#1227 - Efficient subset check between CartesianProducts

[schillic]

Closes #1227.

Additionally this PR adds a subset check between CartesianProductArrays of different block structure by using the fallback implementation.

Tests were already present.

# two non-subset blocks

julia> @time X ⊆ Y
  0.000024 seconds (111 allocations: 7.453 KiB)
false

julia> @time ⊆(X, Y, true)
  0.000032 seconds (151 allocations: 9.891 KiB)
(false,   [1]  =  -0.989371
  [2]  =  0.890032
  [3]  =  1.15984
  [4]  =  2.03698)

### this branch

julia> @time X ⊆ Y
  0.000003 seconds
false

julia> @time ⊆(X, Y, true)
  0.000005 seconds (4 allocations: 272 bytes)
(false, [-0.9893707748634932, 0.6090025573953656, 0.6279955152649557, 0.38677706906864234])


# first block is a subset

julia> @time X ⊆ Y
  0.000025 seconds (129 allocations: 8.672 KiB)
false

julia> @time ⊆(X, Y, true)
  0.000030 seconds (163 allocations: 10.703 KiB)
(false,   [1]  =  -0.427312
  [2]  =  0.890032
  [3]  =  1.15984
  [4]  =  2.03698)

### this branch

julia> @time X ⊆ Y
  0.000003 seconds
false

julia> @time ⊆(X, Y, true)
    0.000005 seconds (6 allocations: 384 bytes)
(false, [-0.23845042779673375, 0.6386840809336349, 1.159841586170895, 0.6386840809336349])


# both blocks are subsets

julia> @time X ⊆ Y
  0.000025 seconds (147 allocations: 9.891 KiB)
true

julia> @time ⊆(X, Y, true)
  0.000026 seconds (149 allocations: 10.000 KiB)
(true, Float64[])

### this branch

julia> @time X ⊆ Y
  0.000003 seconds
true

julia> @time ⊆(X, Y, true)
  0.000004 seconds (6 allocations: 336 bytes)
(true, Float64[])

[mforets]

there is a problem to extend ⊆(::CartesianProduct{N}, ::CartesianProduct{N}, ::Bool=false) .. without dispatch on the algorithm, in the sense that the new method doesn't apply for cases which have different block structure, but which are polytopic, so the fallback implementation (_issubset_constraints_list) applies.

one approach would be to handle the case where the blocks dims don't match instead of throwing an error. otoh, this pr shows that we don't test x ⊆ y if x = rand(Interval) × rand(BallInf) and y = rand(BallInf) × rand(Interval).

[schillic]

True, but the same holds for the CPA method. I copied that behavior as suggested in the issue. I can fix them here.

[mforets]

sorry, what i meant in my comment is that if we merge this PR there is a regression as some subset checks no longer work as the one i gave.

True, but the same holds for the CPA method.

ok, i see. then we should fix that too :)

[schillic]

ok, i see. then we should fix that too :)

done ⛑️

[mforets]

cool!

return invoke(⊆, Tuple{LazySet{N}, LazySet{N}, Bool}, X, Y, witness)

i think that using internal functions is better than invoke, but we can fix that later.

julia> @time ⊆(X, Y, true)
0.000005 seconds (6 allocations: 384 bytes)

when measuring small times (like hundreds of ms or smaller), @btime from BenchmarkTools is better because the differences shown can be just variation between runs.

[schillic]

when measuring small times (like hundreds of ms or smaller), @btime from BenchmarkTools is better because the differences shown can be just variation between runs.

Yeah, I mainly looked at the memory usage here.