40 research outputs found
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
On Euclidean Steiner (1+?)-Spanners
Lightness and sparsity are two natural parameters for Euclidean (1+?)-spanners. Classical results show that, when the dimension d ? ? and ? > 0 are constant, every set S of n points in d-space admits an (1+?)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ? > 0 for constant d ? ? have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+?)-spanner. They gave upper bounds of O?(?^{-(d+1)/2}) for the minimum lightness in dimensions d ? 3, and O?(?^{-(d-1))/2}) for the minimum sparsity in d-space for all d ? 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+?)-spanners of lightness O(?^{-1}log?) in the plane, where ? ? ?(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points.
In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+?)-spanners. Using a new geometric analysis, we establish lower bounds of ?(?^{-d/2}) for the lightness and ?(?^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ? 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+?)-spanners of lightness O(?^{-1}log n) for n points in Euclidean plane
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n â„ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Reconstructing Generalized Staircase Polygons with Uniform Step Length
Visibility graph reconstruction, which asks us to construct a polygon that
has a given visibility graph, is a fundamental problem with unknown complexity
(although visibility graph recognition is known to be in PSPACE). We show that
two classes of uniform step length polygons can be reconstructed efficiently by
finding and removing rectangles formed between consecutive convex boundary
vertices called tabs. In particular, we give an -time reconstruction
algorithm for orthogonally convex polygons, where and are the number of
vertices and edges in the visibility graph, respectively. We further show that
reconstructing a monotone chain of staircases (a histogram) is fixed-parameter
tractable, when parameterized on the number of tabs, and polynomially solvable
in time under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Light Euclidean Steiner Spanners in the Plane
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio
of the spanner weight to the weight of the minimum spanning tree of a finite
set of points in . In a recent breakthrough, Le and Solomon
(2019) established the precise dependencies on and of the minimum lightness of -spanners, and
observed that additional Steiner points can substantially improve the
lightness. Le and Solomon (2020) constructed Steiner -spanners
of lightness in the plane, where is the \emph{spread} of the point set, defined as the ratio
between the maximum and minimum distance between a pair of points. They also
constructed spanners of lightness in
dimensions . Recently, Bhore and T\'{o}th (2020) established a lower
bound of for the lightness of Steiner
-spanners in , for . The central open
problem in this area is to close the gap between the lower and upper bounds in
all dimensions .
In this work, we show that for every finite set of points in the plane and
every , there exists a Euclidean Steiner
-spanner of lightness ; this matches the
lower bound for . We generalize the notion of shallow light trees, which
may be of independent interest, and use directional spanners and a modified
window partitioning scheme to achieve a tight weight analysis.Comment: 29 pages, 14 figures. A 17-page extended abstract will appear in the
Proceedings of the 37th International Symposium on Computational Geometr