K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry," Springer-Verlag, New York#Berlin, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....carried out in time proportional to the change in the size of the incidence graph of the arrangement, whose accumulated cost is only O(m ) It thus suces to describe how to compute i and i eciently. We maintain a weight balanced binary tree T whose jth leftmost leaf stores the bundle P j [26]. For each node v 2 T, let S v P denote the set of points stored at the leaves of the subtree rooted at v. At each node v 2 T, we maintain a data structure D v = D(S v ) that supports the following operations on S v : Empty v ( Is one of the pseudo halfplanes determined by empty (of points ....

....arcs, and (ii) is a set of (portions of the) graphs of polynomials of bounded degree. 4.1 Querying with circular arcs Let be the set of circular arcs whose endpoints lie on the left and right boundaries of W . We construct a weight balanced binary tree T on the y coordinates of the points in S [26]. For a node v 2 T, let S v S be the set of points whose y coordinates are stored at the leaves of the subtree rooted at v, and put m v = jS v j. We map each point p = x p ; y p ) 2 S v to the point p = x p ; y p ; x p y p ) in R . Let S v = f p j p 2 S v g. We preprocess S v ....

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K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, Springer-Verlag, Heidelberg, Germany, 1984.

....the time to compute all tangents is O(n log n) By dynamically maintaining the same (balanced hierarchical) representation for the current polygonal region P , we can compute the (at most two) intersection points between P and the supporting line of s in O(log n) time. The reader is referred to [11], pages 84 88 for this representation and the for two statements below. Lemma 1 ( 11] page 85) A balanced hierarchical representation of a convex polygon on n points can be computed in O(n) time. Lemma 2 ( 11] page 87) Given a balanced hierarchical representation of a convex polygon on n ....

....(balanced hierarchical) representation for the current polygonal region P , we can compute the (at most two) intersection points between P and the supporting line of s in O(log n) time. The reader is referred to [11] pages 84 88 for this representation and the for two statements below. Lemma 1 ([11], page 85) A balanced hierarchical representation of a convex polygon on n points can be computed in O(n) time. Lemma 2 ( 11] page 87) Given a balanced hierarchical representation of a convex polygon on n points and a line l, P l can be computed in O(log n) time. One can obtain a balanced ....

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K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional searching and Computational Geometry, Springer Verlag, Berlin, 1984.

....in Section 8, including improved time bounds for Klee s measure problem [26] in the case of 4 dimensional unit hypercubes, and for the minimum diameter spanning tree problem, which are of independent interest. 2 The strategy for semi online dynamization The most common dynamization strategy [5, 24, 25] is based on decomposing a set of objects into subsets, solving the problem on each subset, and combining the answers. An update a ects only a small number of subsets and thus can be eciently handled. Unfortunately, this simple approach is not viable for any of our problems, because they are not ....

....O(b ) time. The total time for n updates is therefore e O( n=b) n n (b ) 2 = 1 ) 1) if we set the parameter b n = 1 ) This proves an amortized time bound of e O(n ) If the application insists on a worst case time bound, a well known modi cation (e.g. see [24, 25]) is required: spread the work of rebuilding the data structure for S evenly over the next b=2 updates. The data structure for S is available for the j th update whenever j mod b b=2. A similar shifted version of S and S can deal with the other case j mod b b=2. 2 All our e orts are now ....

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K. Mehlhorn, Data Structures and Algorithms 3: Multi-Dimensional Searching and Computational Geometry, Springer-Verlag, Heidelberg, 1984.

....Sharir [2] for the decision problem) In the incremental case, a static version of Theorem 1. 1 is again sucient, and although Eppstein s bound is better, our near O( n) update time bound can be made worst case instead of amortized, by a standard trick of spreading the rebuilding work over time [17]. It is interesting to see whether a faster real time algorithm for width is possible. In the decremental case, can our O(log n) amortized time bound be reduced to near logarithmic The planar width problem lifts to a nonconvex optimization problem over a 3 d halfspace intersection. Both of ....

K. Mehlhorn. Data Structures and Algorithms 3: Multi-Dimensional Searching and Computational Geometry. Springer-Verlag, Heidelberg, 1984.

....topic in algorithms research. It is therefore not surprising that geometric data structures have received a lot of attention in computational geometry. Traditionally, geometric data structures dealt mostly with orthogonal objects. Examples of such data structures are range trees and segment trees [99, 114, 122]. There were also data structures for non orthogonal objects, but their worst case behavior was not very satisfactory. In recent years new tools have been developed, cuttings and simplicial partitions, which can be used as a basis for data structures dealing with non orthogonal objects. It turns ....

K. Mehlhorn. Data structures and algorithms 3: Multi-dimensional searching and computational geometry. Springer-Verlag, Heidelberg, Germany, 1984.

....an interval of color i contains x. Therefore, the structures TL v and TR v store the intervals of each color that are most likely to be an answer to the query. Inserting an interval: To insert a new interval b = l; r] of color i into T , we first add b to T using the standard procedure, see [18] for details. Let v 2 T be the node at which b is stored. First, we insert its color i into TCL v , using the standard insertion procedure for balanced binary search trees. Then we insert the endpoints of b into the lists L v and v . Finally, if l (or r) is the leftmost (resp. rightmost) ....

....endpoint of the intervals in B v of color i, we update TL v (resp. TR v ) Since the secondary data structures of two nodes can be merged in linear time, and an interval can be inserted into a secondary structure in O(log n) time, the total amortized time required for inserting b is O(log n) see [18]) An interval is deleted in O(log n) time using a similar approach. We leave it to the reader to fill in the details. Answering a query: To report the colors of intervals intersected by a query point x, we follow a path in T starting at the root. At each node v we do the following. Suppose x x ....

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Mehlhorn, K., Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, Springer-Verlag, Berlin, 1984.

....with each node there is an associated structure that stores some set of objects. Splitting then normally means rebuilding such associated struc tures, which takes too much time. Hence, more sophisticated techniques are required. In this paper we devise such a method for segment trees (see [3, 4, 8, 9, 11, 14]) This variant of the structure we design works efficiently and might be useful in a number of geometric applications (it was recently used for solving some robotics problem Is] Segment trees are used to store intervals on a line. Such intervals might overlap. When splitting a segment tree we ....

....The segment tree is a one dimeusional data structure used for solving many two dimeusional problems, such as finding rectangle intersections and finding the contour of the union of rectangles. It was introduced by Bentley in 1977 [3] see 2 [a:x) b:y) Figure 1: Insertion in a segment tree also [4, 8, 9, 11, 14]) Suppose n segments (intervals) al. hi] a2: bi] an: bn ] on the real line are given. Segments may intersect or overlap, and left and right endpoints of two segments may be equal. We sort all different endpoints, obtaining an ordered sequence x, x2, x, where rn 2n. These endpoints ....

Mehlhorn, K., Data structures and algorithms $: multi-dimensional searching and computational geometry, Springer-Verlag, Berlin, 1984.

....it is sufficient to compute all intersection points between the segments of S n R and the outer polygonal chains in O. Let JO be an interval corresponding to the projection of the polygoal chain O 2 O onto the x axis, and let J = fJO j O 2 Og. We construct an interval tree T on J ; see Mehlhorn [17] for details on interval trees. T is a minimum height binary tree with at most 2m leaves. Each node v of T is associated with an interval U v , and a point x v . Let W v = U v Theta [ Gamma1; 1] be a vertical strip, and let h v be the vertical 10 line passing through x v . For the root u, W u ....

K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, Springer-Verlag, Berlin, 1984.

....systems, typesetting, computer games and Computer Aided Design (CAD) Besides these application, computational geometry developed a solid theoretical framework and lead to several powerful paradigms for geometric algorithms. The maturity of the field is documented in textbooks [PS85, dBvK 97, Meh84b] and handbooks [GO97, SU00] on computational geometry. One of the first questions computational geometry raises is that of the nature of an algorithm. One of the oldest type of algorithms is the construction with ruler and compass. There one can draw a line through two (already constructed) ....

K. Mehlhorn, Data structures and algorithms 3: Multidimensional searching and computational geometry, Springer-Verlag, Berlin, 1984.

....we present the definitions and main properties of fundamental structures in Computational Geometry, for which algorithms of constructions are presented in the following chapters. The complexity of the best known deterministic algorithms is also given. Most of the results are taken from [Ede87, Meh84, PS85] where more details can be found, especially concerning algorithms, that will not be studied here. denotes the d dimensional Euclidean space, with scalar product h Delta; Deltai, and Euclidean distance ffi. Terms such as vertical , above , below , will be often used : the vertical ....

K. Mehlhorn. Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry. Springer-Verlag, 1984.

....the field is relatively young, but it has since blossomed enormously within the framework of algorithm design, data structures, and complexity theory. In the mid eighties, it achieved a somewhat mature status, as indicated by the publication of several books on this subject (for example, Mehlhorn [58], Preparata and Shamos [67] Toussaint [77] or Edelsbrunner [21] Computational geometry seemed to provide an exceedingly fruitful arena for the more theoretical computer scientist (Forrest [34] and now, in the nineties, it even found its way into the introductory texts of the computer ....

K Mehlhorn. Data Structures and Algorithms 3: Multidimensional Sorting and Computational Geometry. Springer-Verlag, Berlin, 1984.

....is worst case nearly optimal for the problem at hand. For precise definitions of these concepts the reader is referred to [BGH97] The results of this paper are based on kinetizing (i.e. maintaining under continuous point motion) a certain type of multidimensional range search tree (MDRS tree) Meh84] used to query the value of some specified function on subsets of the points [BS80] We apply this MDRS tree to obtain kinetic data structures for the closest pair among n moving points in IR , the closest pair among a collection of n moving objects, and approximate and exact MSTs for n moving ....

....search tree In this section, we study in the kinetic setting the classic range searching problem when the ranges are determined by a fixed set of directions. This version of the range searching problem has received extensive treatment in the literature both in the static and in the dynamic case [Meh84, Mul94] We focus here on a specific version of this problem that will be well suited for the applications to the closest pair and the MST. In the generic range searching problem, a set S of n points in IR is given, together with a (generally infinite) set of ranges which are certain ....

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Kurt Mehlhorn. Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, volume 3 of EATCS Monographs on Theoretical Computer Science. SpringerVerlag, Heidelberg, Germany, 1984.

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K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry," Springer-Verlag, New York#Berlin, 1984.

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K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry," Springer-Verlag, New York#Berlin, 1984.

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K. Mehlhorn. Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry. EATCS Monographs on Theoretical Computer Science, Springer Verlag, Berlin, 1984.

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K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry," Springer-Verlag, New York#Berlin, 1984.

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K. Mehlhorn, Data structures and algorithms 3: multi-dimensional searching and computational geometry. Springer-Verlag, 1984.

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K. Mehlhorn. Data structures and algorithms 3: Multi-dimensional searching and computational geometry. Springer-Verlag, 1984. 91-032.

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K. Mehlhorn. Data structures and algorithms 3: Multi dimensional searching and computational geometry. 1984.

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Kurt Mehlhorn. Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, volume 3 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Heidelberg, Germany, 1984.

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K. Mehlhorn, Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry," Springer-Verlag, New York#Berlin, 1984.

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K. Mehlhorn. Data Structures and Algorithms 3: Multi-Dimensional Searching and Computational Geometry. Springer-Verlag, Heidelberg, 1984.

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K. Mehlhorn. Data Structures and Algorithms 3: Multidimensional Searching and Computational Geometry. Springer, 1984.

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Kurt Mehlhorn. Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1984.

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Mehlhorn, K., Data structures and algorithms 3: multi-dimensional searching and computational geometry, Springer- Verlag, 1984.

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