102 research outputs found
More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time
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
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
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 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
Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time
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 unit disks in the plane, a -approximate maximum
independent set can be maintained with worst-case update time ,
and optimal output-sensitive reporting. Moreover, this result generalizes to
fat objects of comparable sizes in any fixed dimension , 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 -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
Dynamic Connectivity: Connecting to Networks and Geometry
Dynamic connectivity is a well-studied problem, but so far the most
compelling progress has been confined to the edge-update model: maintain an
understanding of connectivity in an undirected graph, subject to edge
insertions and deletions. In this paper, we study two more challenging, yet
equally fundamental problems.
Subgraph connectivity asks to maintain an understanding of connectivity under
vertex updates: updates can turn vertices on and off, and queries refer to the
subgraph induced by "on" vertices. (For instance, this is closer to
applications in networks of routers, where node faults may occur.)
We describe a data structure supporting vertex updates in O (m^{2/3})
amortized time, where m denotes the number of edges in the graph. This greatly
improves over the previous result [Chan, STOC'02], which required fast matrix
multiplication and had an update time of O(m^0.94). The new data structure is
also simpler.
Geometric connectivity asks to maintain a dynamic set of n geometric objects,
and query connectivity in their intersection graph. (For instance, the
intersection graph of balls describes connectivity in a network of sensors with
bounded transmission radius.)
Previously, nontrivial fully dynamic results were known only for special
cases like axis-parallel line segments and rectangles. We provide similarly
improved update times, O (n^{2/3}), for these special cases. Moreover, we show
how to obtain sublinear update bounds for virtually all families of geometric
objects which allow sublinear-time range queries, such as arbitrary 2D line
segments, d-dimensional simplices, and d-dimensional balls.Comment: Full version of a paper to appear in FOCS 200
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization
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
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