H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317--340, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... plane is measured by sampling a set S of points on the surface of O (e.g. using coordinate measurement machines) and computing the width of the thinnest shell containing S [21] Motivated by this and other applications, the problem of computing A (S) in the plane has been studied extensively [2, 6 8, 18, 19, 22, 26, 28, 30 33, 35, 36, 38]. Ebara et al. 18] noticed that in the planar case the center of A (S) is a vertex of the overlay of the nearest and farthest neighbor Voronoi diagrams of S. This property was later re ned and extended in [32, 36] These observations immediately lead to an O(n ) time algorithm for ....

H. Edelsbrunner, L. J. Guibas, and J. Stol , Optimal point location in a monotone subdivision, SIAM J. Comput., 15 (1986), 317-340.

....area was Kirkpatrick s elegant method based on hierarchical triangulations [9] which supported query processing in O(log n) time using O(n) space. This was followed by a number of other optimal methods with better practical performance including the layered DAG of Edelsbrunner, Guibas, and Stolfi [7], searching in similar lists by Cole [6] the method based on persistent search trees by Sarnak and Tarjan [13] and the randomized incremental method of Mulmuley [12] and Seidel [14] The question of the exact constant factor in query time was raised in work by Goodrich, Orletsky # Department of ....

H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317--340, 1986.

....and Kirkpatrick [11] presented a simple approach based on hierarchical triangulations, which reduced the space to O(n) This was followed by a number of other methods with better performance in terms of constant factors and simplicity. These include the methods by Edelsbrunner, Guibas, and Stolfi [8], Cole [5] and Sarnak and Tarjan [17] and randomized methods by Mulmuley [14] and Seidel [18] Recently, the question of the exact constant factor in query time was raised in work by Goodrich, Orletsky and Ramaiyer [9] This question was answered definitively by Adamy and Seidel [1] who showed ....

H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317--340, 1986.

....q can be reported quickly. The first worst case asymptotically optimal algorithm for this problem was due to Kirkpatrick, which achieved O(n) space and O(log n) query time [8] This was followed by a number of related methods with better practical performance by Edelsbrunner, Guibas, and Stolfi [7], Cole [6] and Sarnak and Tarjan [13] In spite of their enhanced practicality, these methods lacked the simplicity of Kirkpatrick s method. A truly simple randomized incremental method was discovered by Mulmuley [11] and Seidel [14] This method is based on randomly inserting the line segments ....

H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317--340, 1986.

....data structure, or prove that the problem is hard in some respect. Computational geometry problems can be motivated by either fundamental issues, such as vhether or not there is a linear time algorithm to triangulate a poly gon [15] or by practical issues, such as point location data structures [8, 16, 32, 55], pattern layout algorithms [62] and industrial part casting algorithms [13, 77] The vork in this dissertation is motivated by a hydrology problem from geographic information systems. A geographic information system (GIS) is a system for apturing, storing, checking, integrating, manipulating, ....

H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Cornput., 15(2):317-340, 1986.

....P from its initial position to a new position according to the vector Bp BQ connecting the two centers of gravity of two sets P and Q. VORONOIDIAGRAM takes as input a set of points and returns the Voronoi Diagram of such a set and the corresponding tree needed to perform nearest neighbor search [6, 13]. Its construction requires O(k log c) time where k: NEAREST NEIGHBOR which takes as input the Voronoi Diagram V of the set Q, a point qi ( and the set of points P and returns the nearest point of P inside the Voronoi polygon having as representative qi. It requires O(log k) for each point of ....

H. Edelsbronner, L.J. Guibas, and J. Stolfi, Optimal Point Location in a Monotone Subdivision, SIAM Journal on Computing, 15(2):317-340, 1986.

....P from its initial position to a new position according to the vector Bp BQ connecting the two centers of gravity of two sets P and Q. VORONOIDIAGRAM takes as input a set of points and returns the Voronoi Diagram of such a set and the corresponding tree needed to perform nearest neighbor search [7, 14]. Its construction requires O(k log k) time where k: IQI; NEAREST NEIGHBOR which takes as input the Voronoi Diagram VQ of the set Q, a point qi ( and the set of points P and returns the nearest point of P inside the Voronoi polygon having as representative qi. It requires O(log k) for each ....

H. Edelsbronner, L.J. Guibas, and J. Stolfi, Optimal Point Location in a Monotone Subdivision, SIAM Journal on Computing, 15(2):317-340, 1986.

....range searching to place our result in a broader perspective. Point location in 2 space has been studied extensively and solved in a satisfactory way for many types of scenes, as several solutions achieve logarithmic query time and (near ) linear storage, after (near )linear preprocessing time [12, 14, 20]. In higher dimensional spaces, on the contrary, efficient solutions are available only for restricted problem instances. In 3 space, Chazelle [5] obtains O(log n) query time and an O(n) storage requirement for the case where the stored geometric objects are the 3 cells of a spatial ....

H. Edelsbrunner, L.J. Guibas, and J. Stolfi, Optimal point location in a monotone subdivision, SIAM Journal on Computing 15 (1986), pp. 317-340.

....points in P. 13 Consider the subdivision of P induced by the cut segments once more. First it has to be determined which regions (rectangles) contain the two query points, that is, the leaves andTt where the paths to s and t end. Using the optimal point location method of Edelsbrunner et al. [6]) this can be done in time O(logn) with a structure that uses O(n) storage. Observe that, since we can compute the vertex edge visisble pairs in linear time, we can turn P into a monotone subdivision in linear time and, hence, the point location structure can he built in linear time. Again a ....

.... region (depending on whether the ray is directed to the right or to the left, of course) Observe that the subdivision is monotone and, hence, the region which contains the starting point of the query ray can be determined in O(log n) time with a structure using O(n) space and preprocessing ([6]) Thus the extra storage and preprocessing that is needed at some node is O(IPI) since Z IHI O(IPI) for the partitioning of P into histograms Hi as used. Because the query time of our ray shooting structure is O(log n) the total query time becomes O(log n I) We conclude: query points in a ....

Edelsbrmner, H., L.J. Guibas and J. Stolid, Optimal point location in a mono- tone subdivision, SIAM J. Cornput. 15(1985), pp. 317-340.

....optimal the consttmts in the triangular refinement method are very high which makes the algorithm unpractical. Hence, people have worked on designing other techniques that are also fast in practice. The best method known today is probably the technique of Edelsbrunner, Guibas and Stolfi [23] (see also [22] chapter 11) Many different types of Voronoi diagrams do exist. The diagram presented in section 3.1 is just the simplest one. First of all one can usedifferent distance functions. Lee and Wong [52] showed how to compute Voronoi diagrams for the L and Lo metric. Lee [48] ....

Edelsbrunner, H., Guibas, L.J., Stolfi, J., Optimal point location in a monotone subdivision. SIAM J. Cornput. 15, 317-340 (1986)

....data structure for queries with starting point outside the polygon can be built in the same way by interpreting the exterior of P as a generalized polygon. Our data structure is based on the hierarchical vertical decomposition 7 of P. We first compute 7 and a point location structure for 7 [7]. A region r 7 has at most three doors, and therefore at most two pairs of doors through which an x monotone path could traverse r. Let dl, d2 be such a pair of doors. Let A be the set of trapezoids connecting dl and d2. We suppose for the moment that we have constructed a data structure that ....

....diagram of A can be computed in O(IA I log IA l) time [14] One can then break up the curved edges into x monotone pieces and compute a vertical decomposition using plane sweep in O(IA l log I A I) time. Afterwards, we build a point location structure for monotone subdivisions with curved edges [7]. To perform the query, we first test whether lies below re(A ) If that is the case, we then locate the center x of our query arc in the Voronoi diagram, and thus obtain the segment s E A closest to x. If and only if lies below re(A ) and the distance of s to x is larger than the radius of ....

H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Cornput., 15:317-340, 1986.

....disks. This results in a subdivision of the plane into O(n) regions, which is monotone in the horizontal direction. Preprocess this subdivision for efficient point location, choosing one of the methods that works for circular arcs too, such as Cole s [5] or the structure of Edelsbrunner et al. [8]. This solves the problem of finding the disks that contain the endpoints of the query segment s in O(n) space and O(log n) query time (there are at most two answers) To find the other disks that intersect s, the following structure is used. Let T be a standard conjugation tree on the centers of ....

....starting point q does not lie in any disk of . So in the sequel we assume that the starting point q for the shooting query lies outside the union of the disks. This can easily be tested in O(log n) time by building a point location structure on the union of the disks, which has linear size (see [8], 12] We use the following lemma to simplify the problem: Lemma 6 Given a set D of (possibly intersecting) disks, and a ray shooting quer with point q and direction d, then the first disk that is hit (i.e. the answer to the query) has its center in the halfplane H, which is bounded by the ....

Edelsbrunner, H., L.J. Guibas, and J. Stolfi, Optimal point location in a monotone subdivision, SIAM J. Cornput. 15 (1986), pp. 317-340.

....O(jP j log jV (P )j) time using O(jP j) space. Proof: Compute the facets of the 3 dimensional upper hull F ( Delta (P ) and store f Delta (r)# : Delta (r) 2 F ( Delta (P ) g, which is a set of O(jV (P )j) triangles in E with disjoint interiors, in a planar point location structure [14, 30, 33]; this takes O(jP j log jV (P )j) time. For each p 2 P , we can then test whether p# 2 int S Delta in logarithmic time by finding a facet Delta (r) of F ( Delta (P ) with Delta (p)# 2 Delta (r)# and then determining which side of r# the point p# lies on. 2 Another operation on the region ....

H. Edelsbrunner, L. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15:317--340, 1986.

....point location on their hierarchical data structure [12] in O( log rO log p) tinhe, which is optimal. Unfortunately, the preprocessing requirements are high: they use O(1 2 r) tinhe with O( 3) processors on a CREW PRAM. In [17] we show how to construct the bridged separator tree data structure [13, 9] for a monotone subdivision in O(1 ) tinhe using log processors on an EREW PRAM. The bridged separator tree is very efficient in practice [ The search path used to process a point location query is highly implicit, due to the spacesaving nature of the bridged separator tree, which makes ....

.... takes O(log n) time using a CREW FRAM with n processors [1, 18, 10] or a CReW PRAM with n log n processors [11] Parallel triangu lation can also be performed using a randomized CREW FRAM algorithm that runs in O(log nloglogn) time and does O(n) expected work [6, 5] The bridged separator tree [13, 9] uses O(n) space and supports point location queries in 8 in O(1 n) time. It is a balanced binary tree T with catalogs where searches are performed implicitly. We recall that a separator cr of is a monotone chain from oc to oc in the vertical direction. Let rl, r2, rf be the regions of ....

H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Cornput., 15:317 340, 1986.

....an orthogonal BSP of size O(m) whose leaf cells are empty. We can construct such a spanning tree in time O ( n rs) log rs) we need O (rs log rs) time to construct the BSP [5] plus O( n rs) log rs) time to locate the sites in the BSP subdivision using an optimal point location structure [6], and O(n rs) for the bottom up construction of the spanning tree. In the next section we will show that a linear cost spanning tree exists for any set of disjoint barrier segments (even if they are not orthogonal) however, we do not know of an equally efficient way to construct the tree in ....

H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Cornput., 15:317-340, 1986.

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H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317--340, 1986.

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Edelsbruner H., Guibas L.,Stolfi J., Optimal point location in a monotone subdivision, Technical Report 2, DEC systems Research Center, 1984.

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H. Edelsbrunner, L. Guibas, and J. Stol . Optimal point location in a monotone subdivision. SIAM J. Computing, 15:317-340, 1986.

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H. Edelsbrunner, L.J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317--340, 1986. 16

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H. Edelsbrunner, L. J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317--340, 1986.

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H. Edelsbrunner, L. J. Guibas, and J. Stol . Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15:317-340, 1986.

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H. Edelsbrunner, L.J. Guibas, and J. Stol . Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317-340, 1986.

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H. Edelsbrunner, L. J. Guibas, and J. Stol . Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317-340, 1986.

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H. Edelsbrunner, L.J. Guibas and J. Stolfi, Optimal point location in a monotone subdivision, SIAM J. Comput., 15:2 (1986), pp. 317-340.

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H. Edelsbrunner, L.J. Guibas, and J. Stol , Optimal point location in a monotone subdivision, SIAM J. Computing 15 (1986), 317-340.

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