Delaunay Triangulations in O(sort(n)) Time and More

作者：Buchin Kevin^*; Mulzer Wolfgang

来源：Journal of the ACM, 2011, 58(2): 6.

DOI：10.1145/1944345.1944347

• Algorithms
• Theory
• Convex hull
• Delaunay triangulation
• word RAM

We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports the shuffle operation in constant time; (ii) if we know the ordering of a planar point set in x-and in y-direction, its DT can be found by a randomized algebraic computation tree of expected linear depth; (iii) given a universe U of points in the plane, we construct a data structure D for Delaunay queries: for any P subset of U, D can find the DT of P in expected time O(vertical bar P vertical bar log log vertical bar U vertical bar); (iv) given a universe U of points in 3-space in general convex position, there is a data structure D for convex hull queries: for any P subset of U, D can find the convex hull of P in expected time O(vertical bar P vertical bar(log log vertical bar U vertical bar)(2)); (v) given a convex polytope in 3-space with n vertices which are colored with chi >= 2 colors, we can split it into the convex hulls of the individual color classes in expected time O(n(log log n)(2)).
The results (i)-(iii) generalize to higher dimensions, where the expected running time now also depends on the complexity of the resulting DT. We need a wide range of techniques. Most prominently, we describe a reduction from DTs to nearest-neighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling.

• 出版日期2011-4