P. J. Green and R. Sibson, "Computing Dirichlet tessellations in the plane," Comput. J. (Cambridge) 21,68--173 (1978).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of science. For relevant surveys and bibliographies consult [1, 16, 54] Various approaches for their con struction are described in the literature. Naturally, it was the planar case, which was solved first, but extensions to 3 and higher dimensions followed. Incremental methods (for example [4, 39]) compete with divide and conquer algorithms (for example [8,42] Newer research studies randomization [41] One approach uses a lifting map (see below) to transform the triangulation problem in R to the problem of constructing the convex hull in R . This idea goes back to [7] details on the ....

....search : p; a) is called. Since T is a Delaunay triangulation, the volume of A=0 a is strictly decreasing, for increasing . Thus, the function always terminates successfully. This simple idea of walking through the triangulation is not new, it was already mentioned in [4] with a reference to [39], where it was used for computing Delaunay triangulations in the plane. It is difficult to give a good estimate for the time complexity of search. In the worst case, it is quadratic in z, the number of points in 7 . A more realistic estimate is that the number of steps required to locate p; is ....

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P J Green and R Sibson. Computing Dirichlet tessellations in the plane. The Compuler Journal, 21(2):168 173, 1978.

....Two steps are performed: rstly, we locate M in order to nd the set R(M) of all the triangles in conAEict with M (location step) secondly, we create the new triangles (in sertion step) Based on this approach, there exists several incremental algorithms. They dioeer mainly in the way they nd R(M) [7, 2, 13, 8, 9, 10, 12]. The complexity of the location step, which strongly depends on the location strategy will be evoked in the next section. We rst focus our interest on the triangulation of R(M) M M R(M) F (M) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 Figure 2: Inserting a site in the ....

P. J. Green and R. R. Sibson. Computing Dirichlet tessellations in the plane. Comput. J., 21:168173, 1978.

....that algorithms use and do an experimental analysis of 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 dimensions and computing their convex hull [2] Which approach is best in practice This paper presents an experimental comparison of a number of these ....

P. Green and R. Sibson. Computing dirichlet tessellations in the plane. Computing Journal, 21:168-- 173, 1977.

....[16] who also pointed out the significance of the number of facets of the Voronoi regions. An iterative algorithm was developed by Hwang [17] and investigated in more detail by Butovitsch [18, pts. D E] The idea, which has also been considered for other point sets and in other applications [19], 20] is basically as follows: i) Select a codeword c and compute its distance d to the input x . ii) Compute the distance d from a new codeword = c c a to x , where a ( N C 0 . If d d , set c c : and go to (ii) iii) If N C 0 ( contains unexamined codewords, go to (ii) ....

P. J. Green and R. Sibson, "Computing Dirichlet tessellations in the plane," The Computer J., vol. 21, no. 2, pp. 168--173, May 1978.

....model (BPM) as given in Holmes, Denison and Mallick (1999) and how to extend it to handle data that arises as counts. The model is based on the assumption that the space can be split into local regions where the responses come from the same distribution. We make use of the Voronoi tessellation (Green and Sibson, 1978) to de ne the distinct regions as it provides a exible method of partitioning and is straightforward to compute. Voronoi tessellations are de ned by a number of generating points which each de ne a region centre. If these centres are given by T = t 1 ; t M ) then we can de ne the region ....

Green, P. J. and Sibson, R. (1978). Computing Dirichlet tessellations in the plane, Comp. J. 21: 168-173.

....Antipolis, France. Olivier.Devillers sophia.inria.fr. algorithms have the two following properties: they are incremental and they do not used complicated data structures in addition to the triangulation itself. Among these algorithms, let us mention the historical algorithm of Green and Sibson [GS78], or some other variants [MSZ96, BD95, Dev98, DLM98, Lem97] All perform a walk in the triangulation to accelerate point location. The advantage of that category of incremental Delaunay algorithms is that they may easily be turned into fully dynamic Delaunay algorithms. Since there is no ....

P. J. Green and R. R. Sibson. Computing Dirichlet tessellations in the plane. Comput. J., 21:168-- 173, 1978.

....smooth function ae then j ae(z) Gamma ae(z)j = O(ffi 2 ) where ffi is the diameter of triangle T j . This, incidentally, explains why we do not want to use just any triangulation: we should pick the one that minimizes diameters on average, which turns out to be the Delaunay triangulation [5, 19, 14, 11]. We can now define an approximation F for the dynamical system from R n to R n ; for example, in the two dimensional case we use F (y t ; y t Gamma1 ) ae(y t ; y t Gamma1 ) y t ) 4) where we are assuming a simple lag 1 embedding. This map takes triangles into triangles; for ....

P. J. Green and R. Sibson. Computing Dirichlet tessellations in the plane. The Computer Journal, 21:168--173, 1978.

....method is to randomize an incremental algorithm. Incremental algorithms are often very simple: objects are inserted one by one with the desired structure updated after each addition; a classic example is the incremental computation of a Delaunay triangulation or a Voronoi diagram in two dimensions[48, 54]. An objection to incremental algorithms is that if the insertion order is chosen badly, then each addition requires a lengthy update to the structure, and the algorithm runs slowly. A remarkable discovery for some problems is that if objects are inserted in random order, then the expected work at ....

P.J. Green, R. Sibson, Computing Dirichlet tessellations in the plane, Computer Journal, 21(22):168-173, 1977.

....there are some examples of dynamic diagram creation in the literature (e.g. Gold, 1998; Icking et al. 1999) there has been little research directed at the supporting algorithms and data structures, particularly for higher order variants. Notable exceptions (for the OVD only) are the work of Green and Sibson (1978), who proposed an edge swapping procedure for node insertion and Heller (1990) who considers the dynamic deletion of points. However the OVD is limited to a collection of nearest neighbourhood operations, such as generating a buffer zone and searching for the nearest facility from a given ....

Green, P. J. and Sibson, R. (1978). Computing Dirichlet tessellations in the plane, The Computing Journal, 21: 168-173.

....This sequential test does not a ect the worst case time complexity of the algorithm, O(n 2 ) since R k may have O(n) edges) In practice, however, it may be the major factor in the running time, and some heuristic should be used instead. One option is the simple walking method suggested in [4], which can be used, because the face of the site of S closest to k must have a contributing edge. More elaborated heuristics exist (see [2] for instance) Figure 7(a) shows, in bold lines, the edges traversed by the procedure for the example of Fig. 6(b) The function processContributingEdge is ....

P. J. Green and R. Sibson, Computing Dirichlet tessellations in the plane, Computer Journal 21 (1977) 168-173.

....for Voronoi diagrams is the manipulation and representation of the edges extending to infinity. Although this may not be a difficult problem, solutions are artificial. For explicitly construction of Voronoi diagrams we list three common solutions: bounding the diagram with a window or a polygon [5]; introducing virtual edges connecting adjacent infinite edges [15, 13] and introducing a virtual vertex connecting all infinite edges [12, 6] The first solution rules out infinite edges. The other two unify the topological representation of finite and infinite edges but introduce meaningless ....

....Fig. 7(b) for e 1 counterClockWiseOrder( e 1 orig , v 1 , v 2 , e 1 left ) true, and for sym(e 1 ) counter ClockWiseOrder( sym(e 1 ) orig , v 1 , v 2 , sym(e 1 ) lef t ) false. Then e 1 contributes v 1 and sym(e 1 ) contributes v 2 . Our traverse procedure has the same goal as the one in [5]. The vertex contributed by an edge e is the point where the boundary of R k crosses e leaving the region of e right and entering the region of e lef t (in a counterclockwise direction) We need to find the next vertex (which is the point where the boundary leaves the region of e lef t ) The ....

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P. J. Green and R. Sibson, Computing Dirichlet tessellations in the plane, Computer Journal 21 (1997) 168--173.

....or static in the sense that only a single Voronoi model is implemented (usually the ordinary Voronoi diagram) and sometimes both. These restrictions severely impede usefulness in a dynamic setting or for exploratory purposes. Furthermore, with the exception of the ordinary Voronoi diagram (e.g. Green Sibson, 1977; Guibas et al. 1991; Gold, 1992a; 1993) it would appear that little research has been conducted on the design and implementation of dynamic algorithms for insertion and deletion of nodes, or for changing from one Voronoi model to another. Supporting dynamic Voronoi modeling and analysis ....

....within the circumcircle of some existing point, thus it violates the Delaunay property that circumcircles of triangles must be empty. Only those existing vertices that define the affected circumcircle need to be changed, all other vertices can be carried over directly into the updated diagram. Green and Sibson (1977) first described this approach, having complexity ( n O per node inserted. Translating this algorithm to a data structure based around the Delaunay triangulation, as opposed to the Voronoi regions, we get the following insertion procedure: 1. Locate the triangle containing the newly inserted ....

Green, P. J. and Sibson, R. (1977). Computing Dirichlet tessellations in the plane, The Computing Journal, 21: 168-173.

....from the non smooth single models. The method, which we shall call the Bayesian partition model (BPM) is based on the simple and intuitive idea of decomposing the design space into a number of homogeneous regions. Specifically, the domain of interest H partitioned using a Voronoi tessellation (Green and Sibson, 1978). Within each region f(x) or p(y = C k jx) is approximated by either a Bayes linear or a Bayes multinomial model. We shall assume that these models are independent between regions. Our method has similarities with tree structured models (Breiman et al. 1984) in that the data space is ....

....partition model (BPM) is that points nearby in the design space have similar values in the response space. On this basis we construct a number of disjoint regions in H where the data within regions is assumed exchangeable. We choose to take these regions to be defined by a Voronoi tessellation (Green and Sibson, 1978). This tessellation is determined by a number of centroids T = t 1 ; t M ) which split H into M disjoint regions, R 1 ; RM , such that points within R i are closer to t i than any other centre t j j 6= i, i.e. R i = fx 2 H : jjx Gamma t i jj jjx Gamma t j jj for all j 6= ig ....

Green, P.J. and Sibson, R. (1978) Computing Dirichlet tessellations in the plane. Comp.

....that is used for solving a large number of problems in many areas. Accordingly, several algorithms have been devised and implemented for constructing it in two and higher dimensions (see Bentley, Weide and Yao (1980) Bowyer (1981) Brostow, Dussault and Fox (1978) Brown (1979) Finney (1979) Green and Sibson (1978), Lee and Schachter (1980) Maus (1984) Ohya, Iri and Murota (1984) Seidel (1986) Shamos (1978) Shamos and Hoey (1975) Tanemura, Ogawa and Ogita (1983) Watson (1981) Witzgall (1973b) and many of its statistical and geometrical properties have been derived (see Bentley, et al. 1980) Klee ....

Green P. J. and Sibson R. (1978), Computing Dirichlet tessellations in the plane, Comput. J. 21, No. 2, 168-173.

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Green, P. and Sibson, R. (1978) Computing Dirichlet tessellations in the plane. Comp. J., 21, 168--173.

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P. J. Green and R. Sibson, "Computing Dirichlet tessellations in the plane," Comput. J. (Cambridge) 21,68--173 (1978).

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Green, P.J. and Sibson, R. (1977) Computing Dirichlet tessellations in the plane. The Computer Journal 21:168-173. (cit. Adlard and Smith 1981).

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P. J. Green and R. R. Sibson. Computing Dirichlet tessellations in the plane. Comput. J., 21:168-- 173, 1978.

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P.J. Green and R. Sibson. Computing Dirichlet tessellations in the plane. Computer Journal, 21(22):168 - 173, 1977.

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P. J. Green and R. R. Sibson. Computing Dirichlet tessellations in the plane. Comput. J., 21:168173, 1978.

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P. J. Green and R. Sibson. Computing Dirichlet tessellation in the plane. Computer Journal, 21:168--173, 1978.

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Green, P. J. and Sibson, R. "Computing Dirichlet tessellations in the plane", Comput. J., 21, 168--173, 1978.

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P. J. Green and R. Sibson. Computing Dirichlet tessellations in the plane. The Computer Journal, 21:168--173, 1978.

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P. J. Green and R. Sibson. Computing dirichlet tessellations in the plane. The Computer Journal, 21(2):168--173, May 1978.

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P.J. Green, R. Sibson, Computing Dirichlet tessellations in the plane, Computer Journal, 21(22):168--173, 1977.

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