## A Comparison of Sequential Delaunay Triangulation Algorithms (1996)

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Citations: | 60 - 0 self |

### BibTeX

@MISC{Su96acomparison,

author = {Peter Su and Robert L. Scot Drysdale},

title = {A Comparison of Sequential Delaunay Triangulation Algorithms},

year = {1996}

}

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### Abstract

This paper presents an experimental comparison of a number of different algorithms for computing the Deluanay triangulation. The algorithms examined are: Dwyer’s divide and conquer algorithm, Fortune’s sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri, and Murota, a new bucketing-based algorithm described in this paper, and Devillers’s version of a Delaunay-tree based algorithm that appears in LEDA), an algorithm that incrementally adds a correct Delaunay triangle adjacent to a current triangle in a manner similar to gift wrapping algorithms for convex hulls, and Barber’s convex hull based algorithm. Most of the algorithms examined are designed for good performance on uniformly distributed sites. However, we also test implementations of these algorithms on a number of non-uniform distibutions. The experiments go beyond measuring total running time, which tends to be machine-dependent. We also analyze the major high-level primitives that algorithms use and do an experimental analysis of how often implementations of these algorithms perform each operation.

### Citations

491 | Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams
- Guibas, Stolfi
- 1985
(Show Context)
Citation Context ... how often implementations of these algorithms perform each operation. 1. Introduction Sequential algorithms for constructing the Delaunay triangulation come in five basic flavors: divideand -conquer =-=[8, 17]-=-, sweepline [11], incremental [7, 15, 17, 16, 20], growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls [9, 19, 25], and lifting the sites into three dimension... |

365 |
A sweepline algorithm for Voronoi diagrams
- Fortune
- 1987
(Show Context)
Citation Context ...tations of these algorithms perform each operation. 1. Introduction Sequential algorithms for constructing the Delaunay triangulation come in five basic flavors: divideand -conquer [8, 17], sweepline =-=[11]-=-, incremental [7, 15, 17, 16, 20], growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls [9, 19, 25], and lifting the sites into three dimensions and computing ... |

263 |
Integral Geometry and Geometric Probability
- Santalo
- 1976
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Citation Context ...ns in detail is that the behavior of Giftwrapping is heavily influenced by the nature of the convex hull of the input. In the square distribution, the expected number of convex hull edges is O(log n) =-=[22]-=-. The graph shows that the number of points tested per site stays constant over our test range, while the number of buckets examined actually decreases. This reflects the fact the number of edges on o... |

249 | Algorithms for parallel memory I: Two-level memories
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- 1994
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Citation Context ...here have been attempts at theoretical analysis of algorithms on complex memory systems, the models involved are generally complicated and the analysis of even simple algorithms is highly challenging =-=[1, 26, 14, 27]. Such ana-=-lysis is also made more difficult by the fact that the "memory system efficiency" of an algorithm is very dynamic and data dependent. Therefore, finding ways to combine a more tractable abst... |

168 |
Randomized incremental construction of Delaunay and Voronoi diagrams, Unpublished manuscript
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- 1990
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Citation Context ...lgorithms perform each operation. 1. Introduction Sequential algorithms for constructing the Delaunay triangulation come in five basic flavors: divideand -conquer [8, 17], sweepline [11], incremental =-=[7, 15, 17, 16, 20]-=-, growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls [9, 19, 25], and lifting the sites into three dimensions and computing their convex hull [2]. Which appr... |

152 |
Applications of random sampling
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ...lgorithms perform each operation. 1. Introduction Sequential algorithms for constructing the Delaunay triangulation come in five basic flavors: divideand -conquer [8, 17], sweepline [11], incremental =-=[7, 15, 17, 16, 20]-=-, growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls [9, 19, 25], and lifting the sites into three dimensions and computing their convex hull [2]. Which appr... |

121 |
Computing Dirichlet tesselation in the plane
- Green, Sibson
- 1977
(Show Context)
Citation Context ...lgorithms perform each operation. 1. Introduction Sequential algorithms for constructing the Delaunay triangulation come in five basic flavors: divideand -conquer [8, 17], sweepline [11], incremental =-=[7, 15, 17, 16, 20]-=-, growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls [9, 19, 25], and lifting the sites into three dimensions and computing their convex hull [2]. Which appr... |

108 |
Convex hulls of finite sets of points in two and three dimensions
- Preparata, Hong
- 1977
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Citation Context ...fi [17] note that their divide and conquer algorithm for computing the Deluanay triangulation can be viewed as a variant of Preparata and Hong's algorithm for computing three dimensional convex hulls =-=[21]-=-. Others have also used this approach. Recently Barber [2] has developed a Delaunay Triangulation algorithm based on a convex hull algorithm that he calls Quickhull. He combines the 2-dimensional divi... |

96 |
K-d trees for semidynamic point sets
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- 1990
(Show Context)
Citation Context ...place the buckets with a nearest-neighbor search structure that is less sensitive to the distribution of sites, so that it can perform non-local searches more efficiently. Bentley's adaptive k-d tree =-=[3]-=- is a good example of such a data structure. 9 2 10 2 11 2 12 2 13 2 14 2 15 2 16 2 17 14.5 15 15.5 16 16.5 17 17.5 CCW Tests Per Site CCW Tests Used by Dwyer Number of Points Figure 4: CCW orientatio... |

95 |
Voronoi diagrams and arrangements
- Edelsbrunner, Seidel
- 1986
(Show Context)
Citation Context ...ate insertions. 1.6. Convex Hull Based Algorithms Brown [6] was the first to establish a connection between Voronoi diagrams in dimension d and convex hulls in dimension d+ 1. Edelsbrunner and Seidel =-=[10]-=- later found a correspondence between Delaunay triangles of a set of sites in dimension 2 and downward-facing faces of the convex hull of those sites lifted onto a paraboloid of rotation in dimension ... |

72 |
Higher-Dimensional Voronoi Diagrams in Linear Expected Time
- Dwyer
- 1989
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Citation Context ... come in five basic flavors: divideand -conquer [8, 17], sweepline [11], incremental [7, 15, 17, 16, 20], growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls =-=[9, 19, 25]-=-, and lifting the sites into three dimensions and computing their convex hull [2]. Which approach is best in practice? This paper presents an experimental comparison of a number of these algorithms. M... |

68 |
Voronoi diagrams from convex hulls
- Brown
- 1979
(Show Context)
Citation Context ...the case when sites are distributed in the unit square and it avoids the extra overhead of managing a priority queue, especially avoiding duplicate insertions. 1.6. Convex Hull Based Algorithms Brown =-=[6]-=- was the first to establish a connection between Voronoi diagrams in dimension d and convex hulls in dimension d+ 1. Edelsbrunner and Seidel [10] later found a correspondence between Delaunay triangle... |

59 |
Faster divide-and-conquer algorithm for constructing Delaunay triangulations
- Dwyer
- 1987
(Show Context)
Citation Context ... how often implementations of these algorithms perform each operation. 1. Introduction Sequential algorithms for constructing the Delaunay triangulation come in five basic flavors: divideand -conquer =-=[8, 17]-=-, sweepline [11], incremental [7, 15, 17, 16, 20], growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls [9, 19, 25], and lifting the sites into three dimension... |

48 |
Stable maintenance of point-set triangulation in two dimensions
- Fortune
- 1990
(Show Context)
Citation Context ...uad-edge data structure and only two geometric primitives, a CCW orientation test and an in-circle test. These primitives are defined in terms of 3 by 3 and 4 by 4 determinants, respectively. Fortune =-=[12, 13]-=- shows how to compute these accurately with finite precision. Dwyer [8] showed that a simple modification of this algorithm runs in O(n log log n) expected time on uniformly distributed sites. Dwyer's... |

41 |
Uniform memory hierarchies
- Alpern, Carter, et al.
- 1990
(Show Context)
Citation Context ...here have been attempts at theoretical analysis of algorithms on complex memory systems, the models involved are generally complicated and the analysis of even simple algorithms is highly challenging =-=[1, 26, 14, 27]. Such ana-=-lysis is also made more difficult by the fact that the "memory system efficiency" of an algorithm is very dynamic and data dependent. Therefore, finding ways to combine a more tractable abst... |

30 |
A new algorithm for three dimensional Voronoi tessellation
- Tanemura, Ogawa, et al.
- 1983
(Show Context)
Citation Context ... come in five basic flavors: divideand -conquer [8, 17], sweepline [11], incremental [7, 15, 17, 16, 20], growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls =-=[9, 19, 25]-=-, and lifting the sites into three dimensions and computing their convex hull [2]. Which approach is best in practice? This paper presents an experimental comparison of a number of these algorithms. M... |

25 |
On the randomized construction of the Delaunay tree. Theoret
- Boissonnat, Teillaud
- 1993
(Show Context)
Citation Context ... total expected cost of Locate will be O(n log n) time. Sharir and Yaniv [23] prove a bound of about 12nH n + O(n). This structure is similar to the Delaunay tree described by Boissonnat and Teillaud =-=[5]-=-. 1.4. A Faster Incremental Construction Algorithm We present a Locate variant that leads to an easily implemented incremental algorithm that seems to perform better than those mentioned above when th... |

24 |
Optimal expected time algorithms for closest point problems
- Bentley, Weide, et al.
- 1980
(Show Context)
Citation Context ...hose mentioned above when the input is uniformly distributed. We use a simple bucketing algorithm similar to the one that Bentley, Weide and Yao used for finding the nearest neighbor of a query point =-=[4]-=-. This leads to an O(n) time algorithm while maintaining the relative simplicity of the incremental algorithm. The bucketing scheme places the sites into a uniform grid as it adds them to the diagram.... |

20 |
Delaunay triangulation and the convex hull of n points in expected linear time
- Maus
- 1984
(Show Context)
Citation Context ... come in five basic flavors: divideand -conquer [8, 17], sweepline [11], incremental [7, 15, 17, 16, 20], growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls =-=[9, 19, 25]-=-, and lifting the sites into three dimensions and computing their convex hull [2]. Which approach is best in practice? This paper presents an experimental comparison of a number of these algorithms. M... |

17 |
Improvements of the incremental method for the Voronoi diagram with computational comparison of various algorithm
- Ohya, Iri, et al.
- 1984
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Citation Context |

13 |
Efficient memory access in large-scale computation
- Vitter
- 1991
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Citation Context ...here have been attempts at theoretical analysis of algorithms on complex memory systems, the models involved are generally complicated and the analysis of even simple algorithms is highly challenging =-=[1, 26, 14, 27]. Such ana-=-lysis is also made more difficult by the fact that the "memory system efficiency" of an algorithm is very dynamic and data dependent. Therefore, finding ways to combine a more tractable abst... |

10 |
Computational Geometry with Imprecise Data and Arithmetic
- Barber
- 1993
(Show Context)
Citation Context ...15, 17, 16, 20], growing a triangle at a time in a manner similar to gift wrapping algorithms for convex hulls [9, 19, 25], and lifting the sites into three dimensions and computing their convex hull =-=[2]-=-. Which approach is best in practice? This paper presents an experimental comparison of a number of these algorithms. Many of these algorithms were designed for good performance on uniformly distibute... |

10 | Efficient Parallel Algorithms for Closest Point Problems
- Su
- 1994
(Show Context)
Citation Context ... of this section briefly describes the various algorithmic approaches. More detailed descriptions of the algorithms, including pseudocode, can be found in Chapter 2 of the first author's Ph.D. thesis =-=[24]-=-. This study was supported in part by the funds of the National Science Foundation, DDM-9015851, and by a Fulbright Foundation fellowship. 1 1.1. Divide-and-Conquer Guibas and Stolfi [17] gave an O(n ... |

8 |
Constructing Delaunay triangulations by merging buckets in quadtree order
- Katajainen, Koppinen
- 1988
(Show Context)
Citation Context ...rges the strips together along vertical lines. His experiments indicate that in practice this algorithm runs in linear expected time. Another version of this algorithm, due to Katajainen and Koppinen =-=[18], merges s-=-quare buckets together in a "quad-tree" order. They show that this algorithm runs in linear expected time for uniformly distributed sites. In fact, their experiments show that the performanc... |

4 |
Numerical stability of algorithms for delaunay triangulations and voronoi diagrams
- Fortune
- 1992
(Show Context)
Citation Context ...uad-edge data structure and only two geometric primitives, a CCW orientation test and an in-circle test. These primitives are defined in terms of 3 by 3 and 4 by 4 determinants, respectively. Fortune =-=[12, 13]-=- shows how to compute these accurately with finite precision. Dwyer [8] showed that a simple modification of this algorithm runs in O(n log log n) expected time on uniformly distributed sites. Dwyer's... |

3 |
Randomized incremental construction of delaunay diagrams: Theory and practice
- Sharir, Yaniv
- 1991
(Show Context)
Citation Context ...e level to one of a constant number of triangles that might contain the site at the next level. It is not hard to show that the total expected cost of Locate will be O(n log n) time. Sharir and Yaniv =-=[23]-=- prove a bound of about 12nH n + O(n). This structure is similar to the Delaunay tree described by Boissonnat and Teillaud [5]. 1.4. A Faster Incremental Construction Algorithm We present a Locate var... |

1 | Delaunay triangulation and the convex hull ofnpoints in expected linear time - Maus - 1984 |