 
Summary: ConstantLevel Greedy Triangulations
Approximate the MWT Well
Oswin Aichholzer 1 , Franz Aurenhammer 1 , G¨unter Rote 2 , YinFeng Xu 3
1 Institute for Theoretical Computer Science, Graz University of Technology,
Klosterwiesgasse 32/2, A8010 Graz, Austria
2 Institut f¨ur Mathematik, Technische Universit¨at Graz,
Steyrergasse 30, A8010 Graz, Austria
3 School of Management, Xi'an Jiaotong University,
Xi'an Shaanxi, 710049, P. R. China
Abstract. The wellknown 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 socalled 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 jGT (S)j Ÿ 3 \Delta 2 k+1 jMWT (S)j, where k is the number of levels
and MWT (S) is the minimumweight triangulation of S.
1 Introduction
A triangulation of a given set S of n points in the plane is a maximal set of non
