Computing the Union of 3-Colored Triangles

Given is a set \s\ of $n$ points, each colored with one of $k \geq 3$ colours. We say that a triangle defined by three points of \s\ is 3-colored if its vertices have distinct colours. We prove in this paper that the problem of constructing the boundary of the union \ts\ of all such 3-colored triangles can be done in optimal $O(n \log n)$ time.