Optimal Halfspace Range Reporting in Three Dimensions

Afshani, Peyman; Chan, Timothy M.

doi:10.1137/1.9781611973068.21

Cited by 50 publications

(119 citation statements)

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“…Define the query cost in the same manner, as in (1). Note the query cost of a partial partition tree upper-bounds the number of internal and leaf cells crossed by a hyperplane, but does not account for the cost of any auxiliary data structures we plan to store at the leaves.…”

Section: Proofmentioning

confidence: 99%

“…We don't need to compute X ∆ i ( R), but rather we try different values of r (powers of 2) until we find a cutting with the right number of subcells. Thus, the expected cost over all iterations is O(t)(b log N ) O (1) . By Markov's inequality, with probability Ω(1), this bound holds; if not, we declare failure.…”

Section: By Lemma 23 the Number Of Subcells Is O(b)mentioning

confidence: 99%

“…We can re-run the algorithm until success, with O(log N ) number of trials w.h.p. The running time remains O(m + t)(b log N ) O (1) . At the end, we can assign all the input points to subcells in O(bn) additional time (this term can actually be reduced to O(n log b) with point location data structures).…”

Section: And Query Time O(logmentioning

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: Proofmentioning

confidence: 99%

Section: By Lemma 23 the Number Of Subcells Is O(b)mentioning

confidence: 99%

Section: And Query Time O(logmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal Partition Trees

Chan

2012

Discrete Comput Geom

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“…Combining with the recent result of Afshani and Chan [1] for 3D halfspace range reporting, we obtain a structure for the variable-threshold version of our problem with O(n log 2 n) space and O(log 3 n + k) query time (Section 4). Although the bounds have extra log factors in this case, our result shows that this problem is still significantly easier than 2D simplex queries.…”

Section: Introductionmentioning

confidence: 78%

Range searching on uncertain data

Agarwal

Cheng

2012

ACM Trans. Algorithms

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Querying uncertain data has emerged as an important problem in data management due to the imprecise nature of many measurement data. In this paper we study answering range queries over uncertain data. Specifically, we are given a collection P of n uncertain points in R, each represented by its one-dimensional probability density function (pdf). The goal is to build a data structure on P such that given a query interval I and a probability threshold τ , we can quickly report all points of P that lie in I with probability at least τ . We present various structures with linear or near-linear space and (poly)logarithmic query time. Our structures support pdf's that are either histograms or more complex ones such as Gaussian or piecewise algebraic.

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Reachability Problems for Transmission Graphs

2021

Lecture Notes in Computer Science

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Optimal Halfspace Range Reporting in Three Dimensions

Cited by 50 publications

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Optimal Partition Trees

Optimal Partition Trees

Range searching on uncertain data

Reachability Problems for Transmission Graphs

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