#1305 - Set difference between axis-aligned boxes (#1790)

* add difference for hyperrectangular sets * update docs and add test for setdiff of hyperrectangles * Update src/ConcreteOperations/difference.jl Co-Authored-By: Christian Schilling <[email protected]> * Update difference.jl * Update unit_Hyperrectangle.jl
JuliaReach · Nov 23, 2019 · 1525fa8 · 1525fa8
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diff --git a/docs/src/lib/binary_functions.md b/docs/src/lib/binary_functions.md
@@ -126,4 +126,5 @@ pontryagin_difference
 ```@docs
 \(::LazySet, ::LazySet)
 difference(::IN, ::IN) where {N, IN<:Interval{N}}
+difference(::AbstractHyperrectangle{N}, ::AbstractHyperrectangle{N}) where {N}
 ```
diff --git a/src/ConcreteOperations/difference.jl b/src/ConcreteOperations/difference.jl
@@ -3,6 +3,8 @@ export difference
 # alias for set difference
 import Base: \
 
+const IA = IntervalArithmetic
+
 """
     \\(X::LazySet, Y::LazySet)
 
@@ -111,3 +113,42 @@ function difference(I1::IN, I2::IN)::Union{EmptySet{N}, IN, UnionSet{N, IN, IN}}
         end
     end
 end
+
+# ============================================
+# Set difference between hyperrectangular sets
+# ============================================
+
+"""
+    difference(X::AbstractHyperrectangle{N}, Y::AbstractHyperrectangle{N}) where {N}
+
+Return the set difference between the given hyperrectangular sets.
+
+### Input
+
+- `X` -- first hyperrectangular set
+- `Y` -- second hyperrectangular set
+
+The set difference is defined as:
+
+```math
+    X \\setminus Y = \\{x: x ∈ X \\text{ and } x ∉ Y \\}
+```
+
+### Output
+
+A `UnionSetArray` consisting of the union of hyperrectangles. Note that this
+union is in general not convex.
+
+### Algorithm
+
+This function calls the implementation in `IntervalArithmetic.setdiff`.
+
+### Notes
+
+The backslash symbol, `\\`, can be used as an alias.
+"""
+function difference(X::AbstractHyperrectangle{N}, Y::AbstractHyperrectangle{N}) where {N}
+    Xib = convert(IA.IntervalBox, X)
+    Yib = convert(IA.IntervalBox, Y)
+    return UnionSetArray(convert.(Hyperrectangle, IA.setdiff(Xib, Yib)))
+end
diff --git a/test/unit_Hyperrectangle.jl b/test/unit_Hyperrectangle.jl
@@ -230,4 +230,13 @@ for N in [Float64, Rational{Int}, Float32]
     H1 = Hyperrectangle(N[0, 1], N[2, 3])
     H2 = Hyperrectangle(N[3, 2], N[1, 0])
     @test minkowski_sum(H1, H2) == Hyperrectangle(N[3, 3], N[3, 3])
+
+    # set difference
+    h = Hyperrectangle(low=N[0], high=N[1])
+    q = Hyperrectangle(low=N[0], high=N[0.5])
+    @test convert(Interval, difference(h, q).array[1]) == Interval(N(0.5), N(1))
+
+    # another set difference test in higher-dimensions
+    b = BallInf(N[0, 0, 0], N(1))
+    @test isempty(difference(b, b))
 end