Improved pointer machine and I/O lower bounds for simplex range reporting and related problems

Afshani, Peyman

doi:10.1145/2261250.2261301

Cited by 12 publications

(38 citation statements)

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“…This lower bound is tight for any h = log d−2+Ω(1) n. To obtain the lower bound, we use a novel geometric argument, rather than the combinatorial framework that was pioneered by Chazelle [12] and which was used in all the previous lower bounds. In fact, our new technique has already led to an improved lower bound for simplex range reporting [1], a long standing open problem. We thus suspect that our new technique might have even further implications.…”

Section: Our Resultsmentioning

confidence: 99%

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Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model

Afshani

Arge

Larsen

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in ddimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported efficiently. Rectangle stabbing is the "dual" problem where a set of n axis-aligned rectangles should be stored in a data structure, such that the t rectangles that contain a query point can be reported efficiently. Very recently an optimal O(log n + t) query time pointer machine data structure was developed for the three-dimensional version of the orthogonal range reporting problem. However, in four dimensions the best known query bound of O(log 2 n/ log log n + t) has not been improved for decades.We describe an orthogonal range reporting data structure that is the first structure to achieve significantly less than O(log 2 n + t) query time in four dimensions. More precisely, we develop a structure that uses O(n(log n/ log log n) d ) space and can answer d-dimensional orthogonal range reporting queries (for d ≥ 4) in O(log n(log n/ log log n) d−4+1/(d−2) +t) time. Ignoring log log n factors, this speeds up the best previous query time by a log 1−1/(d−2) n factor. For the rectangle stabbing problem, we show that any data structure that uses nh space must use Ω(log n(log n/ log h) d−2 + t) time * This research was done while the author was a postdoctoral researcher at Dalhousie University, and was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) through the Postdoctoral Fellowships Program (PDF). † Is supported in part by MADALGO and in part by a Google Fellowship in Search and Information Retrieval ‡ Center for Massive Data Algorithmics, a center of the Danish National Research Foundation.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. to answer a query. This improves the previous results by a log h factor, and is the first lower bound that is optimal for a large range of h, namely for h ≥ log d−2+ε n where ε > 0 is an arbitrarily small constant. By a simple geometric transformation, our result also implies an improved query lower bound for orthogonal range reporting.

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Section: Our Resultsmentioning

confidence: 99%

“…Our idea of using geometry and volume based arguments to prove lower bounds is quite novel, and as mentioned, it has already led to improved simplex range reporting lower bounds [1]. Finding further applications of our technique is still open.…”

Section: Discussionmentioning

confidence: 99%

Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model

Afshani

Arge

Larsen

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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“…With only a few exceptions (see [1,3]), lower bounds for range reporting in the pointer machine model have all been proved by appealing to a theorem initially due to Chazelle [5] and later refined by Chazelle and Rosenberg [10]. Since our lower bounds also rely on this theorem, we introduce it in the following: First, let P be a set of input objects to a range searching problem and let R be a set of query ranges.…”

Section: Previous Resultsmentioning

confidence: 99%

“…. , p tq /ε } independently and uniformly in [0, 1] d . With probability 1−o(1), the query set R is (tq, O(lg(1/ε)))-approximate favorable for the point set P .…”

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confidence: 99%

Improved range searching lower bounds

Larsen

Nguyêݱn

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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In this paper we present a number of improved lower bounds for range searching in the pointer machine and the group model. In the pointer machine, we prove lower bounds for the approximate simplex range reporting problem. In approximate simplex range reporting, points that lie within a distance of ε · diam(s) from the border of a query simplex s, are free to be included or excluded from the output, where ε ≥ 0 is an input parameter to the range searching problem. We prove our lower bounds by constructing a hard input set and query set, and then invoking Chazelle and Rosenberg's [CGTA'96] general theorem on the complexity of navigation in the pointer machine.For the group model, we show that input sets and query sets that are hard for range reporting in the pointer machine (i.e. by Chazelle and Rosenberg's theorem), are also hard for dynamic range searching in the group model. This theorem allows us to reuse decades of research on range reporting lower bounds to immediately obtain a range of new group model lower bounds. Amongst others, this includes an improved lower bound for the fundamental problem of dynamic d-dimensional orthogonal range searching, stating that tqtu = Ω((lg n/ lg lg n) d−1 ). Here tq denotes the query time and tu the update time of the data structure. This is an improvement of a lg 1−δ n factor over the recent lower bound of Larsen [FOCS'11], where δ > 0 is a small constant depending on the dimension.

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“…After this, since the total size of all the arborescences were α|Q|, it easily follows that there exists at least one arborescence T such that |T | ≤ 2αk(T ) with k(T ) ≥ 2β. In this case, by using the same technique as in [1], we can find at least one β-heavy hub of size O(αβ).…”

Section: Lower Boundsmentioning

confidence: 99%

Concurrent Range Reporting in Two-Dimensional Space

Afshani¹,

Cheng²,

Tao³

et al. 2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the concurrent range reporting (CRR) problem, the input is L disjoint sets S 1 , ..., S L of points in R d with a total of N points. The goal is to preprocess the sets into a structure such that, given a query range r and an arbitrary set Q ⊆ {1, ..., L}, we can efficiently report all the points in S i ∩ r for each i ∈ Q. The problem was studied as early as 1986 by Chazelle and Guibas [9] and has recently re-emerged when studying higher-dimensional complexity of orthogonal range reporting [2,3].We focus on the one-and two-dimensional cases of the problem.We prove that in the pointermachine model (as well as comparison models such as the real RAM model), answering queries requires Ω(|Q| log(L/|Q|) + log N + K) time in the worst case, where K is the number of output points. In one dimension, we achieve this query time with a linear-space dynamic data structure that requires optimal O(log N ) time to update. We also achieve this query time in the static case for dominance and halfspace queries in the plane. For three-sided ranges, we get close to within an inverse Ackermann (α(·)) factor: we answer queries in O(|Q| log(L/|Q|)α(L)+ log N + K) time, improving the best previously known query times of O(|Q| log(N/|Q|) + K) and O(2 L L + log N + K). Finally, we give an optimal data structure for three-sided ranges for the case L = O(log N ). IntroductionWe consider concurrent range reporting (CRR) in twodimensional space. The dataset consists of L sets S 1 , ..., S L of points in R d , where L is potentially ω(1). We want to preprocess the sets into a structure such that, given a query range r and an arbitrary set Q ⊆ {1, ..., L}, we can efficiently report all the points in S i ∩ r for each i ∈ Q. See Figure 1. This is

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Improved pointer machine and I/O lower bounds for simplex range reporting and related problems

Cited by 12 publications

References 21 publications

Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model

Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model

Improved range searching lower bounds

Concurrent Range Reporting in Two-Dimensional Space

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