## Linear-time reconstruction of Delaunay triangulations with applications (1997)

Venue: | In Proc. Annu. European Sympos. Algorithms, number 1284 in Lecture Notes Comput. Sci |

Citations: | 21 - 3 self |

### BibTeX

@INPROCEEDINGS{Snoeyink97linear-timereconstruction,

author = {Jack Snoeyink and Marc Van Kreveld},

title = {Linear-time reconstruction of Delaunay triangulations with applications},

booktitle = {In Proc. Annu. European Sympos. Algorithms, number 1284 in Lecture Notes Comput. Sci},

year = {1997},

pages = {459--471},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Many of the computational geometers' favorite data structures are planar graphs, canonically determined by a set of geometric data, that take \Theta(n log n) time to compute. Examples include 2-d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3-d convex hulls. Given such a structure, one can determine a permutation of the data in O(n) time such that the data structure can be reconstructed from the permuted data in O(n) time by a simple incremental algorithm. As a consequence, one can permute a data file to "hide" a geometric structure, such as a terrian model based on the Delaunay triangulation of a set of sampled points, without disrupting other applications. One can even include "importance" in the ordering so the incremental reconstruction produces approximate terrain models as the data is read or received. For the Delaunay triangulation, we can also handle input in degenerate position, even though the data structures may no longer be cano...

### Citations

706 | Algorithms in Combinatorial Geometry - Edelsbrunner - 1987 |

550 |
Computational Geometry in C
- O’Rourke
- 1994
(Show Context)
Citation Context ...rapezoids---by extending vertical segments from each given segment endpoint to the segments above and below---imposes structure on the plane that can be exploited by intersection and other algorithms =-=[17, 15]-=-. 4. The convex hull of n points in 3-space: The convex hull is the smallest convex set enclosing the points [17, 19]. Its surface can be represented as a planar graph. Each of these have construction... |

481 | Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams
- Guibas, Stolfi
- 1985
(Show Context)
Citation Context ...nates of three consecutive vertices in counter-clockwise order. drawTraverse() implements the traversal without making changes to the data structure. These are based on a quadedge-like data structure =-=[11]-=- that uses directed edges: Sym(e) reverses edge e, and Lnx(e) and Lpr(e) give the ccw and cw edges around the face to the left of e. Figure 2 shows the faces numbered in the order that the traversal e... |

450 |
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams
- Okabe, Boots, et al.
- 2000
(Show Context)
Citation Context ...canonically determined by n input data elements. Well-known examples include: 1. The Voronoi diagram of n point or line segment sites in the plane. This oftreinvented data structure records proximity =-=[1, 16]-=-; it is the decomposition of the plane into maximally-connected regions that have the same set of sites as closest neighbors. 2. The Delaunay triangulation of n points. This dual of the Voronoi diagra... |

289 |
Computational Geometry: An Introduction Through Randomized Algorithms
- Mulmuley
- 1994
(Show Context)
Citation Context ...rapezoids---by extending vertical segments from each given segment endpoint to the segments above and below---imposes structure on the plane that can be exploited by intersection and other algorithms =-=[17, 15]-=-. 4. The convex hull of n points in 3-space: The convex hull is the smallest convex set enclosing the points [17, 19]. Its surface can be represented as a planar graph. Each of these have construction... |

280 | Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms
- Edelsbrunner, Mücke
- 1990
(Show Context)
Citation Context ...erturbation methods described for the Delaunay triangulation, however, use a global index or memory address of a point. Since we permute the points, this is not an option. ("Simulation of Simplic=-=ity" [8]-=- can be implemented by using local indices determined by lexicographic ordering.) The Delaunay graph of a set S of Fig. 5: Spanning tree in Delaunay graph of degenerate point set points is a graph wit... |

262 | Optimal search in planar subdivisions
- Kirkpatrick
- 1983
(Show Context)
Citation Context ...x hull or 2-d Delaunay triangulation still requires\Omega\Gamma n log n) time. Djijev and Lingas [6] have shown the same for the Voronoi diagram. We use ideas from Kirkpatrick's point location scheme =-=[13]-=- to find a permutation of the n data elements in linear time so that a simple incremental algorithm can reconstruct the desired planar graph from the permuted data in O(n) time and space. We need only... |

188 | Mesh Generation and Optimal Triangulation
- Bern, Eppstein
- 1992
(Show Context)
Citation Context ...he Delaunay triangulation of n points. This dual of the Voronoi diagram joins two points with an edge if some circle touches those two sites and no other. The Delaunay is important in GIS and meshing =-=[2, 18]-=-. Supported in part by grants from NSERC and Facet Decision Systems y Supported in part by ESPRIT IV LTR project 21957 (CGAL). 3. The trapezoidation (vertical visibility map) of n line segments. Decom... |

181 |
Voronoi diagrams-a survey of a fundamental geometric data structure
- Aurenhammer
- 1991
(Show Context)
Citation Context ...canonically determined by n input data elements. Well-known examples include: 1. The Voronoi diagram of n point or line segment sites in the plane. This oftreinvented data structure records proximity =-=[1, 16]-=-; it is the decomposition of the plane into maximally-connected regions that have the same set of sites as closest neighbors. 2. The Delaunay triangulation of n points. This dual of the Voronoi diagra... |

163 |
Randomized incremental construction of Delaunay and Voronoi diagrams
- Guibas, Knuth, et al.
- 1992
(Show Context)
Citation Context ...ependent insertions. Fig. 1: TIN in Java implementation This result is a natural combination of work in hierarchical representation of planar subdivisions [13] and incremental construction algorithms =-=[10, 21]-=-. It can be seen as a form of geometry compression and progressive expansion, which is of interest in computer graphics [5, 12]. We illustrate the idea with one application: compressing and transmitti... |

160 |
Optimal point location in a monotone subdivision
- Edelsbrunner, Guibas, et al.
- 1986
(Show Context)
Citation Context ...s of any spanning tree of vertices; the duals of the edges that remain form a spanning tree of the faces. This observation can be used to give a simple traversal of any planar graph with convex faces =-=[7, 4]-=-. As a bonus, we can use this traversal to do hidden surface elimination by the painter's algorithm. Observation 4 Any planar graph with convex faces has a canonical traversal that requires no data st... |

59 | Backwards analysis of randomized geometric algorithms
- Seidel
- 1993
(Show Context)
Citation Context ...ependent insertions. Fig. 1: TIN in Java implementation This result is a natural combination of work in hierarchical representation of planar subdivisions [13] and incremental construction algorithms =-=[10, 21]-=-. It can be seen as a form of geometry compression and progressive expansion, which is of interest in computer graphics [5, 12]. We illustrate the idea with one application: compressing and transmitti... |

50 |
Computational Geometry--An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ... structure on the plane that can be exploited by intersection and other algorithms [17, 15]. 4. The convex hull of n points in 3-space: The convex hull is the smallest convex set enclosing the points =-=[17, 19]-=-. Its surface can be represented as a planar graph. Each of these have construction algorithms that run in optimal \Theta(n log n) time; the lower bounds are proved by reduction from sorting [14, 19].... |

44 | Dynamic trees and dynamic point location
- Goodrich, Tamassia
- 1991
(Show Context)
Citation Context ...s of any spanning tree of vertices; the duals of the edges that remain form a spanning tree of the faces. This observation can be used to give a simple traversal of any planar graph with convex faces =-=[4, 9]-=-. As a bonus, we can use this traversal to do hidden surface elimination by the painter's algorithm. Observation 4 Any planar graph with convex faces has a canonical traversal that requires no data st... |

41 |
The triangulated irregular network
- PEUCKER, FOWLER, et al.
- 1978
(Show Context)
Citation Context ...he Delaunay triangulation of n points. This dual of the Voronoi diagram joins two points with an edge if some circle touches those two sites and no other. The Delaunay is important in GIS and meshing =-=[2, 18]-=-. Supported in part by grants from NSERC and Facet Decision Systems y Supported in part by ESPRIT IV LTR project 21957 (CGAL). 3. The trapezoidation (vertical visibility map) of n line segments. Decom... |

31 |
Efficiently Four-Coloring Planar Graphs
- Robertson, Sanders, et al.
- 1996
(Show Context)
Citation Context ...4or 5-colorings could be used to obtain independent sets guaranteeing 1/4 or 1/5 of the vertices. The best algorithm known for 4-coloring a planar graph takes quadratic time and must handle 633 cases =-=[20], but 5-co-=-loring in linear time is not especially difficult [3, 20]. The greedy algorithm above does avoid high degree vertices; it also gives us more freedom to select "important points" for our terr... |

28 | Simple traversal of a subdivision without extra storage”, Simple traversal of a subdivision without extra storage, Int
- Berg, Kreveld, et al.
- 1997
(Show Context)
Citation Context ...s of any spanning tree of vertices; the duals of the edges that remain form a spanning tree of the faces. This observation can be used to give a simple traversal of any planar graph with convex faces =-=[4, 9]-=-. As a bonus, we can use this traversal to do hidden surface elimination by the painter's algorithm. Observation 4 Any planar graph with convex faces has a canonical traversal that requires no data st... |

21 | Computational Geometry – A Survey
- Lee, Preparata
- 1984
(Show Context)
Citation Context ... [17, 19]. Its surface can be represented as a planar graph. Each of these have construction algorithms that run in optimal \Theta(n log n) time; the lower bounds are proved by reduction from sorting =-=[14, 19]-=-. Presorting the data in natural orders does not necessarily reduce the construction time. Seidel [22] has shown that after sorting by x-coordinate, computing a 3-d convex hull or 2-d Delaunay triangu... |

17 | Finding the Constrained Delaunay Triangulation and Constrained Voronoi Diagram of a Simple Polygon
- Wang, Chin
- 1995
(Show Context)
Citation Context ...t set to low-degree vertices, then Step P3 is linear by independent retriangulation of polygons of constant size. Otherwise, we can use the linear-time constrained-Delaunay algorithm of Wang and Chin =-=[23]-=-. Step P2, which is standard in hierarchical triangulations, implies that the number of vertices decreases by a constant in each phase. Using the 1/6 guaranteed in Section 4, the total time for all ph... |

15 |
Progressive meshes, Computer Graphics
- Hoppe
- 1996
(Show Context)
Citation Context ...ion of planar subdivisions [13] and incremental construction algorithms [10, 21]. It can be seen as a form of geometry compression and progressive expansion, which is of interest in computer graphics =-=[5, 12]-=-. We illustrate the idea with one application: compressing and transmitting terrain models. TIN (triangulated irregular network) terrain models [18] in geographic information systems (GIS) often fit a... |

14 |
A Method for Proving Lower Bounds for certain Geometric
- Seidel
- 1985
(Show Context)
Citation Context ...at run in optimal \Theta(n log n) time; the lower bounds are proved by reduction from sorting [14, 19]. Presorting the data in natural orders does not necessarily reduce the construction time. Seidel =-=[22]-=- has shown that after sorting by x-coordinate, computing a 3-d convex hull or 2-d Delaunay triangulation still requires\Omega\Gamma n log n) time. Djijev and Lingas [6] have shown the same for the Vor... |

11 |
On computing the Voronoi diagram for restricted planar figures
- Djidjev, Lingas
- 1991
(Show Context)
Citation Context ... the construction time. Seidel [22] has shown that after sorting by x-coordinate, computing a 3-d convex hull or 2-d Delaunay triangulation still requires\Omega\Gamma n log n) time. Djijev and Lingas =-=[6]-=- have shown the same for the Voronoi diagram. We use ideas from Kirkpatrick's point location scheme [13] to find a permutation of the n data elements in linear time so that a simple incremental algori... |

10 | A linear 5-coloring algorithm of planar graphs
- Chiba, Nishizeki, et al.
- 1981
(Show Context)
Citation Context ...eeing 1/4 or 1/5 of the vertices. The best algorithm known for 4-coloring a planar graph takes quadratic time and must handle 633 cases [20], but 5-coloring in linear time is not especially difficult =-=[3, 20]. The gree-=-dy algorithm above does avoid high degree vertices; it also gives us more freedom to select "important points" for our terrain model application. 5 Dealing with degenerate point sets Thus fa... |

5 |
Geometry compression, Computer Graphics
- Deering
- 1995
(Show Context)
Citation Context ...ion of planar subdivisions [13] and incremental construction algorithms [10, 21]. It can be seen as a form of geometry compression and progressive expansion, which is of interest in computer graphics =-=[5, 12]-=-. We illustrate the idea with one application: compressing and transmitting terrain models. TIN (triangulated irregular network) terrain models [18] in geographic information systems (GIS) often fit a... |