On range reporting, ray shooting and
            <i>k</i>
            -level construction

Ramos, Edgar A.

doi:10.1145/304893.304993

Cited by 62 publications

(59 citation statements)

References 18 publications

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“…(Note that for d ≥ 6, the auxiliary data structure for R is unnecessary as δ = 0 still works.) Part (ii) follows from a direct application of hierarchical cuttings in the shallow context where we only need to cover a lower envelope of n halfspaces (although no references explicitly state this result, see [10,45] for the general idea, which involves just repeated applications of an X-sensitive shallow cutting lemma as in Lemma 6.1).…”

Section: Shallow Applicationsmentioning

confidence: 99%

“…Matoušek [36] described a shallow version of his partition theorem to obtain a data structure with O(n log n) preprocessing time, O(n log log n) space, and O(n 1−1/ d/2 log O(1) n + k) query time for any d ≥ 4, where k denotes the output size in the reporting case. (The space was reduced to O(n) by Ramos [45] for even d; see also [1] for d = 3.) The same paper by Matoušek also gave an alternative data structure with O(n 1+ε ) preprocessing time, O(n) space, and O(n 1−1/ d/2 2 O(log * n) ) query time for halfspace range emptiness for any d ≥ 4.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Partition Trees

Chan

2012

Discrete Comput Geom

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We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG'92, Matoušek gave a partition tree method for ddimensional simplex range searching achieving O(n) space and O(n 1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n 1+ε ) preprocessing time for any fixed ε > 0. An earlier method by Matoušek (SoCG'91) requires O(n log n) preprocessing time but O(n 1−1/d log O(1) n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n 1−1/d ) query time with high probability. Our method has several advantages:• It is conceptually simpler than Matoušek's SoCG'92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children's cells at each node).• It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods).• A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n 1−1/ d/2 ) query time for halfspace range emptiness in even dimensions d ≥ 4.Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points, . . . ).

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Section: Shallow Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal Partition Trees

Chan

2012

Discrete Comput Geom

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“…For this purpose we can use a recent algorithm of Chan [8] that preprocesses n points in O(n log n) expected time into a data structure of size O(n log n), such that a range reporting query can be answered in O(log n + k) expected time, where k is the number of points reported. An improved data structure of Ramos [27] reduces the space bound to O(n log log n) and makes the query time worst case.…”

Section: Introductionmentioning

confidence: 99%

Randomized incremental constructions of three-dimensional convex hulls and planar voronoi diagrams, and approximate range counting

Kaplan¹,

Sharir²

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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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.The first bound leads to an algorithm that, for a query point x ∈ R 2 , efficiently retrieves the sequence of nearest neighbors of x in P , over the random insertion process. A query takes O(log n) expected time, and the expected storage size is O(n log n). Similarly, the other bounds lead to an algorithm that, for a query direction ω ∈ S 2 , efficiently retrieves the sequence of the convex hull vertices that are touched by the planes with outward direction ω that support the convex hull during the random insertion process. Again, a query takes O(log n) expected time, and the expected storage size is O(n log n).These algorithms are used as the main component in the approximate range counting technique that we present, for ranges that are halfspaces in R 3 or disks in the plane. Our algorithms have the best performance known to date; they slightly improve upon the previous technique of Aronov and Har-Peled [5], and appear to be conceptually simpler. *

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“…In Chan et al's algorithm, randomization was used to construct combinatorial objects that have properties similar to those of shallow cuttings for 3D dominance ranges. Shallow cuttings were introduced by Matoušek [18], and a complicated randomized O(n log n)-time algorithm was given by Ramos [20] for constructing shallow cuttings in the more general setting of 3D halfspace ranges. In a recent SODA'14 paper, Afshani and K. Tsakalidis [2] presented the first deterministic O(n log n)-time algorithm for constructing shallow cuttings for 3D dominance ranges using linear space in the pointer machine model.…”

Section: Introductionmentioning

confidence: 99%

Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM

Afshani

Chan

Tsakalidis

2014

Automata, Languages, and Programming

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Abstract. We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(n log n + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((n log n + k) log log n) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Pȃtraşcu [SoCG'11] that attains the same time complexity but in expectation. The 2D rectangle enclosure problem is related to the offline dominance range reporting problem in 4D, and our result leads to the currently fastest deterministic algorithm for offline dominance reporting in any constant dimension d ≥ 4. A key tool behind Chan et al.'s previous randomized algorithm is shallow cuttings for 3D dominance ranges. Recently, Afshani and Tsakalidis [SODA'14] obtained a deterministic O(n log n)-time algorithm to construct such cuttings. We first present an improved deterministic construction algorithm that runs in O(n log log n) time in the word-RAM; this result is of independent interest. Many additional ideas are then incorporated, including a linear-time algorithm for merging shallow cuttings and an algorithm for an offline tree point location problem.

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On range reporting, ray shooting and k -level construction

Cited by 62 publications

References 18 publications

Optimal Partition Trees

Optimal Partition Trees

Randomized incremental constructions of three-dimensional convex hulls and planar voronoi diagrams, and approximate range counting

Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM

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