1,971 research outputs found
Dynamic Geometric Data Structures via Shallow Cuttings
We present new results on a number of fundamental problems about dynamic geometric data structures:
1) We describe the first fully dynamic data structures with sublinear amortized update time for maintaining (i) the number of vertices or the volume of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the discrete 1-center of a 2D point set, (v) the number of maximal (i.e., skyline) points in a 3D point set. The update times are near n^{11/12} for (i) and (ii), n^{7/8} for (iii) and (iv), and n^{2/3} for (v). Previously, sublinear bounds were known only for restricted "semi-online" settings [Chan, SODA 2002].
2) We slightly improve previous fully dynamic data structures for answering extreme point queries for the convex hull of a 3D point set and nearest neighbor search for a 2D point set. The query time is O(log^2n), and the amortized update time is O(log^4n) instead of O(log^5n) [Chan, SODA 2006; Kaplan et al., SODA 2017].
3) We also improve previous fully dynamic data structures for maintaining the bichromatic closest pair between two 2D point sets and the diameter of a 2D point set. The amortized update time is O(log^4n) instead of O(log^7n) [Eppstein 1995; Chan, SODA 2006; Kaplan et al., SODA 2017]
Minimum Cuts in Geometric Intersection Graphs
Let be a set of disks in the plane. The disk graph
for is the undirected graph with vertex set
in which two disks are joined by an edge if and only if they
intersect. The directed transmission graph for
is the directed graph with vertex set in which
there is an edge from a disk to a disk if and only if contains the center of .
Given and two non-intersecting disks , we
show that a minimum - vertex cut in or in
can be found in
expected time. To obtain our result, we combine an algorithm for the maximum
flow problem in general graphs with dynamic geometric data structures to
manipulate the disks.
As an application, we consider the barrier resilience problem in a
rectangular domain. In this problem, we have a vertical strip bounded by
two vertical lines, and , and a collection of
disks. Let be a point in above all disks of , and let
a point in below all disks of . The task is to find a curve
from to that lies in and that intersects as few disks of
as possible. Using our improved algorithm for minimum cuts in
disk graphs, we can solve the barrier resilience problem in
expected time.Comment: 11 pages, 4 figure
Approximate Range Counting Revisited
We study range-searching for colored objects, where one has to count (approximately) the number of colors present in a query range. The problems studied mostly involve orthogonal range-searching in two and three dimensions, and the dual setting of rectangle stabbing by points. We present optimal and near-optimal solutions for these problems. Most of the results are obtained via reductions to the approximate uncolored version, and improved data-structures for them. An additional contribution of this work is the introduction of nested shallow cuttings
An Optimal Algorithm for Higher-Order Voronoi Diagrams in the Plane: The Usefulness of Nondeterminism
We present the first optimal randomized algorithm for constructing the
order- Voronoi diagram of points in two dimensions. The expected running
time is , which improves the previous, two-decades-old result
of Ramos (SoCG'99) by a factor. To obtain our result, we (i)
use a recent decision-tree technique of Chan and Zheng (SODA'22) in combination
with Ramos's cutting construction, to reduce the problem to verifying an
order- Voronoi diagram, and (ii) solve the verification problem by a new
divide-and-conquer algorithm using planar-graph separators.
We also describe a deterministic algorithm for constructing the -level of
lines in two dimensions in time, and constructing
the -level of planes in three dimensions in
time. These time bounds (ignoring the term) match the current best
upper bounds on the combinatorial complexity of the -level. Previously, the
same time bound in two dimensions was obtained by Chan (1999) but with
randomization.Comment: To appear in SODA 2024. 16 pages, 1 figur
Dynamic Data Structures for k-Nearest Neighbor Queries
Our aim is to develop dynamic data structures that support -nearest neighbors (-NN) queries for a set of point sites in the plane in time, where is some polylogarithmic function of . The key component is a general query algorithm that allows us to find the -NN spread over substructures simultaneously, thus reducing an term in the query time to . Combining this technique with the logarithmic method allows us to turn any static -NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic \emph{geodesic} -NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only -NN data structure. More generally, we obtain a dynamic planar -NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions
Independent Range Sampling, Revisited
In the independent range sampling (IRS) problem, given an input set P of n points in R^d, the task is to build a data structure, such that given a range R and an integer t >= 1, it returns t points that are uniformly and independently drawn from P cap R. The samples must satisfy inter-query independence, that is, the samples returned by every query must be independent of the samples returned by all the previous queries. This problem was first tackled by Hu, Qiao and Tao in 2014, who proposed optimal structures for one-dimensional dynamic IRS problem in internal memory and one-dimensional static IRS problem in external memory.
In this paper, we study two natural extensions of the independent range sampling problem. In the first extension, we consider the static IRS problem in two and three dimensions in internal memory. We obtain data structures with optimal space-query tradeoffs for 3D halfspace, 3D dominance, and 2D three-sided queries. The second extension considers weighted IRS problem. Each point is associated with a real-valued weight, and given a query range R, a sample is drawn independently such that each point in P cap R is selected with probability proportional to its weight. Walker\u27s alias method is a classic solution to this problem when no query range is specified. We obtain optimal data structure for one dimensional weighted range sampling problem, thereby extending the alias method to allow range queries
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]
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