9 research outputs found
Stabbing line segments with disks: complexity and approximation algorithms
Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least cardinality
set of disks of fixed radii where the set of segments forms a straight
line drawing of a planar graph without edge crossings. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for and some constant where and
are Euclidean lengths of the longest and shortest graph edges
respectively. Fast -time -approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality holds uniformly for some constant
i.e. when lengths of edges of are uniformly bounded from above by
some linear function of Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017
The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data
We present a new multi-dimensional data structure, which we call the skip
quadtree (for point data in R^2) or the skip octree (for point data in R^d,
with constant d>2). Our data structure combines the best features of two
well-known data structures, in that it has the well-defined "box"-shaped
regions of region quadtrees and the logarithmic-height search and update
hierarchical structure of skip lists. Indeed, the bottom level of our structure
is exactly a region quadtree (or octree for higher dimensional data). We
describe efficient algorithms for inserting and deleting points in a skip
quadtree, as well as fast methods for performing point location and approximate
range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in
the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30
Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles
In this paper we study two geometric data structure problems in the special
case when input objects or queries are fat rectangles. We show that in this
case a significant improvement compared to the general case can be achieved.
We describe data structures that answer two- and three-dimensional orthogonal
range reporting queries in the case when the query range is a \emph{fat}
rectangle. Our two-dimensional data structure uses words and supports
queries in time, where is the number of points in the
data structure, is the size of the universe and is the number of points
in the query range. Our three-dimensional data structure needs
words of space and answers queries in time. We also consider the rectangle stabbing problem on a set of
three-dimensional fat rectangles. Our data structure uses space and
answers stabbing queries in time.Comment: extended version of a WADS'19 pape
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
Approximation algorithms for the mobile piercing set problem with applications to clustering,
Abstract. The main contributions of this paper are two-fold. First, we present a simple, general framework for obtaining efficient constantfactor approximation algorithms for the mobile piercing set (MPS) problem on unit-disks for standard metrics in fixed dimension vector spaces. More specifically, we provide low constant approximations for L 1 and L ∞ norms on a d-dimensional space, for any fixed d > 0, and for the L 2 norm on two-and three-dimensional spaces. Our framework provides a family of fully-distributed and decentralized algorithms, which adapt (asymptotically) optimally to the mobility of disks, at the expense of a low degradation on the best known approximation factors of the respective centralized algorithms: Our algorithms take O(1) time to update the piercing set maintained, per movement of a disk. We also present a family of fully-distributed algorithms for the MPS problem which either match or improve the best known approximation bounds of centralized algorithms for the respective norms and space dimensions. Second, we show how the proposed algorithms can be directly applied to provide theoretical performance analyses for two popular 1-hop clustering algorithms in ad-hoc networks: the lowest-id algorithm and the Least Cluster Change (LCC) algorithm. More specifically, we formally prove that the LCC algorithm adapts in constant time to the mobility of the network nodes, and minimizes (up to low constant factors) the number of 1-hop clusters maintained. While there is a vast literature on simulation results for the LCC and the lowest-id algorithms, these had not been formally analyzed prior to this work. We also present an O(log n)-approximation algorithm for the mobile piercing set problem for nonuniform disks (i.e., disks that may have different radii), with constant update time
Dynamic Data Structures for Fat Objects and Their Applications
We present several efficient dynamic data structures for point-enclosure queries, involving convex fat objects in R2 or R3. Our planar structures are actually fitted for a more general class of objects-- (fi; ffi)-covered objects-- which are not necessarily convex, see definition below. These structures are more efficient than alternative known structures, because they exploit the fatness of the objects. We then apply these structures to obtain efficient solutions to two problems (i) Finding a perfect containment matching between a set of points and a set of convex fat objects. (ii) Finding a piercing set for a collection of convex fat objects, whose size is optimal up to some constant factor