L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381--413, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a partition of S into a left subset S 1 and a right subset S 2 . Then we search over all entries of the matrix. When passing from one entry to an adjacent one, one point is moved from S 1 to S 2 or vice versa. We then update the farthest point Voronoi diagrams and center hulls of S 1 and S 2 as in [4, 5, 9]. We refer the readers to [5] for additional details concerning the dynamic construction of FPVD and center hulls. And we conclude that Step 2 can be done in O(log n) time. Step 3 computes the distance between two center hulls, it is nothing but the minimum distance problem as in [3] The only ....

....S 1 and a right subset S 2 . Then we search over all entries of the matrix. When passing from one entry to an adjacent one, one point is moved from S 1 to S 2 or vice versa. We then update the farthest point Voronoi diagrams and center hulls of S 1 and S 2 as in [4, 5, 9] We refer the readers to [5] for additional details concerning the dynamic construction of FPVD and center hulls. And we conclude that Step 2 can be done in O(log n) time. Step 3 computes the distance between two center hulls, it is nothing but the minimum distance problem as in [3] The only difference is the algorithm ....

L.J. Guibas, D.E. Knuth and M. Sharir, Randomized Incremental Construction of Delaunay and Voronoi Diagrams, Algorithmica, 7(1992), pp.381413.

....locate point routine searches where in the triangulation p is. Point location is an important issue for any incremental algorithm, and several approaches have been proposed. Most e#cient methods rely on dedicated data structures, reaching the expected time of O(log n) to locate one point [6] 4] [11]. Alternatively, bucketing has also been used [16] with good performances for well distributed points. We follow the simpler jump and walk approach [14] which takes expected O(n ) time (in DTs) It has the advantage that no additional data structures are needed, which is an important issue in ....

....or a quadrilateral, and during the whole flipping process F (p) continues to be a star shaped polygon. The insertion of one point may require O(n) edge flips, however for DTs with a random input it is known that the expected number of edge flips is constant, no matter how they are distributed [11]. Note that if p is inserted in a constrained edge e, the crep list of indices of e is copied into the crep lists of the two new sub edges created after the division of e at point p. The final pseudo codes of the insertion routines are as follows: vertex insert point in edge ( p, e ) if p is ....

Guibas, L.J., Knuth, D.E., Sharir, M. (1992): Randomized Incremental Construction of Delaunay and Voronoi Diagrams. Algorithmica, 7, 381--413

....triangulation of P, denoted Del(P) is a PLC that contains all Delaunay simplices. If the points are in general position, that is, if no d 2 points in P are co spherical, then Del(P) is a simplicial complex. The Delaunay triangulation of a point set can be constructed in O(nlogn) time in 2D [10, 17, 16] and in O(n [a 2q) time in d dimensions [10, 31] A nice survey of these algorithms can be found in [16] One way to obtain a triangulation that conforms to the boundary of a PSLG domain is to use a constrained Delaunay triangulation. Let P be the set of vertices ofa PSLG q. Two pointsp and q in ....

L. J. Guibas, D. E. Knuth, and M. Sharir. Ran- domized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7:381-413, 1992.

....of a set of n point sites in the plane was a rather complicated divide and conquer algorithm by Shamos and Hoey [132] Since then many other algorithms have been developed. To illustrate the power of randomized algorithms, we describe a simple yet optimal randomized algorithm due to Guibas et al. [78]. The algorithm constructs the Delaunay triangulation of the set S. From this one can easily compute the Voronoi diagram of S in linear time, if needed. As for linear programming, we use the randomized incremental approach. Thus we insert the sites in S in a random order, and we maintain the ....

....current triangulation that contains p. 5. Replace A by three new triangles by connecting pi to the vertices of A. 6. Turn the new triangulation into a Delaunay triangulation by edge flipping. The simple procedure to find out which edge flips to perform is described in detail by Guibas et al. [78]. The only thing we have swept under the rug so far is step 4 of the algorithm, where we have to locate the triangle that contains p. Of course we could simply check all triangles, but this would lead to a quadratic algorithm. This shows that one still has to be careful when designing randomized ....

L. J. Guibas, D. E. Knuth, and M. Shaxir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381-413, 1992.

....tz sites in the plane in time O(kn ) with a constant factor that depends on s. More recently, in order to gain simplicity, several authors have designed algorithms which are incremental and randomized. Such an approach has been applied successfully for constructing Vorono diagrams in the plane [8,9] and in d space [2,10,11] A common point to all these randomized algorithms is that no distribution assumptions are made as it is the case, for example, in [12] Hence the results remain valid for any set of points, provided that the points are inserted at random. The algorithms in [10,9,11] are ....

....order k VoronoY diagrams is O(k[ nc ) Recently, Mulmuley [11,14] has obtained a randomized algorithm whose expected complexity meets this bound for d 2 and whose complexity is O(nk 2 nlog n) for d = 2. This algorithm also uses a conflict graph. None of the previous algorithms, except [2,8] are semi dynamic. If one new site is to be added, the VoronoY diagram has to be entirely reconstructed. In this paper, we present an algorithm that is semi dynamic. After each insertion of a new site, the algorithm updates a data structure, called the k Delaunay tree. This structure generalizes ....

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L.J. Guibas, D.E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi' diagrams. Algorithmica. To be published. Abstract published in LNCS 443 (ICALP 90).

....that correspond to averaging over the n possible permutations of the n inserted objects, each supposed to be equally likely to occur. 4.4.2 Amortization We have been able to bound the cost of inserting the k object in the I DAG. This cost is not amortized as opposed to the results in [BDT, GKS92] but the object may be any one of the inserted objects with the same probability. It must be noted however that the bound given in Theorem 4.3 cannot be a bound for the cost of inserting a given object. Indeed let us consider the construction of the Delaunay triangulation (the dual of the ....

L.J. Guibas, D.E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7(4):381--413, 1992.

....algorithms have been proposed, non optimal in the worst case but with a good randomized complexity. Some of these algorithms [9,15] use a conflict graph and so are static. The others are on line; a first idea of online algorithms was presented in [6] and a randomized analysis can be found in [12,4]. Recently, a very simple kind of analysis has been proposed for randomized geometric algorithms [22] Incremental randomized algorithms have also been used for constructing higher order Vorono diagrams [16,5,3] None of the above algorithms allows deletion. Using the algorithm of [1] a site can ....

L.J. Guibas, D.E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7(4):381 413, 1992. 26

.... geometry, and the algorith mic complexity is known to be at most as dimcult as that of construction convex hulls in one higher dimension[23] Commonly used techniques for lower dimensional Delau nay triangulation are divide and conquer, plane sweep and randomized incremental algorithms[23, 32], all of which have similar bounds on running time (or expected running time) of O(nlogn) in 2 and O(n 2) in 3, and can be generalized to higher dimensions. However the incremental Delaunay construction is known to be extremely useful for mesh generation, especially adaptive mesh generation. Point ....

....refinement method. This method by Ruppert[47] generates good quality Delaunay meshes for PSLGs in 2, and is almost the simplest 2D meshing algorithm in terms of implementation. The method is mainly based on the random ized incremental Delaunay triangulation algorithm by Guibas, Knuth and Sharir[32], and a refinement algorithm by Chew for used for constructing quality constrained Delaunay triangulations[13] It first generates a Delaunay triangulation for the ver tices of the input PSLG and then successively inserts Steiner points and updates the Delaunay triangulation until all the ....

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L. J. Guibas, D. E. Knuth and M. Sharir, "Randomized incremental construction of Delaunay and Voronoi diagrams," Algorithmica, 7, 381-413, 1992.

....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 con structions of convex hulls in d dimensions can be found in [21] Another approach is based on local ....

....of S. Joe [46] avoids this problem by adding the points in a sequence sorted along a fixed direction, for example, the x axis, and shows that the resulting algorithm has a time complexity of O(2) if S is a set of points in R s. This is optimal in the worst case. Guibas, Knuth, and Sharir [41] study the complexity of a similar algorithm in where points are added in a random sequence. They show that, while (2) flips are required in the worst case, the expected number of flips is only O( Edelsbrunner and Shah [28] extend this result to a randomized incremental flip algorithm in R E and ....

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L J Guibas, D E Knuth, and M Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. In G Goos and J Hartmanis, editors, Lecture Notes in Computer S'cience 3: ICALP 90 Proceedings, pages 414 431. Springer-Verlag, New York, 1990.

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L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381--413, 1992.

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L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381--413, 1992.

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L. Guibas, D. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7(4):381-413, 1992.

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L. Guibas, D. Knuth, and M. Sharir. Randomized Incremental Construction of Delaunay and Voronoi Diagrams. Algorithmica 7: 381-413, 1992.

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Guibas, L. J., Knuth, D. E., and Sharir, M. Randomized incremental construction of Delaunay and Vorono diagrams, in: Proc. 17th Int. Colloq.---Automata, Languages and Programming, Lecture Notes in Computer Science (LNCS) 443 (1990), Springer Verlag, Berlin, pp. 414-- 431 33

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L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381413, 1992.

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L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381--413, 1992.

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L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381--413, 1992.

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L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381413, 1992.

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L. Guibas, D. Knuth and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams, Algorithmica 7 (1992), pp. 381--413.

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L. J. Guibas, D. E. Knuth, and M. Sharir, "Randomized incremental construction of Delaunay and Voronoi diagrams," Algorithmica, vol. 7, pp. 381--413, 1992.

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L. J. Guibas, D. E. Knuth, and M. Sharir, "Randomized incremental construction of Delaunay and Voronoi diagrams," Algorithmica, vol. 7, pp. 381--413, 1992.

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Leonidas J. Guibas, D. E. Knuth, and Micha Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7:381413, 1992.

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L. J. Guibas, D. E. Knuth, and M. Sharir, "Randomized incremental construction of Delaunay and Voronoi diagrams," Algorithmica, vol. 7, pp. 381--413, 1992.

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L. Guibas, D. Knuth, and M. Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7(4):381-413, 1992.

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Guibas, L. J., Knuth, D. E., and Sharir, M. Randomized Incremental Construction of Delaunay and Vorono Diagrams, in Proc. 17th Int. Colloq.---Automata, Languages and Programming, vol. 443 of Springer Verlag LNCS (1990), Springer Verlag, Berlin, pp. 414--431

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