Two- and three-dimensional point location in rectangular subdivisions
Journal of Algorithms
Randomized algorithms
Separators for sphere-packings and nearest neighbor graphs
Journal of the ACM (JACM)
Improved parallel integer sorting without concurrent writing
Information and Computation
A comparison of sequential Delaunay triangulation algorithms
Computational Geometry: Theory and Applications
Algorithmic geometry
Journal of Computer and System Sciences
Faster deterministic sorting and priority queues in linear space
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Closest-point problems simplified on the RAM
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Integer Sorting in 0(n sqrt (log log n)) Expected Time and Linear Space
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Priority Queues: Small, Monotone and Trans-dichotomous
ESA '96 Proceedings of the Fourth Annual European Symposium on Algorithms
Incremental constructions con BRIO
Proceedings of the nineteenth annual symposium on Computational geometry
Deterministic sorting in O(nlog logn) time and linear space
Journal of Algorithms
A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces
Discrete & Computational Geometry
Streaming computation of Delaunay triangulations
ACM SIGGRAPH 2006 Papers
Complexity of Delaunay triangulation for points on lower-dimensional polyhedra
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Well-separated pair decomposition in linear time?
Information Processing Letters
Preprocessing Imprecise Points and Splitting Triangulations
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Computing hereditary convex structures
Proceedings of the twenty-fifth annual symposium on Computational geometry
Markov Incremental Constructions
Discrete & Computational Geometry - Special Issue: 24th Annual Symposium on Computational Geometry
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
SIAM Journal on Computing
HCPO: an efficient insertion order for incremental Delaunay triangulation
Information Processing Letters
SIAM Journal on Computing
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
A faster algorithm for computing motorcycle graphs
Proceedings of the twenty-ninth annual symposium on Computational geometry
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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.