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26
External Memory Data Structures
, 2001
"... In many massive dataset applications the data must be stored in space and query efficient data structures on external storage devices. Often the data needs to be changed dynamically. In this chapter we discuss recent advances in the development of provably worstcase efficient external memory dynami ..."
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Cited by 81 (36 self)
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In many massive dataset applications the data must be stored in space and query efficient data structures on external storage devices. Often the data needs to be changed dynamically. In this chapter we discuss recent advances in the development of provably worstcase efficient external memory dynamic data structures. We also briefly discuss some of the most popular external data structures used in practice.
On approximating the depth and related problems
 SIAM J. Comput
"... We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear pr ..."
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Cited by 63 (11 self)
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We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries. 1
Efficient Searching with Linear Constraints (Extended Abstract)
"... ) Pankaj K. Agarwal Lars Arge y Jeff Erickson z Paolo G. Franciosa x Jeffrey Scott Vitter  Abstract We show how to preprocess a set S of points in R d to get an external memory data structure that efficiently supports linearconstraint queries. Each query is in the form of a linear c ..."
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Cited by 58 (17 self)
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) Pankaj K. Agarwal Lars Arge y Jeff Erickson z Paolo G. Franciosa x Jeffrey Scott Vitter  Abstract We show how to preprocess a set S of points in R d to get an external memory data structure that efficiently supports linearconstraint queries. Each query is in the form of a linear constraint a \Delta x b; the data structure must report all the points of S that satisfy the query. Our goal is to minimize the number of disk blocks required to store the data structure and the number of disk accesses (I/Os) required to answer a query. For d = 2, we present the first nearlinear size data structures that can answer linearconstraint queries using an optimal number of I/Os. We also present a linearsize data structure that can answer queries efficiently in the worst case. We combine these two approaches to obtain tradeoffs between space and query time. Finally, we show that some of our techniques extend to higher dimensions d. Center for Geometric Computing, Computer...
LowDimensional Linear Programming with Violations
 In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci
, 2002
"... Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given half ..."
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Cited by 46 (3 self)
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Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3d runs in near O(n + k ) expected time. Interestingly, the idea is based on concavechain decompositions (or covers) of the ( k)level, previously used in proving combinatorial klevel bounds.
Dynamic Planar Convex Hull Operations in NearLogarithmic Amortized Time
 JOURNAL OF THE ACM
, 1999
"... We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangentfinding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of ..."
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Cited by 35 (6 self)
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We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P , such as membership and tangentfinding. Updates take O(log 1+" n) amortized time and queries take O(log n) time each, where n is the maximum size of P and " is any fixed positive constant. For some advanced queries such as bridgefinding, both our bounds increase to O(log 3=2 n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log 2 n) time per update. 1 Introduction Although the algorithmic study of convex hulls is as old as computational geometry itself, the basic problem of optimally maintaining the planar convex hull under insertions and deletions of points [30, 34] remains unsolved and has been regarded by some as one of the foremost open problems in the area [14, 26]. Besides its natural appeal, such a dynamic data structure has a wide range of applications, as it is often us...
On Range Reporting, Ray Shooting and klevel Construction
"... We describe the following data structures. For halfspace range reporting, in 3space using expected preprocessing time O(n log n), worstcase storage O(n log log n) and worstcase reporting time O(log n + k), where n is the number of data points and k the number of points reported; in dspace, with ..."
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Cited by 24 (0 self)
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We describe the following data structures. For halfspace range reporting, in 3space using expected preprocessing time O(n log n), worstcase storage O(n log log n) and worstcase reporting time O(log n + k), where n is the number of data points and k the number of points reported; in dspace, with d even, using worstcase preprocessing time O(n log n), storage O(n) and reporting time O(n 1 1=bd=2c log c n + k), where c is a constant. For ray shooting in a convex polytope in dspace determined by n facets, using deterministic preprocessing time O((n= log n) bd=2c log c n) and storage O((n= log n) bd=2c 2 c log n ) and with query time O(log n). For ray shooting in arbitrary direction among n hyperplanes using preprocessing O(n d = log bd=2c n) and query time O(log n). We also describe a randomized algorithm for constructing the klevel of n planes in 3space. In the case of planes dual to points in convex position, in which the size of the klevel is O(nk), the a...
Randomized incremental construction of threedimensional convex hulls and planar Voronoi diagrams, and approximate range counting
 in Proceedings of the Seventeenth Annual ACMSIAM Symposium on Discrete Algorithms
, 2006
"... We present new algorithms for approximate range counting, where, for a specified ε> 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [12], for this task, and then specialize it to two important instan ..."
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Cited by 19 (7 self)
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We present new algorithms for approximate range counting, where, for a specified ε> 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [12], for this task, and then specialize it to two important instances of range counting: halfspaces in R 3 and disks in the plane. The technique reduces the approximate range counting problem to that of finding the minimum rank of a data object in the range, with respect to a random permutation of the input. A major technical step in our analysis, which we believe to be of independent interest, is a bound of O(n log n) on the expected complexity of the overlay of all the Voronoi faces that are generated during a randomized incremental construction of the Voronoi diagram of n points in the plane. The same bound holds for the expected complexity of the overlay of all the faces of the minimization diagram of the lower envelope of n planes in R 3, or for the expected complexity of the overlay of all the normal (or Gaussian) diagram faces of the convex hull of n points in R 3, that are generated during a randomized incremental construction of the lower envelope or of the hull, respectively. All these bounds are tight in the worst case.
Optimal halfspace range reporting in three dimensions
 In Proceedings of the 20 th ACMSIAM Symposium on Discrete Algorithms
, 2009
"... We give the first optimal solution to a standard problem in computational geometry: threedimensional halfspace range reporting. We show that n points in 3d can be stored in a linearspace data structure so that all k points inside a query halfspace can be reported in O(log n + k) time. The data st ..."
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Cited by 14 (7 self)
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We give the first optimal solution to a standard problem in computational geometry: threedimensional halfspace range reporting. We show that n points in 3d can be stored in a linearspace data structure so that all k points inside a query halfspace can be reported in O(log n + k) time. The data structure can be built in O(n log n) expected time. The previous methods with optimal query time required superlinear (O(n log log n)) space. We also mention consequences, for example, to higher dimensions and to externalmemory data structures. As an aside, we partially answer another open question concerning the crossing number in Matouˇsek’s shallow partition theorem in the 3d case (a tool used in many known halfspace range reporting methods). 1
Remarks on kLevel Algorithms in the Plane
, 1999
"... In light of recent developments, this paper reexamines the fundamental geometric problem of how to construct the klevel in an arrangement of n lines in the plane. ffl The author's recent dynamic data structure for planar convex hulls improves a decadeold sweepline algorithm by Edelsbrunner and ..."
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Cited by 12 (6 self)
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In light of recent developments, this paper reexamines the fundamental geometric problem of how to construct the klevel in an arrangement of n lines in the plane. ffl The author's recent dynamic data structure for planar convex hulls improves a decadeold sweepline algorithm by Edelsbrunner and Welzl, which now runs in O(n log m+m log 1+" n) deterministic time and O(n) space, where m is the output size and " is any positive constant. We discuss simplification of the data structure in this particular application, by viewing the problem kinetically. ffl HarPeled recently announced a randomized algorithm with an expected running time of O((n + m)ff(n) log n). We observe that a version of an earlier randomized incremental algorithm by Agarwal, de Berg, Matousek, and Schwarzkopf yields almost the same result. ffl The current combinatorial bound by Dey shows that m = O(nk 1=3 ) in the worst case. We give an algorithm that guarantees O(n log n + nk 1=3 ) expected time. 1 Introd...
On Enumerating and Selecting Distances
 Int. J. Comput. Geom. Appl
, 1999
"... Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selec ..."
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Cited by 9 (2 self)
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Given an npoint set, the problems of enumerating the k closest pairs and selecting the kth smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n + k) running time in any fixeddimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n + n 2=3 k 1=3 log 5=3 n). We also describe outputsensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search. Keywords: distance enumeration, distance selection, closest pairs, range counting, randomized algorithms. 1 Introduction Finding the closest pair of an npoint set has a long history in computational geometry (see [34] for a nice survey). In the plane, the problem can be solved in O(n log n) time using the Delaunay triangulation. In an arbitrary fixed dimension d, the first O(n log n) algorithm, based on di...