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 new way of defining GT (S). The new definition does not rely on the length ordering of the edges. 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 jGT (S)j 3 \Delta 2 k+1 jMWT (S)j, where k is the number of levels and MWT (S) is the minimum-weight triangulation of S. This constitutes the first non-trivial upper bound on jGT (S)j for general points sets S