Constant-Level Greedy Triangulations Approximate the MWT Well

The well-known greedy triangulation GT(S) of a finite point set S is obtained by inserting compatible edges in increasing length order, where an edge is compatible if it does not cross previously inserted ones. Exploiting the concept of so-called light edges, we introduce a definition of GT(S) that does not rely on the length ordering of the edges. Rather, it provides a decomposition of GT(S) into levels, and the number of levels allows us to bound the total edge length of GT(S). In particular, we show |GT(S)| ≤ 3 · 2k+1|MWT(S)|, where k is the number of levels and MWT(S) is the minimum-weight triangulation of S.