## Robust Proximity Queries in Implicit Voronoi Diagrams (1996)

Venue: | IN PROC. 8TH CANAD. CONF. COMPUT. GEOM |

Citations: | 11 - 3 self |

### BibTeX

@TECHREPORT{Liotta96robustproximity,

author = {Giuseppe Liotta and Franco P. Preparata and Roberto Tamassia},

title = {Robust Proximity Queries in Implicit Voronoi Diagrams},

institution = {IN PROC. 8TH CANAD. CONF. COMPUT. GEOM},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, as a worst-case quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach, we consider the important classical problem of proximity queries in 2 and 3 dimensions, and develop a new technique for the efficient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique gives both low degree and fast query time, and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations and arithmetic degree.

### Citations

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Citation Context ...e circle with center q passing through r. 9 It is well known that such queries are efficiently solved by performing point location in the Voronoi diagram (possibly of higher order) V (S) of the sites =-=[50]-=-. For nearest neighbor search, the alternative extremal-search method [25] also exists. We begin by examining in Section 3.1 the geometric test primitives used by the theoretically optimal and practic... |

674 | LEDA: A Platform for Combinatorial and Geometric Computing
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Citation Context ...S) by duality. Next, we compute the approximations x (v) and y (v) for each vertex v of V (S) by means of integer division. For effective procedures that perform the integer division, see, e.g., LEDA =-=[45]-=-. Let a, b, and c be the three sites of S that define vertex v. Adopting the same notation as in the proof of Lemma 9, the y-coordinate y(v) of v is given by the ratio y(v) = Y 1 2W 1 , where Y 1 is a... |

621 | Voronoi diagrams - a survey of a fundamental geometric data structure
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Citation Context ...ee distinct sites (points and segments) of the plane contains a given query site. The incircle test is a basic operation for many algorithms that construct the Voronoi diagram of the sites (see, e.g. =-=[36, 40, 32, 3]-=-). The degree of the incircle test has been extensively studied by Burnikel [8] and by Burnikel, Mehlhorn and Schirra [10]. Following the notation of Burnikel [8], an incircle test is conveniently exp... |

491 | Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams
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(Show Context)
Citation Context ...ee distinct sites (points and segments) of the plane contains a given query site. The incircle test is a basic operation for many algorithms that construct the Voronoi diagram of the sites (see, e.g. =-=[36, 40, 32, 3]-=-). The degree of the incircle test has been extensively studied by Burnikel [8] and by Burnikel, Mehlhorn and Schirra [10]. Following the notation of Burnikel [8], an incircle test is conveniently exp... |

365 |
A sweepline algorithm for Voronoi diagrams
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Citation Context ...ee distinct sites (points and segments) of the plane contains a given query site. The incircle test is a basic operation for many algorithms that construct the Voronoi diagram of the sites (see, e.g. =-=[36, 40, 32, 3]-=-). The degree of the incircle test has been extensively studied by Burnikel [8] and by Burnikel, Mehlhorn and Schirra [10]. Following the notation of Burnikel [8], an incircle test is conveniently exp... |

282 | Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms
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Citation Context ...here geometric computations can be carried out either exactly or with a guaranteed error bound. This has motivated a great deal of research on the subject of robust computational geometry (see, e.g., =-=[4, 11, 10, 18, 26, 27, 30, 35, 33, 38, 47, 53, 56, 20, 29, 31]). Also, e-=-fficiency must be evaluated in a finer framework than the conventional "big-Oh" analysis. In particular, constant factors dependent on the precision requirement of the numerical computations... |

264 | Optimal search in planar subdivisions
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Citation Context ...ies with optimal degree 2. 3.1 Test Primitives and Methods for Planar Point Location The chain method [43], the bridged chain method [24], the trapezoid method [49], the subdivision refinement method =-=[41]-=-, and the persistent search tree method [52] are popular deterministic techniques for point location in a planar map that combine theoretical efficiency with good performance in practice (see, e.g., [... |

171 | Planar point location using persistent search trees
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Citation Context ...es and Methods for Planar Point Location The chain method [43], the bridged chain method [24], the trapezoid method [49], the subdivision refinement method [41], and the persistent search tree method =-=[52]-=- are popular deterministic techniques for point location in a planar map that combine theoretical efficiency with good performance in practice (see, e.g., [23, 50]). Namely, denoting with n the size o... |

168 |
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Citation Context ...(n) for the other methods. For monotone maps, the preprocessing time is O(n) for the chain method and the bridged chain method, and O(n log n) for the other methods. The randomized-incremental method =-=[34]-=- also exists. Such method is specialized for point location in Voronoi diagrams, uses expected space O(n), and has expected query time O(log 2 n). By a careful examination of the query algorithms used... |

159 |
Optimal point location in a monotone subdivision
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Citation Context ...point sites in the plane, which allows us to perform proximity queries with optimal degree 2. 3.1 Test Primitives and Methods for Planar Point Location The chain method [43], the bridged chain method =-=[24]-=-, the trapezoid method [49], the subdivision refinement method [41], and the persistent search tree method [52] are popular deterministic techniques for point location in a planar map that combine the... |

155 | Fractional cascading I: A data structuring technique, Algorithmica 1
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Citation Context ... in M is reduced to point location in the resulting refinement M 0 of M . Hence, the chain method is ordinary for general maps. In the bridged chain method [24], the technique of fractional cascading =-=[16, 17] is applie-=-d to the sets of y-coordinates of the separators. Fractional cascading establishes "bridges" between the separator of a node and the separators of its children such that there are O(1) verti... |

99 | Epsilon Geometry : Building robust algorithms from imprecise computations
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- 1989
(Show Context)
Citation Context ...here geometric computations can be carried out either exactly or with a guaranteed error bound. This has motivated a great deal of research on the subject of robust computational geometry (see, e.g., =-=[4, 11, 10, 18, 26, 27, 30, 35, 33, 38, 47, 53, 56, 20, 29, 31]). Also, e-=-fficiency must be evaluated in a finer framework than the conventional "big-Oh" analysis. In particular, constant factors dependent on the precision requirement of the numerical computations... |

98 | Efficient Exact Arithmetic for Computational Geometry
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(Show Context)
Citation Context ...here geometric computations can be carried out either exactly or with a guaranteed error bound. This has motivated a great deal of research on the subject of robust computational geometry (see, e.g., =-=[4, 11, 10, 18, 26, 27, 30, 35, 33, 38, 47, 53, 56, 20, 29, 31]). Also, e-=-fficiency must be evaluated in a finer framework than the conventional "big-Oh" analysis. In particular, constant factors dependent on the precision requirement of the numerical computations... |

92 | Towards exact geometric computation
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Citation Context ...r than "approximate") computations, with the assumption that (bounded-length) input data are error free. In this category falls the exact geometric computation paradigm independently advocat=-=ed by Yap [57]-=- and by the Saarbrucken school [9], and so does our approach. Within this paradigm, we introduce the concept of degree of a geometric algorithm, which characterizes, up to a small additive constant, t... |

89 |
Primitives for the Manipulation of Three-dimensional Subdvisions
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- 1987
(Show Context)
Citation Context ... extension of the definition for two-dimensional Voronoi diagrams given in Section 3.4. Namely V (S) stores the topological structure of the 3D Voronoi diagram V (S) of S (e.g., the data structure of =-=[22]-=-) and the following geometric information for each vertex and facet: ffl For each vertex v of V (S), V (S) stores the semi-integer (b + 1)-bit approximations x (v), y (v) and z (v) of the x-, y-, and ... |

81 |
Fast detection of polyhedral intersection
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- 1983
(Show Context)
Citation Context ...ances comparisons, fails to achieve optimal degree because the search is also guided by geometric primitives requiring 4b bits of precision. Moreover, this method uses the hierarchical representation =-=[21]-=- of the 3D convex hull of the sites lifted to a 3D paraboloid. Searching the hierarchical representation involves high constant factors and, by the authors' own admission, casts some doubts on the pra... |

80 |
P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete Comput. Geom
- Aggarwal, Guibas, et al.
- 1989
(Show Context)
Citation Context .... The space and preprocessing time of Theorems 5--6 and the query time of Theorem 6 can be improved while preserving the same degree bounds by more complicated procedures along the lines suggested in =-=[1, 2, 14]-=-. 4 Proximity Queries for Point Sites in 3D Space In this section, we consider the following proximity query on a set S of point sites in threedimensional (3D) space: nearest neighbor search: given qu... |

80 |
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Citation Context |

77 |
Construction of three-dimensional delaunay triangulations using local transformations
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Citation Context ... yields the following result: The implicit Voronoi diagram V (S) of a set S of n points in 3D space can be constructed by computing the 3D Delaunay triangulation with the incremental algorithm by Joe =-=[39]-=-, whose time complexity and storage is O(n 2 ) (see also [48]). Since the most demanding operation of the algorithm in terms of degree is the 3D insphere test, from Lemma 6 we have that the degree of ... |

75 |
Verifiable implementations of geometric algorithms using finite precision arithmetic
- Milenkovic
- 1989
(Show Context)
Citation Context |

74 | Efficient Delaunay triangulation using rational arithmetic
- Karasick, Lieber, et al.
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(Show Context)
Citation Context ...tions with bit precision db +O(1). Theoretical analysis and experimental results show that multiprecision numerical computations take up most of the CPU time of exact geometric algorithms (see, e.g., =-=[40, 48]-=-). Thus, we believe that, in defining the efficiency of a geometric algorithm, the degree should be considered as important as the asymptotic time complexity and should correspondingly play a major ro... |

65 | Static analysis yields efficient exact integer arithmetic for computational geometry
- Fortune, Wyk
- 1996
(Show Context)
Citation Context |

64 |
On k-nearest neighbor Voronoi diagrams in the plane
- Lee
(Show Context)
Citation Context ... be obtained from the order k \Gamma 1 implicit Voronoi diagram V k\Gamma1 (S) by intersecting each face of V k\Gamma1 (S) with the (order 1) Voronoi diagram of a suitable subset of the vertices of S =-=[42]-=-. As shown in [42, 15], V k (S) can be computed in O(k(n \Gamma k) p n log n) time. Since the construction is based on iteratively computing Voronoi diagrams by using the incircle test, which is the m... |

64 |
Location of a point in a planar subdivision and its applications
- Lee, Preparata
- 1977
(Show Context)
Citation Context ...tation of Voronoi diagrams for point sites in the plane, which allows us to perform proximity queries with optimal degree 2. 3.1 Test Primitives and Methods for Planar Point Location The chain method =-=[43]-=-, the bridged chain method [24], the trapezoid method [49], the subdivision refinement method [41], and the persistent search tree method [52] are popular deterministic techniques for point location i... |

59 |
How to compute the Voronoi Diagram of Line Segments
- Burnikel, K
(Show Context)
Citation Context ...cently attracted considerable attention because, due to its demands of high precision for exact computation, it is particularly appropriate in assessing effectiveness of robust approaches (see, e.g., =-=[8, 10, 19, 28, 30, 26, 54, 31]-=-). While Voronoi diagrams are interesting in their own right, the main reason for con1 structing and storing them is to efficiently answer fundamental proximity queries such as nearest-neighbor and ci... |

55 | Safe and effective determinant evaluation
- Clarkson
- 1992
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Citation Context |

53 | On Degeneracy in Geometric Computations - Burnikel, Mehlhorn, et al. - 1994 |

48 |
Stable maintenance of point-set triangulation in two dimensions
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Citation Context |

47 |
The problems of accuracy and robustness in geometric computation
- Hoffmann
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(Show Context)
Citation Context ...ent on the precision requirement of the numerical computations should be taken into account. For an early survey of the different approaches to robust computational geometry the reader is referred to =-=[37]-=-. To a first, rough, approximation, robustness approaches are of two main types: perturbing and nonperturbing. Perturbing approaches transform the given problem into one that is intended not to suffer... |

45 |
Symbolic treatment of geometric degeneracies
- Yap
- 1990
(Show Context)
Citation Context |

42 |
A strong and easily computable separation bound for arithmetic expressions involving square roots
- Burnikel, Fleischer, et al.
- 1997
(Show Context)
Citation Context ...bed in Section 2 should be extended to the computation of the degree of other classes of geometric primitives. Recently, motivated in part by a preliminary version of this paper [44], Burnikel et al. =-=[12]-=- have presented a new separation bound for arithmetic expressions involving square roots. Also, since the degree of an algorithm expresses worst-case computational requirement occurring in degenerate ... |

41 | Evaluating signs of determinants using single-precision arithmetic
- Avnaim, Boissonnat, et al.
- 1997
(Show Context)
Citation Context |

41 |
Fractional cascading
- Chazelle, Guibas
- 1986
(Show Context)
Citation Context ... in M is reduced to point location in the resulting refinement M 0 of M . Hence, the chain method is ordinary for general maps. In the bridged chain method [24], the technique of fractional cascading =-=[16, 17] is applie-=-d to the sets of y-coordinates of the separators. Fractional cascading establishes "bridges" between the separator of a node and the separators of its children such that there are O(1) verti... |

41 |
Constmction of the Voronoi diagram for one million generators in single-precision arithmetic
- SUGIHARA, IRI
- 1989
(Show Context)
Citation Context |

38 | Exact geometric computation in LEDA
- Burnikel, Könemann, et al.
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(Show Context)
Citation Context ...segment sites. The extended Voronoi diagram V 0 (S) of S is said to be explicit if the coordinates of the vertices of V 0 (S) are computed and stored with exact arithmetic, i.e., as algebraic numbers =-=[9, 57]-=-. In the following lemma, we analyze the degree of test primitive left-right(q; e) for a straight-line edge e of an explicit extended Voronoi diagram. Lemma 24 The left-right(q; e) test primitive for ... |

38 |
Numerical stability of algorithms for 2-d Delaunay triangulations
- Fortune
- 1995
(Show Context)
Citation Context ...cently attracted considerable attention because, due to its demands of high precision for exact computation, it is particularly appropriate in assessing effectiveness of robust approaches (see, e.g., =-=[8, 10, 19, 28, 30, 26, 54, 31]-=-). While Voronoi diagrams are interesting in their own right, the main reason for con1 structing and storing them is to efficiently answer fundamental proximity queries such as nearest-neighbor and ci... |

36 |
Robust set operations on polyhedral solids
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- 1989
(Show Context)
Citation Context |

32 |
A new approach to planar point location
- Preparata
- 1981
(Show Context)
Citation Context ...hich allows us to perform proximity queries with optimal degree 2. 3.1 Test Primitives and Methods for Planar Point Location The chain method [43], the bridged chain method [24], the trapezoid method =-=[49]-=-, the subdivision refinement method [41], and the persistent search tree method [52] are popular deterministic techniques for point location in a planar map that combine theoretical efficiency with go... |

31 | Checking Geometric Programs or Verification of Geometric
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(Show Context)
Citation Context ...i diagram must have the topology of a convex map) as illustrated in [53], but more specifically verifies its topological agreement with the structure emerging from the tests executed by the algorithm =-=[46]-=-. 28 Beyond these general methodological issues, the investigation reported in these pages leaves some interesting open problems, such as answering nearest neighbor queries in subquadratic time and op... |

28 |
Solving query-retrieval problems by compacting Voronoi diagrams
- Aggarwal, Hansen, et al.
- 1990
(Show Context)
Citation Context .... The space and preprocessing time of Theorems 5--6 and the query time of Theorem 6 can be improved while preserving the same degree bounds by more complicated procedures along the lines suggested in =-=[1, 2, 14]-=-. 4 Proximity Queries for Point Sites in 3D Space In this section, we consider the following proximity query on a set S of point sites in threedimensional (3D) space: nearest neighbor search: given qu... |

27 | Polyhedral modelling with multiprecision integer arithmetic. Comput-Aided Des
- Fortune
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(Show Context)
Citation Context |

26 |
How to search in history
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(Show Context)
Citation Context ... and Methods for Spatial Point Location There are only two known efficient spatial point location methods for cell-complexes that are applicable to 3D Voronoi diagrams: the separating surfaces method =-=[13, 55]-=-, which extends the chain-method [43], and the persistent planar location method [51], which extends the persistent search tree method [52]. Let N be the number of facets of a cell-complex C. The quer... |

20 |
upper bounds for neighbor searching
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(Show Context)
Citation Context .... The space and preprocessing time of Theorems 5--6 and the query time of Theorem 6 can be improved while preserving the same degree bounds by more complicated procedures along the lines suggested in =-=[1, 2, 14]-=-. 4 Proximity Queries for Point Sites in 3D Space In this section, we consider the following proximity query on a set S of point sites in threedimensional (3D) space: nearest neighbor search: given qu... |

18 |
Efficient point location in a convex spatial cell-complex
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(Show Context)
Citation Context ...location methods for cell-complexes that are applicable to 3D Voronoi diagrams: the separating surfaces method [13, 55], which extends the chain-method [43], and the persistent planar location method =-=[51]-=-, which extends the persistent search tree method [52]. Let N be the number of facets of a cell-complex C. The query time is O(log 2 N) for both methods. The space used is O(N) for the separating surf... |

17 |
A new point-location algorithm and its practical efficiency — Comparison with existing algorithms
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(Show Context)
Citation Context ...], and the persistent search tree method [52] are popular deterministic techniques for point location in a planar map that combine theoretical efficiency with good performance in practice (see, e.g., =-=[23, 50]-=-). Namely, denoting with n the size of the map, all the above point location methods have O(log n) query time and O(n log n) preprocessing time. The space used is O(n log n) for the trapezoid method a... |

13 |
Topologyoriented approach to robustness and its applications to several Voronoi-diagram algorithms
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- 1990
(Show Context)
Citation Context ...cently attracted considerable attention because, due to its demands of high precision for exact computation, it is particularly appropriate in assessing effectiveness of robust approaches (see, e.g., =-=[8, 10, 19, 28, 30, 26, 54, 31]-=-). While Voronoi diagrams are interesting in their own right, the main reason for con1 structing and storing them is to efficiently answer fundamental proximity queries such as nearest-neighbor and ci... |

12 |
Edelsbrunner, An Improved Algorithm for Constructing k–th Order Voronoi Diagrams
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(Show Context)
Citation Context ...the order k \Gamma 1 implicit Voronoi diagram V k\Gamma1 (S) by intersecting each face of V k\Gamma1 (S) with the (order 1) Voronoi diagram of a suitable subset of the vertices of S [42]. As shown in =-=[42, 15]-=-, V k (S) can be computed in O(k(n \Gamma k) p n log n) time. Since the construction is based on iteratively computing Voronoi diagrams by using the incircle test, which is the most expensive operatio... |

11 |
A note on euclidean near neighbor searching
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(Show Context)
Citation Context ...) time with optimal degree 2. Circular range search queries in a set S of n point sites can be reduced to a sequence of 2 i -nearest neighbors queries in V 2 i(S), i = 0; \Delta \Delta \Delta ; log n =-=[7]-=-. This approach yields a data structure with O(n 3 ) space and preprocessing time, and O(log n log log n + k) query time, where k is the size of the output. Hence, with analogous reasoning as above, w... |

10 |
Delaunay triangulations in three dimensions with finite precision arithmetic
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Citation Context |

10 |
Finding Extreme Points in Three Dimensions and Solving the Post-Office Problem in the Plane
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Citation Context ...s optimal). It appears therefore realistic to expect that proximity-query methods with optimal precision be based on the use of distance comparisons. In this context, the early extremal-search method =-=[25]-=-, which uses distances comparisons, fails to achieve optimal degree because the search is also guided by geometric primitives requiring 4b bits of precision. Moreover, this method uses the hierarchica... |