## Voronoi Diagrams (0)

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Venue: | Handbook of Computational Geometry |

Citations: | 144 - 20 self |

### BibTeX

@INPROCEEDINGS{Aurenhammer_voronoidiagrams,

author = {Franz Aurenhammer and Technische Universität Graz and Rolf Klein and Fernuniversität Hagen and Praktische Informatik Vi},

title = {Voronoi Diagrams},

booktitle = {Handbook of Computational Geometry},

year = {},

pages = {201--290},

publisher = {Elsevier Science Publishers B.V. North-Holland}

}

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### Abstract

Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3-space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.

### Citations

1760 | Computational Geometry: An Introduction - Preparata - 1985 |

565 | Voronoi diagrams – a survey of a fundamental geometric data structure
- Aurenhammer
(Show Context)
Citation Context ...hem to the field. The reader interested in a complete overview over the existing literature should consult the book by Okabe et al. [210] who list more than 600 papers, and the surveys by Aurenhammer =-=[27]-=-, Bernal [39], and Fortune [124]. Also, chapters 5 and 6 of Preparata and Shamos [215] and chapter 13 of Edelsbrunner [104] could be consulted. Within this treatise, we cannot review all known results... |

429 | Spatial tessellations: concepts and applications of Voronoi diagrams - Okabe, Boots, et al. - 2000 |

387 | Applications of random sampling in computational geometry
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ... efforts have been made to compute the single planar order-k Voronoi diagram efficiently. Different approaches have been taken in Lee [180], Chazelle and Edelsbrunner [59], Aurenhammer [26], Clarkson =-=[71]-=-, and Agarwal et al. [2]. In the last two papers randomized runtimes of O(kn1+ε ) and (roughly) O(k(n − k)logn) are achieved, respectively, which is close to optimal. Below we describe a (roughly) O(k... |

333 | A sweepline algorithm for Voronoi diagrams - Fortune - 1986 |

271 |
Algorithms for reporting and counting geometric intersections
- Bentley, Ottmann
- 1979
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Citation Context ... edges that divides S. They show experimentally that their implementation is comparable in work to the best sequential algorithms. 3.4 Sweep The well–known line sweep algorithm by Bentley and Ottmann =-=[34]-=- computes the intersections of n line segments in the plane by moving a vertical line, H, acrossthe plane. The line segments currently intersected by H are stored in bottom-up order. This order must b... |

271 |
Worst-case analysis of a new heuristic for the travelling salesman problem
- Christofides
- 1976
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Citation Context ...t neccessarily the minimum, spanning tree of S. ✷ A(S) can be constructed in linear time from MST(S). A more sophisticated construction of a spanning cycle by means of MST(S) is given in Christofides =-=[69]-=-. An approximation factor of 1.5 is achieved, at an expense of roughly O(n 2√ n)inconstruction time. Another NP-complete problem, which has a factor-2 approximation by means of MST(S), is the construc... |

186 | Provably good mesh generation
- Bern, Eppstein, et al.
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Citation Context ...space can be solved by means of DT(S). The prohibitive size of DT(S) in d-space is reduced by augmenting S to have a linear-size, bounded degree Delaunay triangulation, with the method of Bern et al. =-=[38]-=-. 63s5.1.3 Largest empty and smallest enclosing circle Suppose someone wants to build a new residence within a given area, as far away as possible from each of n sources of disturbance. If the area is... |

179 | Mesh generation and optimal triangulation
- Bern, Eppstein
- 1992
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Citation Context ... in DT(S ∪ C), where C is the total set of added sites. For several site adding algorithms, |C| depends on the size as well as on the geometry of L. See, e.g., the survey article by Bern and Eppstein =-=[37]-=- and references therein. Edelsbrunner and Tan [118] show that |C| = O(k 2 n) is always sufficient, and construct a set of sites with this size in time O(k 2 n + n 2 ), for k = |L|. A different, and mo... |

171 | Constrained Delaunay Triangulation
- Chew
- 1987
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Citation Context ...ve b(p, q) < ∞ and that lie on a circle enclosing only sites r ∈ S with at least one of b(r, p),b(r, q) =∞. Algorithms for computing V (S, L) andDT(S, L) have been proposed in Lee and Lin [182], Chew =-=[63]-=-, Wang and Schubert [255], Wang [254], Seidel [228], and Kao and Mount [155]. The last two methods seem best suited to implementation. For an application of DT(S, L) to quality mesh generation see Che... |

152 | The discrete geodesic problem - Mitchell, Mount, et al. - 1987 |

148 |
Guaranteed-quality mesh generation for curved surfaces
- Chew
- 1993
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Citation Context ...Wang and Schubert [255], Wang [254], Seidel [228], and Kao and Mount [155]. The last two methods seem best suited to implementation. For an application of DT(S, L) to quality mesh generation see Chew =-=[65]-=-. We sketch the O(n log n) time plane sweep approach in [228]. If only V (S, L) is required then the plane sweep algorithm described in subsection 3.4 can be applied without much modification. If DT(S... |

119 |
There are planar graphs almost as good as the complete graph
- Chew
- 1989
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Citation Context ... Delaunay dual of generalized Voronoi diagrams. The convex distance function (subsection 4.5.2) whose unit circle is an equilateral triangle leads to a triangulation with t =2. This was shown in Chew =-=[64]-=-, along with the following result concerning the constrained Delaunay triangulation (subsection 4.4) for the L1-metric: There is always a path between two given sites, whose length is at most √ 10 tim... |

96 |
Representing Geometric Structures in d Dimensions: Topology and Order
- Brisson
- 1993
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Citation Context ...egenerate case, exactly d +1edges, d+1 facets (faces of dimension d − 1), and d + 1 cells meet at each vertex of PD(S). 2 For storing a d-dimensional cell complex, the cell-tuple structure in Brisson =-=[48]-=- seems appropriate. This data structure represents the incidence and ordering information in a cell complex in a simple uniform way. � When each weighted site p ∈ S is interpreted as the sphere σp =(p... |

92 |
The geometry of geodesics
- Busemann
- 1955
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Citation Context ...rcle, and the shortest paths are segments of great circles, too. (One can show that the only other metric space in which all bisector segments are shortest paths is the hyperbolic space; see Busemann =-=[52]-=-). For each pair of antipodal points on the sphere there is a continuum of shortest paths connecting them. But this does not affect the Voronoi diagram of n points; 43sit can be computed in optimal O(... |

90 | Four results on randomized incremental constructions
- Clarkson, Mehlhorn, et al.
- 1993
(Show Context)
Citation Context ...ng the methods used. To mention but a few results, Boissonnat et. al. [43] andGuibaset.al.[134] have refined the methods of storing the past in order to locate new conflicts quickly, Clarkson et. al. =-=[73]-=- have generalized and simplified the analytic framework, and Seidel [230] systematically applied the technique of backward analysis first used by Chew [62]. Themethodin[134] for storing the past is br... |

88 | A.C.: Optimal expected-time algorithms for closest point problems
- Bentley, Weide, et al.
- 1980
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Citation Context ...e triangle that contains a new site pi have been used for speed-up. Joe [152], who implemented Sloan’s algorithm [238], and Su and Drysdale [240], who used a variant of Bentley et al.’s spiral search =-=[36]-=-, report on fast experimental runtimes. The arising issues of numerical stability have been addressed in Fortune [123], Sugihara [241], and Jünger et al. [153]. A technique similar to incremental inse... |

88 | C.H.: The weighted region problem: finding shortest paths through a weighted planar subdivision - Mitchell, Papadimitriou - 1991 |

84 |
diagrams: properties, algorithms and applications
- Power
- 1987
(Show Context)
Citation Context ...onvex polyhedra in one dimension higher. This is a fact with far-ranging implications and has been observed in Brown [49], Klee [164], Edelsbrunner and Seidel [113], Paschinger [213], and Aurenhammer =-=[22]-=-. The power function pow(x, p) can be expressed by the hyperplane π(p) :xd+1 =2x T p − p T p + w(p) in (d + 1)-space, in the sense that a point x lies in cell(p) ofPD(S) iff,atx, π(p) is vertically ab... |

80 |
A linear-time algorithm for computing the Voronoi diagram of a convex polygon, Discrete Comput
- Aggarwal, Guibas, et al.
- 1989
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Citation Context ...he expected number of edges is less than 4. So an O(n) randomized construction algorithm is obtained. The diagrams V (S) andM(C) can also be computed in deterministic linear time; see Aggarwal et al. =-=[4]-=-. The details of this algorithm are much more involved, however. 4.4.4 Constrained Voronoi diagrams and Delaunay triangulations In certain situations, unconstrained proximity among a set of sites is n... |

79 | New sparseness results on graph spanners
- Chandra, Das, et al.
- 1995
(Show Context)
Citation Context .... In higher dimensions, DT(S) may loose its sparseness and thus is of minor interest for computing spanners. Different techniques have been used with success; see e.g. Vaidya [251] and Chandra et al. =-=[57]-=-. 5.3 Geometric clustering Clustering a set of data means finding subsets (called clusters) whose in-class members are similar, and whose cross-class members are dissimilar, according to a predefined ... |

76 |
An optimal convex hull algorithm in any fixed dimension
- Chazelle
- 1993
(Show Context)
Citation Context ...-point set in d-space can be computed in Cd+1(n) time. Worst-case optimal convex hull algorithms working in general dimensions have been designed by Clarkson and Shor [74], Seidel [229], and Chazelle =-=[58]-=-, yielding d ⌈ Cd+1(n) =O(n log n + n 2 ⌉ ). So theorem 4.5 is asymptotically optimal in the worst case. Note, however, that power diagrams in d-space may as well have a fairly small size, O(n), which... |

71 |
Voronoi diagrams based on convex distance functions
- Chew, Drysdale
- 1985
(Show Context)
Citation Context ... sites at the same speed, the fate of each point x of the plane is determined by those sites whose circles reach x first. This ”expanding waves” view has been systematically used by Chew and Drysdale =-=[66]-=- and Thurston [248]. The Voronoi vertices are of degree at least three, by lemma 2.1. Vertices of degree higher than three do not occur if no four point sites are cocircular. The Voronoi diagram V (S)... |

68 | Finding the medial axis of a simple polygon in linear time
- Chin, Snoeyink, et al.
- 1999
(Show Context)
Citation Context ...d time O(n log ∗ n); see Devillers [88]. Recently, O(n) time randomized, and deterministic, algorithms for the medial axis of a simple polygon have been designed by Klein and Lingas [168] andChinetal.=-=[68]-=-, settling open questions of long standing. The case of a convex polygon is considerably easier; see subsection 4.4.3. Some of the algorithms above also work for curved objects. The plane-sweep algori... |

67 |
Voronoi diagrams from convex hulls
- Brown
- 1979
(Show Context)
Citation Context ...ace, see e. g. Preparata and Shamos [215], we have obtained another optimal algorithm for the Voronoi diagram. The connection between Voronoi diagrams and convex hulls has first been studied by Brown =-=[49]-=- who used the inversion transform. The simpler lifting mapping has 17 E q' q C' p' r' p C rsbeen used, e.g., in Edelsbrunner and Seidel [113]. We shall see several applications and generalizations in ... |

65 | Taschenbuch der Mathematik - Bronstein |

63 |
Euclidean minimum spanning trees and bichromatic closest pairs. Discrete and Computational Geometry
- Agarwal, Edelsbrunner, et al.
- 1991
(Show Context)
Citation Context ...(S) maynotbeofmuchusefor the construction of MST(S), because of the possibly quadratic size. Subquadratic worst-case time algorithms for computing MST(S) ind-space exist (Yao [259] and Agarwal et al. =-=[3]-=-), but it still remains open whether an O(n log n) time algorithm can be developed, at least for 3-space. A traveling salesman tour, TST(S), for a set S of point sites in the plane is a minimum length... |

62 | Generalized Voronoi diagrams in the plane - Lee, Drysdale - 1981 |

61 | Concrete and Abstract Voronoi diagrams - Klein - 1990 |

57 |
Exact Computation of Voronoi Diagrams and Line Segment Intersections
- Burnikel
- 1996
(Show Context)
Citation Context ...space by divide & conquer (Kirkpatrick [160], Lee [179], and Yap [261]), plane sweep (Fortune [125]), and randomized incremental insertion (Boissonnat et al. [43] andKleinetal.[169]). Burnikel et al. =-=[51]-=- give an overview of existing methods, and discuss implementation details of an algorithm in Sugihara et al. [243] that first inserts all segment endpoints, and then all the segments, of G in random o... |

57 |
Which triangulations approximate the complete graph? International Sym[24
- Das, Joseph
- 1991
(Show Context)
Citation Context ...of polygonal obstacles. Alternative triangulations of S are known to exhibit good spanner properties. Examples are the greedy triangulation and the minimum-weight triangulation; see 69sDas and Joseph =-=[78]-=-. However, only for DT(S) are there algorithms available that are worst-case efficient and easy to implement. In higher dimensions, DT(S) may loose its sparseness and thus is of minor interest for com... |

56 | A novel type of skeleton for polygons
- Aichholzer, Alberts, et al.
- 1995
(Show Context)
Citation Context ...epresents a corresponding terrain with fixed slope. ΣG has the nice property that every raindrop that hits a terrain facet f runs off to the segment or terminal of G defining f; see Aichholzer et al. =-=[8]-=-. This may have applications in the study of rain water fall and its impact on the floodings caused by rivers in a given geographic area. The concept of S(G) can be generalized by tuning the propagati... |

55 |
Three-dimensional reconstruction of complex shapes based on the Delaunay triangulation
- Boissonnat, Geiger
- 1993
(Show Context)
Citation Context ...ighted α-shapes are similar to that of their unweighted counterparts. A different approach for reconstructing shapes in 3-space by means of Delaunay triangulations is persued in Boissonnat and Geiger =-=[44]-=-. They exploit additional information on the sites available in certain applications, namely that S is contained in k parallel planes (corresponding to cross sections of the object to be reconstructed... |

54 | Voronoi diagrams in higher dimensions under certain polyhedral distance functions, Discrete Comput
- Boissonnat, Sharir, et al.
- 1998
(Show Context)
Citation Context ...an upper bound to the number of Lp spheres in d-space that can pass through d +1 given points, which depends only on d but not on p. More is known only for special cases. Recently, Boissonnat et. al. =-=[45]-=- haveshown the following. Theorem 4.11 The Voronoi diagram of n points in 3-space based on the L1 norm is of complexity Θ(n 2 ). The Voronoi diagrams of n points in d-space based on L∞ or on a simplex... |

48 |
Some dynamic computational geometry problems
- Atallah
- 1985
(Show Context)
Citation Context ...wavefront is bounded by λ2(n) =2n − 1, where λs(n) denotes the maximum length of a Davenport–Schinzel sequence over n symbols in which no two symbols appear s times each in alternating positions; see =-=[21]-=-. ✷ The wavefront can be implemented by a balanced binary tree that stores the segments in bottom–up order. This enables us to insert a wave, or remove a wave segment, in time O(log n). Before the swe... |

46 |
Finding k points with minimum diameter and related problems
- Aggarwal, Imai, et al.
(Show Context)
Citation Context ...he order-k Voronoi diagram, Vk(P ), of P is also useful for selecting from P a k-sized cluster C ∗ of minimal dissimilarity µ(C ∗ ). This approach is persued in Dobkin et al. [97] and Aggarwal et al. =-=[6]-=-. For instance, if µ is variance, C ∗ has a non-empty region in Vk(P ). If µ is diameter, C ∗ is contained in a subset of P having a non-empty region in V3k−3(P ). These properties hold in arbitrary d... |

46 | Applications of random sampling to on-line algorithm in computational geometry
- Boissonnat, Devillers, et al.
- 1992
(Show Context)
Citation Context ...analyze. Since Clarkson and Shor [74] introduced their technique, many researchers have been working on generalizing and simplifying the methods used. To mention but a few results, Boissonnat et. al. =-=[43]-=- andGuibaset.al.[134] have refined the methods of storing the past in order to locate new conflicts quickly, Clarkson et. al. [73] have generalized and simplified the analytic framework, and Seidel [2... |

43 |
Real algebraic and semialgebraic sets, Actualités Mathématiques
- Benedetti, Risler
- 1990
(Show Context)
Citation Context ...176] has obtained the following result on the number of Lp spheres that can pass through a set of given points. The proof uses results from the theory of additive complexity, see Benedetti and Risler =-=[33]-=-. This and the subsequent results require that the sites be in general position. Theorem 4.10 There exists an upper bound to the number of Lp spheres in d-space that can pass through d +1 given points... |

42 | Constructing levels in arrangements and higher order voronoi diagrams
- Agarwal, Berg, et al.
- 1998
(Show Context)
Citation Context ...ls up to a given value of k is O(n ⌊d/2⌋ k ⌈d/2⌉ ). The collection of these cells can be constructed within this time for d ≥ 4, with the algorithm in Mulmuley [202]. A modification by Agarwal et al. =-=[2]-=- achieves roughly O(nk 2 ) time also in 3-space. An output-sensitive construction algorithm is given in Mulmuley [203]. All these results apply to families of order-k diagrams in one dimension lower. ... |

41 |
H.: An optimal algorithm for constructing the weighted Voronoi diagram in the plane
- Aurenhammer, Edelsbrunner
- 1984
(Show Context)
Citation Context ...ce, and the multiplicatively weighted Voronoi diagram in the plane, can both be computed in O(n 2 ) time which is optimal. The latter diagram is investigated in detail in Aurenhammer and Edelsbrunner =-=[28]-=- and in Sakamoto and Takagi [223]. 4.3.3 Higher-order Voronoi diagrams and arrangements Higher-order Voronoi diagrams are natural and useful generalizations of classical Voronoi diagrams. Given a set ... |

38 |
Building Voronoi diagrams for convex polygons in linear expected time
- Chew
- 1990
(Show Context)
Citation Context ...cate new conflicts quickly, Clarkson et. al. [73] have generalized and simplified the analytic framework, and Seidel [230] systematically applied the technique of backward analysis first used by Chew =-=[62]-=-. Themethodin[134] for storing the past is briefly described in subsection 4.3.3 for constructing a generalized planar Voronoi diagram. If the set S of sites can be expected to be well distributed in ... |

37 |
F.: Clustering algorithms based on minimum and maximum spanning trees
- Asano, Bhattacharya, et al.
- 1988
(Show Context)
Citation Context ...n − k steps in constructing the hierarchy (that is, after having added the n − k shortest edges to MST(S)), the intermediate k-clustering C1,...,Ck is optimal in the following sense; see Asano et al. =-=[18]-=-: It maximizes the minimum single-linkage distance between the clusters, for all possible k-clusterings of S. The practical value of single-linkage clusterings is, however, restricted by the fact that... |

37 |
Geometric transforms for fast geometric algorithms
- Brown
- 1980
(Show Context)
Citation Context ...distance between two points on the surface is the minimum Euclidean length of a curve that connects the points and runs entirely inside the surface. Such a curve will be called a shortest path. Brown =-=[50]-=- has addressed the Voronoi diagram of points on the surface of the twodimensional sphere. Here great circles play the role of lines in the Euclidean plane. In fact, the bisector of two points is a gre... |

35 | skeletons for general polygonal figures in the plane, in
- Aichholzer, Aurenhammer, et al.
- 1996
(Show Context)
Citation Context ...nt alternative to V (G), we now describe the straight skeleton, S(G), of a planar straight line graph G. This structure is introduced, and discussed in much more detail, in Aichholzer and Aurenhammer =-=[9]-=-. S(G) is composed of angular bisectors and thus does not contain curved edges. In general, its size is even less than that of V (G). Beside its use as a type of skeleton for G, S(G) applies, for exam... |

33 |
The Voronoi diagram of curved objects
- Alt, Cheong, et al.
- 2005
(Show Context)
Citation Context ...tion from the point site case. Yap [261] allows sets of disjoint segments of arbitrary degree-two curves. A randomized incremental algorithm for general curved objects is given in Alt and Schwarzkopf =-=[12]-=-. They show that complicated curved objects can be partitioned into ’harmless’ ones by introducing new points. All these algorithms achieve an optimal running time, O(n log n). In dimensions more than... |

33 |
Output-sensitive construction of polytopes in four dimensions T.M. Chan and clipped Voronoi diagrams in three
- Chan, Snoeyink, et al.
(Show Context)
Citation Context ...thms. The algorithm in Seidel [226] achievesCd+1(n) =O(n2 + f log f), where f is the total number of faces of the convex hull constructed. The latest achievements are C4 = O((n + f)log 2 f)inChanetal.=-=[56]-=- andC5 = O((n + f)log 3 f)inAmatoand Ramos [15]. Space constraints preclude our discussion of power diagrams. Still, some remarks are in order to point out their central role within the context of Vor... |

32 | Simplified Voronoi diagrams
- Canny, Donald
- 1988
(Show Context)
Citation Context ...ometimes also in the application, of V (G). There have been several attempts to linearize and simplify V (G), mainly for the sake of efficient point location and motion planning; see Canny and Donald =-=[54]-=-, Kao and Mount [154], de Berg et al. [80], and McAllister et al. [190]. The compact Voronoi diagram in [190] is particularly suited to these applications. It is defined for the case where G is a set ... |

30 | Intersection and closest-pair problems for a set of circular discs - Sharir - 1985 |

28 |
Solving query-retrieval problems by compacting voronoi diagrams
- Aggarwal, Hansen, et al.
- 1990
(Show Context)
Citation Context ...that allows both insertion and deletion of sites, at a space requirement of O(k(n − k)); see also subsection 4.3.3. A sophisticated technique of compacting order-k Voronoi diagrams in Aggarwal et al. =-=[5]-=- achieves optimal query time O(k +logn) andspaceO(n), but with high constants. The locus approach also works for more general sites and distance measures. Let us assume that the site set, S, consists ... |

28 | Woeginger G.: Geometric clusterings
- Capoyleas, Rote
- 1991
(Show Context)
Citation Context ......,µ(Ck)) is minimal for all possible k-clusterings of P . If k is part of the input, the problem of finding an optimal k-clustering is NPcomplete in general, even in the plane; see Capoyleas et al. =-=[55]-=- for references. For fixed k, polynomial-time algorithms are known for various criteria. This is due to the fact that optimal clusters are separable in a certain sense, and in several cases are induce... |