A Simple Linear Time Greedy Triangulation Algorithm for Uniformly Distributed Points

The greedy triangulation (GT) of a set S of n points in the plane is the triangulation obtained by starting with the empty set and at each step adding the shortest compatible edge between two of the points, where a compatible edge is de ned to be an edge that crosses none of the previously added edges. In this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time O(n) and space O(n), for n points drawn independently from a uniform distribution over some xed convex shape C. This algorithm is an improvement of the O(n logn) algorithmof Dickerson, Drysdale, McElfresh, and Welzl [7]. It uses their basic approach, but generates onlyO(n) plausible greedy edges instead of O(n logn). It uses some ideas similar to those presented in Levcopoulos and Lingas's O(n) expected time algorithm [18]. Since we use more knowledge about the structure of a random point set and its greedy triangulation, our algorithm needs only elementary data structures and simple bucketing techniques. Thus it is a good deal simpler to explain and to implement than the algorithm of [18].