By Topic

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

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

2 Author(s)
Buchin, K. ; Dept. of Math. & Comput. Sci., Tech. Univ. Eindhoven, Eindhoven, Netherlands ; Mulzer, W.

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 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 time O(|P|(log log|U|)2); (v) given a convex polytope in 3-space with n vertices which are colored with ¿ > 2 colors, we can split it into the convex hulls of the individual color classes in time O(n(log log n)2). The results (i)-(iii) generalize to higher dimensions. 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.

Published in:

Foundations of Computer Science, 2009. FOCS '09. 50th Annual IEEE Symposium on

Date of Conference:

25-27 Oct. 2009