Optimal Point Location in a Monotone Subdivision

Edelsbrunner, Herbert; Guibas, Leo J.; Stolfi, Jorge

doi:10.1137/0215023

Cited by 435 publications

(256 citation statements)

References 14 publications

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“…As an arrangement of m lines in the plane results in O(m 2 ) edges, the number of half-edges in DCEL(S) also is O(m 2 ). We can, in time O(m 2 ), preprocess U (S) into a structure that allows point location in O(log m) time [33]. Therefor, testing for each of the O(m 2 ) centers of mass whether they are part of the input takes O(m 2 log m).…”

Section: Property 2 (Quadratic Output Complexity)mentioning

confidence: 99%

“…The time needed for checking whether an atomic object exists at some time moment and computing the snapshot (a triangle) is z(d, ). Because of the preprocessing on the snapshot at time moment J 1 , testing the barycenter of the triangle against that snapshot can be done in O(log n K ) time [33]. In case the snapshots are the same, adjusting the time domains of all atomic objects takes time O( n K ).…”

mentioning

confidence: 99%

“…This follows from Property 8. The preprocessing of the snapshot takes O( n K ) time [33]. The for-loop, starting at Line 9 of Algorithm 4 is executed at most O( n K ) times.…”

mentioning

confidence: 99%

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Affine-Invariant Triangulation of Spatio-Temporal Data with an Application to Image Retrieval

Haesevoets¹,

Kuijpers

Révész

2017

IJGI

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Abstract:In the geometric data model for spatio-temporal data, introduced by Chomicki and Revesz , spatio-temporal data are modelled as a finite collection of triangles that are transformed by time-dependent affinities of the plane. To facilitate querying and animation of spatio-temporal data, we present a normal form for data in the geometric data model. We propose an algorithm for constructing this normal form via a spatio-temporal triangulation of geometric data objects. This triangulation algorithm generates new geometric data objects that partition the given objects both in space and in time. A particular property of the proposed partition is that it is invariant under time-dependent affine transformations, and hence independent of the particular choice of coordinate system used to describe the spatio-temporal data in. We can show that our algorithm works correctly and has a polynomial time complexity (of reasonably low degree in the number of input triangles and the maximal degree of the polynomial functions that describe the transformation functions). We also discuss several possible applications of this spatio-temporal triangulation. The application of our affine-invariant spatial triangulation method to image indexing and retrieval is discussed and an experimental evaluation is given in the context of bird images.

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Section: Property 2 (Quadratic Output Complexity)mentioning

confidence: 99%

mentioning

confidence: 99%

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Affine-Invariant Triangulation of Spatio-Temporal Data with an Application to Image Retrieval

Haesevoets¹,

Kuijpers

Révész

2017

IJGI

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“…At node ν, we store a point-location data structure [12] Proof To report all o * i intersecting s we walk down T , only visiting the nodes ν such that s intersects U(ν). This way we end up in the leaves corresponding to the o * i intersecting s. To decide if we have to visit a child ν of an already visited node, we do a point location with both endpoints of s in the trapezoidal map of U(ν).…”

Section: Objects With Small Union Complexity In Rmentioning

confidence: 99%

Finding Pairwise Intersections Inside a Query Range

2017

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We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q. We present data structures of size O(n · polylog n) and with query time O((k + 1) · polylog n) time, where k is the number of reported pairs, for two classes of objects in R 2 : axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R 3 , we present a data structure of size O(n √ n · polylog n) and query time O(( √ n + k) · polylog n). When the objects and query are fat, we obtain O((k + 1) · polylog n) query time using O(n · polylog n) storage.

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“…When the points are in convex position and given in clockwise order, Aggarwal et al [AGSS89] showed that either Voronoi diagram can be constructed in linear time. Answering point-location queries in either Voronoi diagram of points in convex position can be done in O(log n) time using O(n) preprocessing and space [EGS86].…”

Section: A Simple Data Structurementioning

confidence: 99%

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

Aronov¹,

Bose

Demaine³

et al. 2006

LATIN 2006: Theoretical Informatics

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We consider preprocessing a set S of n points in convex position in the plane into a data structure supporting queries of the following form: given a point q and a directed line in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q among all points to the left of line . 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, for any 0 < ε < 1. 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.The second data structure uses a new representation of nearest-and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only O(log n) amortized pointer changes, in addition to O(log n)-time point-location 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) amortized pointer changes per operation while keeping O(log n)-time point-location queries.

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Optimal Point Location in a Monotone Subdivision

Cited by 435 publications

References 14 publications

Affine-Invariant Triangulation of Spatio-Temporal Data with an Application to Image Retrieval

Affine-Invariant Triangulation of Spatio-Temporal Data with an Application to Image Retrieval

Finding Pairwise Intersections Inside a Query Range

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

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