Output-Sensitive Construction of the Union of Triangles

We present an efficient algorithm for the following problem: Given a collection T = {δ ₁,...,δ _n} of n triangles in the plane, such that there exists a subset S ⊂ T (unknown to us), of ξ ≫ n triangles, such that ∪ _{δ∈ S}δ = ∪ _δ∈Tδ construct efficiently the union of the triangles in T. We show that this problem can be solved in subquadratic time. In our solution, we use the approximate Disjoint-Cover (DC) algorithm, presented as a heuristics in [9]. We present a detailed implementation of this method, which combines a variety of techniques related to range-searching in two dimensions. We provide a rigorous analysis of its performance in the above setting, showing that it does indeed run in subquadratic time (for a reasonable range of ξ).