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11
Applications of Random Sampling in Computational Geometry, II
 Discrete Comput. Geom
, 1995
"... We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms ..."
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Cited by 396 (12 self)
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We use random sampling for several new geometric algorithms. The algorithms are "Las Vegas," and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divideandconquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of n points in E d in O(n log n) expected time for d = 3, and O(n bd=2c ) expected time for d ? 3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set of n...
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Derandomization in Computational Geometry
, 1996
"... We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, repla ..."
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Cited by 17 (1 self)
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We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, nonrandom constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...
Motorcycle Graphs and Straight Skeletons
, 2002
"... We present a new algorithm to compute a motorcycle graph. It runs in O(n p n log n) time when n is the size of the input. We give a new characterization of the straight skeleton of a polygon possibly with holes. ..."
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Cited by 13 (1 self)
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We present a new algorithm to compute a motorcycle graph. It runs in O(n p n log n) time when n is the size of the input. We give a new characterization of the straight skeleton of a polygon possibly with holes.
Finding Stabbing Lines in 3Space
, 1992
"... A line intersecting all polyhedra in a set B is called a "stabber" for the set B. This paper addresses some combinatorial and algorithmic questions about the set S(B) of all lines stabbing B. We prove that the combinatorial complexity of S(B) has an O(n 3 2 c p log n ) upper bound, where n is the to ..."
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Cited by 13 (3 self)
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A line intersecting all polyhedra in a set B is called a "stabber" for the set B. This paper addresses some combinatorial and algorithmic questions about the set S(B) of all lines stabbing B. We prove that the combinatorial complexity of S(B) has an O(n 3 2 c p log n ) upper bound, where n is the total number of facets in B, and c a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one.
Fast Construction of Near Optimal Probing Strategies
 Algorithms for Robotics Motion and Manipulation
, 1999
"... Probing is a common operation employed to reduce the position uncertainty of objects. This thesis demonstrates a technique for constructing provably near optimal probing strategies for precisely localizing polygonal parts. This problem is shown to be dual to the well studied grasping problem of comp ..."
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Cited by 11 (0 self)
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Probing is a common operation employed to reduce the position uncertainty of objects. This thesis demonstrates a technique for constructing provably near optimal probing strategies for precisely localizing polygonal parts. This problem is shown to be dual to the well studied grasping problem of computing optimal nger placements as dened by Mishra et al. [18] and others [11, 17]. A useful quality metric of any given probing strategy can easily be computed from simple geometric constructions in the displacement space of the polygon. The approach will always nd a minimal set of probes that is guaranteed to be near optimal for constraining the position of the polygon. The size of the resulting set of probes is within O(1) of the optimal number of probes and can be computed in O(n log 2 n) time whereas the exact optimal solution is in NPhard [8]. The result of this work is a probing strategy useful in practice for rening part poses. Acknowledgments I would like to express my most s...
On Lines Missing Polyhedral Sets in 3Space
 Discrete Comput. Geom
, 1993
"... We show some combinatorial and algorithmic results concerning sets of lines and polyhedral objects in 3space. Our main results include: (1) An O(n 3 2 c p log n ) upper bound on the worst case complexity of the set of lines missing a starshaped compact polyhedron with n edges, where c is a ..."
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Cited by 8 (0 self)
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We show some combinatorial and algorithmic results concerning sets of lines and polyhedral objects in 3space. Our main results include: (1) An O(n 3 2 c p log n ) upper bound on the worst case complexity of the set of lines missing a starshaped compact polyhedron with n edges, where c is a suitable constant. (2) An O(n 3 2 c p log n ) upper bound on the worst case complexity of the set of lines that can be moved to infinity without intersecting a set of n given lines, where c is a suitable constant. This bound is almost tight. (3) An O(n 1:5+ffl ) randomized expected time algorithm that tests whether a direction v exists along which a set of n red lines can be translated away from a set of n blue lines without collisions. (4) Computing the intersection of two polyhedral terrains in 3space with n total edges in time O(n 4=3+ffl + k 1=3 n 1+ffl + k log 2 n), where k is the size of the output, and ffl ? 0 an arbitrary small but fixed constant. This algorithm ...
Online Maintenance of Visibility and ShortestPath Information
, 1994
"... Given a simple polygon P and a point p 2 P , we show how to maintain the visibility polygon from p, the shortest path tree from p, and the corresponding shortest path partition as p is translated inside P . Given a direction of motion of p, we can determine how far p can move until the first combin ..."
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Cited by 2 (1 self)
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Given a simple polygon P and a point p 2 P , we show how to maintain the visibility polygon from p, the shortest path tree from p, and the corresponding shortest path partition as p is translated inside P . Given a direction of motion of p, we can determine how far p can move until the first combinatorial change in the above structures occurs in O(logn) time, and update the data structure in O(log 2 n) time; preprocessing requires O(n log n) time and linear space. For polygons with holes, we can maintain the visibility information online, but the preprocessing time and space increase to O(E + n log n) and O(E), respectively, where E is the size of the vertex visibility graph. We apply these results to construct a method for answering queries for the visibility polygon for a given viewpoint in O((n 3 =m) log 2 n + k) time after preprocessing which takes O(m log 2 n) time and O(m) space, where m is any number between n 2 and n 3 , and k is the output size. This provides a t...
Algorithms for Weak EpsilonNets
, 1995
"... In the plane, we can find a weak "net for convex sets consisting of O(" \Gamma2 ) points, in time O(n" \Gamma2 log 2 (1=")). We can determine the smallest " for which a given planar set is an "net in time O(n 3 ). In IR d , we [3] can find weak "nets of size O i 1 " d log O(1) 1 " ..."
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In the plane, we can find a weak "net for convex sets consisting of O(" \Gamma2 ) points, in time O(n" \Gamma2 log 2 (1=")). We can determine the smallest " for which a given planar set is an "net in time O(n 3 ). In IR d , we [3] can find weak "nets of size O i 1 " d log O(1) 1 " j in time O(n (1=") O(1) ) (both exponents depending on d). Work by Bernard Chazelle has been supported by NSF Grant CCR9002352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR8921421. Work by David Eppstein has been supported by by NSF grant CCR9258355. Work by Michelangelo Grigni has been supported by National Science and Engineering Research Council Operating Grants and BC Advanced Systems Institute Fellowship Awards. Work by Leonidas Guibas was supported by NSF grant CCR9215219. Work by Leonidas Guibas and Micha Sharir has also been supported by a grant from the U.S.Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sh...