Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM

Afshani, Peyman; Chan, Timothy M.; Tsakalidis, Konstantinos

doi:10.1007/978-3-662-43948-7_7

Cited by 10 publications

(6 citation statements)

References 26 publications

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“…Following work by Cabello and Jej£i£ [8] on SSSP in unit-disk graphs, a previous paper by the authors [15] gave an O n 2 log log n log n -time algorithm therein, but the approach cannot be extended to arbitrary disk graphs. For arbitrary disk graphs, the dynamic Voronoi diagrams data structure of Kaplan et al [28] can be employed to solve the problem in nearly O n 2 log 12 n time, 1 as explained in the next page.…”

Section: Introductionmentioning

confidence: 99%

All-Pairs Shortest Paths in Geometric Intersection Graphs

Chan

Skrepetos

2017

Lecture Notes in Computer Science

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We present a simple and general algorithm for the all-pairs shortest paths (APSP) problem in unweighted geometric intersection graphs. Specically we reduce the problem to the design of static data structures for oine intersection detection. Consequently we can solve APSP in unweighted intersection graphs of n arbitrary disks in O n 2 log n time, axis-aligned line segments in O n 2 log log n time, arbitrary line segmentsWe also reduce the single-source shortest paths (SSSP) problem in unweighted geometric intersection graphs to decremental intersection detection. Thus, we obtain an O (n log n)-time SSSP algorithm in unweighted intersection graphs of n axis-aligned line segments.

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Section: Introductionmentioning

confidence: 99%

All-Pairs Shortest Paths in Geometric Intersection Graphs

Chan

Skrepetos

2017

Lecture Notes in Computer Science

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“…In this paper we described data structures with linear and almost-linear space usage that answer four-dimensional range reporting queries in poly-logarithmic time provided that the query range is bounded on 5 sides. This scenario includes an important special case of dominance range reporting queries that was studied in a number of previous works [16,35,26,1,11,10]; for instance, the offline variant of four-dimensional dominance reporting is used to solve the rectangle enclosure problem [11,2]. Our result immediately leads to better data structures in d ≥ 4 dimensions.…”

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confidence: 89%

Four-Dimensional Dominance Range Reporting in Linear Space

Nekrich¹

2020

Preprint

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In this paper we study the four-dimensional dominance range reporting problem and present data structures with linear or almost-linear space usage. Our results can be also used to answer fourdimensional queries that are bounded on five sides. The first data structure presented in this paper uses linear space and answers queries inwhere k is the number of reported points, n is the number of points in the data structure, and ε is an arbitrarily small positive constant. Our second data structure uses O(n log ε n) space and answers queries in O(log n + k) time.These are the first data structures for this problem that use linear (resp. O(n log ε n)) space and answer queries in poly-logarithmic time. For comparison the fastest previously known linear-space or O(n log ε n)-space data structure supports queries in O(n ε + k) time (Bentley and Mauer, 1980). Our results can be generalized to d ≥ 4 dimensions. For example, we can answer d-dimensional dominance range reporting queries in O(log log n(log n/ log log n) d−3 + k) time using O(n log d−4+ε n) space. Compared to the fastest previously known result (Chan, 2013), our data structure reduces the space usage by O(log n) without increasing the query time.

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“…While working on the 4D dominance range reporting problem, Afshani et al [3] are implicitly performing iterative point location queries along a path of a balanced binary tree on somewhat specialized subdivsions in O(log 3/2 n) total time. Later Afshani et al [4] studied an offline variant of this problem, and they presented a linear sized data structure that achieves optimal query time. The same idea is used to improve the result of 3D point location in orthogonal subdivisions.…”

Section: Related Workmentioning

confidence: 99%

“…We observe that the lower bound of Chazelle and Liu does not apply to orthogonal subdivisions, a very important special case of planar point location problem. Many geometric problems need to solve this base problem, e.g., 4D orthogonal dominance range reporting [3,4], 3D point location in orthogonal subdivisions [27], some 3D vertical ray-shooting problems [21]. In geographic information systems, it is very common to overlay planar subdivisions describing different features of a region to generate a complete map.…”

Section: Introductionmentioning

confidence: 99%

2D Fractional Cascading on Axis-aligned Planar Subdivisions

Afshani,

Cheng

2020

Preprint

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Fractional cascading is one of the influential and important techniques in data structures, as it provides a general framework for solving a common important problem: the iterative search problem. In the problem, the input is a graph G with constant degree. Also as input, we are given a set of values for every vertex of G. The goal is to preprocess G such that when we are given a query value q, and a connected subgraph π of G, we can find the predecessor of q in all the sets associated with the vertices of π. The fundamental result of fractional cascading, by Chazelle and Guibas, is that there exists a data structure that uses linear space and it can answer queries in O(log n + |π|) time, at essentially constant time per predecessor [16]. While this technique has received plenty of attention in the past decades, an almost quadratic space lower bound for "two-dimensional fractional cascading" by Chazelle and Liu in STOC 2001 [18] has convinced the researchers that fractional cascading is fundamentally a one-dimensional technique.In two-dimensional fractional cascading, the input includes a planar subdivision for every vertex of G and the query is a point q and a subgraph π and the goal is to locate the cell containing q in all the subdivisions associated with the vertices of π. In this paper, we show that it is actually possible to circumvent the lower bound of Chazelle and Liu for axis-aligned planar subdivisions. We present a number of upper and lower bounds which reveal that in two-dimensions, the problem has a much richer structure. When G is a tree and π is a path, then queries can be answered in O(log n + |π| + min{|π| √ log n, α(n) |π| log n}) time using linear space where α is an inverse Ackermann function; surprisingly, we show both branches of this bound are tight, up to the inverse Ackermann factor. When G is a general graph or when π is a general subgraph, then the query bound becomes O(log n + |π| √ log n) and this bound is once again tight in both cases.

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Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM

Cited by 10 publications

References 26 publications

All-Pairs Shortest Paths in Geometric Intersection Graphs

All-Pairs Shortest Paths in Geometric Intersection Graphs

Four-Dimensional Dominance Range Reporting in Linear Space

2D Fractional Cascading on Axis-aligned Planar Subdivisions

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