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Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 76 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Geometric Applications of a Randomized Optimization Technique
 Discrete Comput. Geom
, 1999
"... We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, ..."
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Cited by 49 (6 self)
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We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal kpoint subsets, matching point sets under translation, computing rectilinear pcenters and discrete 1centers, and solving linear programs with k violations. 1 Introduction Consider the classic randomized algorithm for finding the minimum of r numbers minfA[1]; : : : ; A[r]g: Algorithm randmin 1. randomly pick a permutation hi 1 ; : : : ; i r i of h1; : : : ; ri 2. t /1 3. for k = 1; : : : ; r do 4. if A[i k ] ! t then 5. t / A[i k ] 6. return t By a wellknown fact [27, 44], the expected number of times that step 5 is execut...
Taking a Walk in a Planar Arrangement
 SIAM J. Comput
, 1999
"... We present a new randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve crosses, where the curve is g ..."
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Cited by 26 (6 self)
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We present a new randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve crosses, where the curve is given in an online manner), and to compute a level in an arrangement, both in an outputsensitive manner. The expected running time of the algorithm is O( t+2 (m+n) log n), where m is the number of intersections between the walk and the given arcs. No algorithm with similar performance is known for the general case of arcs. For the case of lines and segments, our algorithm improves the best known algorithm of [OvL81] by almost a logarithmic factor. 1 Introduction Let S be a set of n xmonotone arcs in the plane. Computing the whole (or parts of the) arrangement A( S), induced by the arcs of S, is one of the fundamental problems in computational geometry, and has received a lot o...
An improved bound for joints in arrangements of lines in space
, 2003
"... Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n ..."
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Cited by 9 (3 self)
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Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three noncoplanar lines meet. We show that the number of joints of L is O(n
On the complexity of many faces in arrangements of pseudosegments and of circles
 IN DISCRETE AND COMPUTATIONAL GEOMETRY: THE GOODMANPOLLACK FESTSCHRIFT
"... We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudosegments, n circles, or n unit circles. The bounds are worstcase optimal for unit circles; they are also worstcase optimal for the case of pseudosegments, except when the number of faces is very small, i ..."
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Cited by 8 (4 self)
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We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudosegments, n circles, or n unit circles. The bounds are worstcase optimal for unit circles; they are also worstcase optimal for the case of pseudosegments, except when the number of faces is very small, in which case our upper bound is a polylogarithmic factor from the bestknown lower bound. For general circles, the bounds nearly coincide with the bestknown bounds for the number of incidences between m points and n circles, recently obtained in [9].
Online Zone Construction in Arrangements of Lines in the Plane
 In Proc. of the 3rd Workshop of Algorithm Engineering
, 1999
"... Given a finite set L of lines in the plane we wish to compute the zone of an additional curve fl in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by fl, where fl is not given in advance but rather provided online portion by por ..."
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Cited by 6 (4 self)
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Given a finite set L of lines in the plane we wish to compute the zone of an additional curve fl in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by fl, where fl is not given in advance but rather provided online portion by portion. This problem is motivated by the computation of the area bisectors of a polygonal set in the plane. We present four algorithms which solve this problem efficiently and exactly (giving precise results even on degenerate input). We implemented the four algorithms. We present implementation details, comparison of performance, and a discussion of the advantages and shortcomings of each of the proposed algorithms. 1 Introduction Given a finite collection L of lines in the plane, the arrangement A(L) is the subdivision of the plane into vertices, edges and faces induced by L. Arrangements of lines in the plane, as well as arrangements of other objects and in higher dimensional spaces, ...
PointLine Incidences in Space
, 2002
"... Given a set L of n lines in R , joints are points in R that are incident to at least three noncoplanar lines in L. We show that there are at most O(n ) incidences between L and the set of its joints. ..."
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Cited by 5 (4 self)
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Given a set L of n lines in R , joints are points in R that are incident to at least three noncoplanar lines in L. We show that there are at most O(n ) incidences between L and the set of its joints.
Online zone construction in arrangements of lines in the plane
 Proc. of the 3rd Workshop of Algorithm Engineering
, 1999
"... Given a finite set L of lines in the plane we wish to compute the zone of an additional curve
in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by
, where
is not given in advance but rather provided online portion by portion. ..."
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Cited by 1 (0 self)
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Given a finite set L of lines in the plane we wish to compute the zone of an additional curve
in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by
, where
is not given in advance but rather provided online portion by portion. This problem is motivated by the computation of the area bisectors of a polygonal set in the plane. We present four algorithms which solve this problem efficiently and exactly (giving precise results even on degenerate input). Our main algorithm is a novel approach based on the binary plane partition technique. We implemented all four algorithms. We present implementation details, comparison of performance, and a discussion of the advantages and shortcomings of each of the proposed algorithms.