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16
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
Data structures for halfplane proximity queries and incremental Voronoi diagrams
 In Proceedings of the 7th Latin American Symposium on Theoretical Informatics, volume 3887 of Lecture Notes in Computer Science
, 2006
"... We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line ℓ in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject t ..."
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Cited by 10 (3 self)
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We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line ℓ in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ. We present two data structures for this problem. The first data structure uses O(n 1+ε) space and preprocessing time, and answers queries in O(2 1/ε log n) time. The second data structure uses O(n log 3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n 2) space. In the process of developing the second data structure, we develop a new representation of nearestpoint and farthestpoint Voronoi diagrams of points in convex position. This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)time pointlocation queries, even though every such update may make Θ(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)time pointlocation queries. 1
Online geometric reconstruction
 Proc. of 22nd SOCG
, 2006
"... We investigate a new class of geometric problems based on the idea of online error correction. Suppose one is given access to a large geometric dataset though a query mechanism; for example, the dataset could be a terrain and a query might ask for the coordinates of a particular vertex or for the ed ..."
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Cited by 7 (2 self)
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We investigate a new class of geometric problems based on the idea of online error correction. Suppose one is given access to a large geometric dataset though a query mechanism; for example, the dataset could be a terrain and a query might ask for the coordinates of a particular vertex or for the edges incident to it. Suppose, in addition, that the dataset satisfies some known structural property P (eg, monotonicity or convexity) but that, because of errors and noise, the queries occasionally provide answers that violate P. Can one design a filter that modifies the query’s answers so that (i) the output satisfies P; (ii) the amount of data modification is minimized? We provide upper and lower bounds on the complexity of online reconstruction for convexity in 2D and 3D. 1
Dynamic connectivity: Connecting to networks and geometry
 In Proceedings 49th FOCS
, 2008
"... Dynamic connectivity is a wellstudied problem, but so far the most compelling progress has been confined to the edgeupdate model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fu ..."
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Cited by 4 (0 self)
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Dynamic connectivity is a wellstudied problem, but so far the most compelling progress has been confined to the edgeupdate model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems: Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by on vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in Õ(m2/3) amortized time, wheremdenotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC’02], which required fast matrix multiplication and had an update time of O(m 0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axisparallel line segments and rectangles. We provide similarly improved update times, Õ(n2/3), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublineartime range queries. In particular, we obtain the first sublinear update time for arbitrary 2D line segments: O ∗ (n9/10); for ddimensional simplices: O ∗ 1 1− (n d(2d+1)); and for ddimensional balls: O ∗ (n 1 − 1
InPlace 2d Nearest Neighbor Search
, 2007
"... Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the inpu ..."
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Cited by 1 (1 self)
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Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the input array holding a permutation of the points. The best query time known for such "inplace data structures " is O(log 2 n). In this paper, we break the O(log 2 n) barrier by providing a method that answers nearest neighbor queries in time O((log n) log3=2 2 log log n) = O(log
Three Problems about Dynamic Convex Hulls
, 2011
"... We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log 1+ε n) for an arbitrarily small co ..."
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Cited by 1 (0 self)
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We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log 1+ε n) for an arbitrarily small constant ε> 0. This improves the previous bound of O(log 3/2 n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n+k) time with O(polylog n) expected amortized update time. A similar result holds for 3dimensional orthogonal range reporting. For 3dimensional halfspace range reporting, the query time increases to O(log 2 n / log log n+k). • A semionline dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O(log n) and amortized update time O(n ε). As a corollary, we can solve the following problem in O(n 1+ε) time: given a triangulated terrain in 3d of size n, identify all faces that are partially visible from a fixed viewpoint. 1
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"... Abstract. This paper studies a discrepancysensitive approach to dynamic fractional cascading. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest ..."
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Abstract. This paper studies a discrepancysensitive approach to dynamic fractional cascading. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric. 1
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"... Abstract. This paper studies a discrepancysensitive approach to dynamic fractional cascading. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest ..."
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Abstract. This paper studies a discrepancysensitive approach to dynamic fractional cascading. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric. 1