Greedy is as Good as Delaunay (Almost)

TL;DR

This paper demonstrates that from a greedy triangulation of a planar point set, one can efficiently compute its Delaunay triangulation, establishing a near-equivalence between these structures with implications for geometric algorithms.

Contribution

It provides a partial converse to a known result, showing that greedy and Delaunay triangulations are essentially interchangeable in linear expected time under certain conditions.

Findings

Greedy triangulation allows linear expected time computation of Delaunay triangulation.

The algorithm generalizes to triangulations with bounded dilation.

Properties of greedy triangulation ensure linear time complexity.

Abstract

Let S be a planar point set. Krznaric and Levcopoulos proved that given the Delaunay triangulation DT(S) for S, one can find the greedy triangulation GT(S) in linear time. We provide a (partial) converse of this result: given GT(S), it is possible to compute DT(S) in linear expected time. Thus, these structures are basically equivalent. To obtain our result, we generalize another algorithm by Krznaric and Levcopoulos to find a hierarchical clustering for S in linear time, once DT(S) is known. We show that their algorithm remains (almost) correct for any triangulation of bounded dilation, i.e., any triangulation in which the shortest path distance between any two points approximates their Euclidean distance. In general, however, the resulting running time may be superlinear. Nonetheless, we can show that the properties of the greedy triangulation suffice to guarantee a linear time bound.