Which Triangulations Approximate the Complete Graph?

A b s t r a c t. Chew and Dobkin et. al. have shown that the Delaunay triangulation and its variants are sparse approximations of the complete graph, in that the shortest distance between two sites within the triangulation is bounded by a constant multiple of their Euclidean separation. In this paper, we show that other classical triangulation algorithms, such as the greedy triangulation, and more notably, the minimum weight triangulation, also approximate the complete graph in this sense. We also design an algorithm for constructing extremely sparse (nontriangular) planar graphs that approximate the complete graph. We define a sufficiency condition and show that any Euclidean planar graph constructing algorithm which satisfies this condition always produces good approximations of the complete graph. This condition is quite general because it is satisfied by all the tri-angulation algorithms mentioned above, and probably by many other graph algorithms as well. We thus partially answer the question posed by the title. From a theoretical standpoint, our results are interesting because we prove non-trivial properties of minimum weight triangulations, of which little is currently known. From a practical standpoint, the graph algorithms we consider are good alternatives to the Delaunay triangulation, particularly when designing a sparse network under severe constraints on the total edge length. Finally, our general approach may help in identifying or designing other algorithms for constructing sparse networks. The complete graph represents an ideal communication network between n sites. However , to conserve resources, sparse networks are often designed which approximate the complete graph in some sense. This problem has also been studied from a geometric context, where the graphs axe Euclidean. In particular, recent research [C, DFS] has shown that the Delaunay triangulation and its variants are good approximations