, 0.4982882233941053], [0.48865435385715217, 0.49312239110020895])
julia> @time box_approximation(P3)
  0.000024 seconds (5 allocations: 416 bytes)
Hyperrectangle{Float64,Array{Float64,1},Array{Float64,1}}([0.4991620587806421, 0.4983966049474178], [0.49909969477169713, 0.4978833437247012])

# master

julia> @time box_approximation(P1)
  0.000004 seconds (3 allocations: 224 bytes)
Hyperrectangle{Float64,Array{Float64,1},Array{Float64,1}}([1.0, 1.0], [1.0, 1.0])
julia> @time box_approximation(P2)
  0.000008 seconds (3 allocations: 224 bytes)
Hyperrectangle{Float64,Array{Float64,1},Array{Float64,1}}([0.49048462923764613, 0.4982882233941053], [0.48865435385715217, 0.49312239110020895])
julia> @time box_approximation(P3)
  0.000049 seconds (3 allocations: 224 bytes)
Hyperrectangle{Float64,Array{Float64,1},Array{Float64,1}}([0.4991620587806421, 0.4983966049474178], [0.49909969477169713, 0.4978833437247012])

[mforets]

i had a look, see this notebook. in low dims (usually <= 10) static arrays are much faster that regular arrays so it'd be nice that our code supports them. it is not the case with the new method but see the alternative implementation ba2 in that notebook. it is also faster for dim 2 and different number of vertices. for dimension 10, ba1 is faster, though i didn't make exhaustive checks!

[mforets]

In 2D this implementation is less efficient for a moderate number of vertices

here ba1 is faster than the version in master, while it allocates a bit more. see cells [15] and [9] in the notebook.

applies same comment as in #2241 (comment) for presenting benchmark results

[schillic]

I added the ideas from your implementation with minor modifications.

Side note: I always get one more allocation than in your notebook (but even when I use your exact version, so this is not a problem of my modifications). Maybe you want to double-check...

[mforets]

lgtm 🎉