Delaunay triangulations in O (sort( n )) time and more

Abstract

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 ⊆ U , D can find the DT of P in expected time O (| P | log log | U |); (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 ⊆ U , D can find the convex hull of P in expected time O (| P | (log log | U |) 2 ); (v) given a convex polytope in 3-space with n vertices which are colored with χ ≥ 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.