Splitting a Delaunay Triangulation in Linear Time

Chazelle, Bernard; Devillers, Olivier; Hurtado, Ferrán; Mora, Mercè; Sacristán, Vera; Teillaud, Monique

doi:10.1007/s00453-002-0939-8

Cited by 26 publications

(8 citation statements)

References 21 publications

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“…The problem is then to "split" P and compute both monochromatic convex hulls. We show how to do this in linear time, which answers the main open question in [10]. We extend our techniques to an arbitrary number of colors by showing how to compute the convex hulls of all the color classes in O(n √ log n ) time.…”

Section: Introductionmentioning

confidence: 77%

“…A simple geometric argument now shows that the merged list can be derived from the convex hull of the union in linear time, see Figure 6. Figure 6: The lower bound for (5,9,12,14), (1,8), (2,4,6,7,10,13), and (3,11). Edges between consecutive elements are bold.…”

Section: Union Of Hullsmentioning

confidence: 99%

“…Given a set of n points in the Euclidean plane and its Voronoi diagram, it was shown in [10] how to compute the Voronoi diagrams of any given subset in linear time. 1 The authors asked whether the convex hull of an arbitrary subset of the vertices of a convex 3-polytope can be computed in linear time, too.…”

Section: Introductionmentioning

confidence: 99%

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Computing hereditary convex structures

Chazelle

Mulzer

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

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Color red and blue the n vertices of a convex polytope P in R 3 . Can we compute the convex hull of each color class in o(n log n)? What if we have χ > 2 colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of P inside it blue: can the convex hull of the blue points be computed in time linear in their number? More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them. In particular, we resolve an eight-year old open problem by showing how to split a convex polytope in linear expected time.

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Section: Introductionmentioning

confidence: 77%

Section: Union Of Hullsmentioning

confidence: 99%

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Computing hereditary convex structures

Chazelle

Mulzer

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

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“…So the total time for computing all nearest unrelated points is O(n log 2 n). (It is possible to build the Voronoi diagrams more quickly by computing a single Voronoi diagram of all of the points and using it to guide the construction of the Voronoi diagrams of the subsets-see [4,9]-but this would not speed up the overall algorithm because of the point location time. )…”

Section: Borůvka's Algorithm and Binary Numberingmentioning

confidence: 99%

Spanning Trees in Multipartite Geometric Graphs

et al. 2017

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Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let n = |R ∪ B|. A bichromatic spanning tree is a spanning tree in the complete bipartite geometric graph with bipartition (R, B). The minimum (respectively maximum) bichromatic spanning tree problem is the problem of computing a bichromatic spanning tree of minimum (respectively maximum) total edge length.1. We present a simple algorithm that solves the minimum bichromatic spanning tree problem in O(n log 3 n) time. This algorithm can easily be extended to solve the maximum bichromatic spanning tree problem within the same time bound. It also can easily be generalized to multicolored point sets. 2. We present Θ(n log n)-time algorithms that solve the minimum and the maximum bichromatic spanning tree problems. 3. We extend the bichromatic spanning tree algorithms and solve the multicolored version of these problems in O(n log n log k) time, where k is the number of different colors (or the size of the multipartition in a complete multipartite geometric graph).

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“…The key point is how to connect the related discrete points. The Delaunay grid (Chazelle et al, 2002;Clarkson & Varadarajan 2007) has an excellent spatial neighboring relationship. In this chapter, the Delaunay triangle or tetrahedron is used as the fundamental geometrical element.…”

Section: Cluster Construction and Analysis Methodsmentioning

confidence: 99%