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Indexing moving points
, 2003
"... We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t; report all K points of S that lie inside R at time t: We first present an in ..."
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Cited by 191 (13 self)
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We propose three indexing schemes for storing a set S of N points in the plane, each moving along a linear trajectory, so that any query of the following form can be answered quickly: Given a rectangle R and a real value t; report all K points of S that lie inside R at time t: We first present an indexing structure that, for any given constant e> 0; uses OðN=BÞ disk blocks and answers a query in OððN=BÞ 1=2þe þ K=BÞ I/Os, where B is the block size. It can also report all the points of S that lie inside R during a given time interval. A point can be inserted or deleted, or the trajectory of a point can be changed, in Oðlog 2 B NÞ I/Os. Next, we present a general approach that improves the query time if the queries arrive in chronological order, by allowing the index to evolve over time. We obtain a tradeoff between the query time and the number of times the index needs to be updated as the points move. We also describe an indexing scheme in which the number of I/Os required to answer a query depends monotonically on the difference between the query time stamp t and the current time. Finally, we develop an efficient indexing scheme to answer approximate
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
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 89 (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
, 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 57 (11 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.
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 54 (3 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.
Efficient Randomized Algorithms for Some Geometric Optimization Problems
 DISCRETE COMPUT. GEOM
, 1995
"... In this paper we first prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let F be a collection of n totally or partially defined algebraic trivariate functions of constant maximum degree, with the add ..."
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Cited by 44 (15 self)
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In this paper we first prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let F be a collection of n totally or partially defined algebraic trivariate functions of constant maximum degree, with the additional property that, for a given pair of functions f; f 0 2 F , the surface f(x; y; z) = f 0 (x; y; z) is xymonotone (actually, we need a somewhat weaker propertysee below). We show that the vertical decomposition of the minimization diagram of F consists of O(n 3+" ) cells (each of constant complexity), for any " ? 0. In the second part of the paper we present a general technique that yields faster randomized algorithms for solving a number of geometric optimization problems, including (i) computing the width of a point set in 3space, (ii) computing the minimumwidth annulus enclosing a set of n points in the plane, and (iii) computing the `biggest stick' inside a simple polyg...
An ExpanderBased Approach to Geometric Optimization
 IN PROC. 9TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1993
"... We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach ..."
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Cited by 42 (15 self)
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We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo [34]. Our new approach
A NearLinear Algorithm for the Planar 2Center Problem
 Discrete & Computational Geometry
, 1996
"... We present an O(n log n)time algorithm for computing the 2center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous O(n log n)time algorithm of [10]. ..."
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Cited by 42 (6 self)
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We present an O(n log n)time algorithm for computing the 2center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous O(n log n)time algorithm of [10].
Optimal Slope Selection via Expanders
, 1993
"... Given n points in the plane and an integer k, the slope selection problem is to find the pair of points whose connecting line has the kth smallest slope. (In dual setting, given n lines in the plane, we want to find the vertex of their arrangement with the kth smallest xcoordinate.) Cole et al. ..."
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Cited by 29 (7 self)
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Given n points in the plane and an integer k, the slope selection problem is to find the pair of points whose connecting line has the kth smallest slope. (In dual setting, given n lines in the plane, we want to find the vertex of their arrangement with the kth smallest xcoordinate.) Cole et al. [6] have given an O(n log n) solution (which is optimal), using the parametric searching technique of Megiddo. We obtain another optimal (deterministic) solution that does not depend on parametric searching and uses expander graphs instead. Our solution is somewhat simpler than that of [6] and has a more explicit geometric interpretation. keywords: computational geometry, algorithms, design of algorithms 1 Introduction In this paper we consider the slope selection problem, as defined in the abstract. For convenience, we prefer to study its dual version. We thus have a collection L = f` 1 ; : : : ; ` n g of n lines in the plane, which we assume to be in general position, meaning that no li...
Optimal slope selection via cuttings
 Computational Geometry: Theory and Applications
, 1998
"... Abstract We give an optimal deterministic O(n log n)time algorithm for slope selection. The algorithm borrows from the optimal solution given in [?], but avoids the complicated machinery of the AKS sorting network and parametric searching. This is achieved by redesigning and refining the O(n log2 n ..."
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Cited by 24 (0 self)
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Abstract We give an optimal deterministic O(n log n)time algorithm for slope selection. The algorithm borrows from the optimal solution given in [?], but avoids the complicated machinery of the AKS sorting network and parametric searching. This is achieved by redesigning and refining the O(n log2 n)time algorithm of [?] with the help of additional approximation tools. 1 Optimal Slope Selection The problem is computing the line defined by two of n given points that has the median slope among all \Gamma n2 \Delta such lines. Equivalently, the problem can be stated as that of selecting the medianabscissa vertex of the arrangement A(L) of a set L of n lines [?]. For generality, we set out to compute the vertex with rank I\Lambda from left to right, for any given 1 ^ I \Lambda ^ \Gamma n2 \Delta.