95 research outputs found

    More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time

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    We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an O(1)-approximation in sublinear update time for set cover for axis-aligned squares in 2D . More precisely, we obtain randomized update time O(n^{2/3+?}) for an arbitrarily small constant ? > 0. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D . As a byproduct, our techniques for dynamic set cover also yield an optimal randomized O(nlog n)-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier O(nlog n(log log n)^{O(1)}) result [SoCG 2020]

    Restricted Strip Covering and the Sensor Cover Problem

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    Given a set of objects with durations (jobs) that cover a base region, can we schedule the jobs to maximize the duration the original region remains covered? We call this problem the sensor cover problem. This problem arises in the context of covering a region with sensors. For example, suppose you wish to monitor activity along a fence by sensors placed at various fixed locations. Each sensor has a range and limited battery life. The problem is to schedule when to turn on the sensors so that the fence is fully monitored for as long as possible. This one dimensional problem involves intervals on the real line. Associating a duration to each yields a set of rectangles in space and time, each specified by a pair of fixed horizontal endpoints and a height. The objective is to assign a position to each rectangle to maximize the height at which the spanning interval is fully covered. We call this one dimensional problem restricted strip covering. If we replace the covering constraint by a packing constraint, the problem is identical to dynamic storage allocation, a scheduling problem that is a restricted case of the strip packing problem. We show that the restricted strip covering problem is NP-hard and present an O(log log n)-approximation algorithm. We present better approximations or exact algorithms for some special cases. For the uniform-duration case of restricted strip covering we give a polynomial-time, exact algorithm but prove that the uniform-duration case for higher-dimensional regions is NP-hard. Finally, we consider regions that are arbitrary sets, and we present an O(log n)-approximation algorithm.Comment: 14 pages, 6 figure

    Conditional Lower Bounds for Dynamic Geometric Measure Problems

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    We give new polynomial lower bounds for a number of dynamic measure problems in computational geometry. These lower bounds hold in the Word-RAM model, conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector Multiplication problem [Henzinger et al., STOC 2015]. In particular we get lower bounds in the incremental and fully-dynamic settings for counting maximal or extremal points in R^3, different variants of Klee's Measure Problem, problems related to finding the largest empty disk in a set of points, and querying the size of the i'th convex layer in a planar set of points. We also answer a question of Chan et al. [SODA 2022] by giving a conditional lower bound for dynamic approximate square set cover. While many conditional lower bounds for dynamic data structures have been proven since the seminal work of Patrascu [STOC 2010], few of them relate to computational geometry problems. This is the first paper focusing on this topic. Most problems we consider can be solved in O(n log n) time in the static case and their dynamic versions have only been approached from the perspective of improving known upper bounds. One exception to this is Klee's measure problem in R^2, for which Chan [CGTA 2010] gave an unconditional Ω(n){\Omega}(\sqrt{n}) lower bound on the worst-case update time. By a similar approach, we show that such a lower bound also holds for an important special case of Klee's measure problem in R^3 known as the Hypervolume Indicator problem, even for amortized runtime in the incremental setting.Comment: Improved presentation, improved the reduction for the Hypervolume Indicator problem and added a reduction for dynamic approximate square set cove

    Conditional Lower Bounds for Dynamic Geometric Measure Problems

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    Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

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    Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem

    Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

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    A fundamental question in computational geometry is for a dynamic collection of geometric objects in Euclidean space, whether it is possible to maintain a maximum independent set in polylogarithmic update time. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set with sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic update time. First, we show that for a fully dynamic set of nn unit disks in the plane, a 1212-approximate maximum independent set can be maintained with worst-case update time O(log2n)O(\log^2 n), and optimal output-sensitive reporting. Moreover, this result generalizes to fat objects of comparable sizes in any fixed dimension dd, where the approximation ratio depends on the dimension and the fatness parameter. Our main result is that for a fully dynamic set of disks of arbitrary radii in the plane, an O(1)O(1)-approximate maximum independent set can be maintained in polylogarithmic expected amortized update time.Comment: Abstract is shortened to meet Arxiv's requirement on the number of character

    Stabbing boxes with finitely many axis-parallel lines and flats

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    We give necessary and sufficient condition for an infinite collection of axis-parallel boxes in Rd\mathbb{R}^{d} to be pierceable by finitely many axis-parallel kk-flats, where 0k<d0 \leq k < d. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner (p,q)(p,q)-problem.Comment: 13 page

    Algorithms and Data Structures for Geometric Intersection Query Problems

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    University of Minnesota Ph.D. dissertation. September 2017. Major: Computer Science. Advisor: Ravi Janardan. 1 computer file (PDF); xi, 126 pages.The focus of this thesis is the topic of geometric intersection queries (GIQ) which has been very well studied by the computational geometry community and the database community. In a GIQ problem, the user is not interested in the entire input geometric dataset, but only in a small subset of it and requests an informative summary of that small subset of data. Formally, the goal is to preprocess a set A of n geometric objects into a data structure so that given a query geometric object q, a certain aggregation function can be applied efficiently on the objects of A intersecting q. The classical aggregation functions studied in the literature are reporting or counting the objects of A intersecting q. In many applications, the same set A is queried several times, in which case one would like to answer a query faster by preprocessing A into a data structure. The goal is to organize the data into a data structure which occupies a small amount of space and yet responds to any user query in real-time. In this thesis the study of the GIQ problems was conducted from the point-of-view of a computational geometry researcher. Given a model of computation and a GIQ problem, what are the best possible upper bounds (resp., lower bounds) on the space and the query time that can be achieved by a data structure? Also, what is the relative hardness of various GIQ problems and aggregate functions. Here relative hardness means that given two GIQ problems A and B (or, two aggregate functions f(A, q) and g(A, q)), which of them can be answered faster by a computer (assuming data structures for both of them occupy asymptotically the same amount of space)? This thesis presents results which increase our understanding of the above questions. For many GIQ problems, data structures with optimal (or near-optimal) space and query time bounds have been achieved. The geometric settings studied are primarily orthogonal range searching where the input is points and the query is an axes-aligned rectangle, and the dual setting of rectangle stabbing where the input is a set of axes-aligned rectangles and the query is a point. The aggregation functions studied are primarily reporting, top-k, and approximate counting. Most of the data structures are built for the internal memory model (word-RAM or pointer machine model), but in some settings they are generic enough to be efficient in the I/O-model as well

    Selection and experimental evaluation of low-cost porous materials for regenerator applications in thermoacoustic engines

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    This paper aims at evaluating three selected low-cost porous materials from the point of view of their suitability as regenerator materials in the design of thermoacoustic travelling wave engines. The materials tested include: a cellular ceramic substrate with regular square channels; steel “scourers”; and stainless steel “wool”. Comparisons are made against a widely used regenerator material: stainless steel woven wire mesh screen. For meaningful comparisons, the materials are selected to have similar hydraulic radii. One set of regenerators was designed around the hydraulic radius of 200 μm. This included the ceramic substrate, steel “scourers”, stainless steel “wool” and stacked wire screens (as a reference). This set was complemented by steel “scourers” and stacked wire screens (as a reference) with hydraulic radii of 120 μm. Therefore six regenerators were produced to carry out the testing. Initial tests were made in a steady air flow to estimate their relative pressure drop due to viscous dissipation. Subsequently, they were installed in a looped-tube travelling-wave thermoacoustic engine to test their relative performance. Testing included the onset temperature difference, the maximum pressure amplitude generated and the acoustic power output as a function of mean pressure between 0 and 10 bar above atmospheric. It appears that the performance of regenerators made out of “scourers” and steel “wool” is much worse than their mesh-screen counterparts of the same hydraulic radius. However cellular ceramics may offer an alternative to traditional regenerator materials to reduce the overall system costs. Detailed discussions are provided
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