4,720 research outputs found
Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams
We consider preprocessing a set of points in convex position in the
plane into a data structure supporting queries of the following form: given a
point and a directed line in the plane, report the point of that
is farthest from (or, alternatively, nearest to) the point among all points
to the left of line . We present two data structures for this problem.
The first data structure uses space and preprocessing
time, and answers queries in time, for any . The second data structure uses space and
polynomial preprocessing time, and answers queries in time. These
are the first solutions to the problem with query time and
space.
The second data structure uses a new representation of nearest- and
farthest-point Voronoi diagrams of points in convex position. This
representation supports the insertion of new points in clockwise order using
only amortized pointer changes, in addition to -time
point-location queries, even though every such update may make
combinatorial changes to the Voronoi diagram. This data structure is the first
demonstration that deterministically and incrementally constructed Voronoi
diagrams can be maintained in amortized pointer changes per operation
while keeping -time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in
Algorithmic
A Robust Zero-Calibration RF-based Localization System for Realistic Environments
Due to the noisy indoor radio propagation channel, Radio Frequency (RF)-based
location determination systems usually require a tedious calibration phase to
construct an RF fingerprint of the area of interest. This fingerprint varies
with the used mobile device, changes of the transmit power of smart access
points (APs), and dynamic changes in the environment; requiring re-calibration
of the area of interest; which reduces the technology ease of use. In this
paper, we present IncVoronoi: a novel system that can provide zero-calibration
accurate RF-based indoor localization that works in realistic environments. The
basic idea is that the relative relation between the received signal strength
from two APs at a certain location reflects the relative distance from this
location to the respective APs. Building on this, IncVoronoi incrementally
reduces the user ambiguity region based on refining the Voronoi tessellation of
the area of interest. IncVoronoi also includes a number of modules to
efficiently run in realtime as well as to handle practical deployment issues
including the noisy wireless environment, obstacles in the environment,
heterogeneous devices hardware, and smart APs. We have deployed IncVoronoi on
different Android phones using the iBeacons technology in a university campus.
Evaluation of IncVoronoi with a side-by-side comparison with traditional
fingerprinting techniques shows that it can achieve a consistent median
accuracy of 2.8m under different scenarios with a low beacon density of one
beacon every 44m2. Compared to fingerprinting techniques, whose accuracy
degrades by at least 156%, this accuracy comes with no training overhead and is
robust to the different user devices, different transmit powers, and over
temporal changes in the environment. This highlights the promise of IncVoronoi
as a next generation indoor localization system.Comment: 9 pages, 13 figures, published in SECON 201
Intersection of paraboloids and application to Minkowski-type problems
In this article, we study the intersection (or union) of the convex hull of N
confocal paraboloids (or ellipsoids) of revolution. This study is motivated by
a Minkowski-type problem arising in geometric optics. We show that in each of
the four cases, the combinatorics is given by the intersection of a power
diagram with the unit sphere. We prove the complexity is O(N) for the
intersection of paraboloids and Omega(N^2) for the intersection and the union
of ellipsoids. We provide an algorithm to compute these intersections using the
exact geometric computation paradigm. This algorithm is optimal in the case of
the intersection of ellipsoids and is used to solve numerically the far-field
reflector problem
Regular triangulations of dynamic sets of points
The Delaunay triangulations of a set of points are a class of
triangulations which play an important role in a variety of
different disciplines of science. Regular triangulations are a
generalization of Delaunay triangulations that maintain both their
relationship with convex hulls and with Voronoi diagrams. In regular
triangulations, a real value, its weight, is assigned to each point.
In this paper a simple data structure is presented that allows
regular triangulations of sets of points to be dynamically updated,
that is, new points can be incrementally inserted in the set and old
points can be deleted from it. The algorithms we propose for
insertion and deletion are based on a geometrical interpretation of
the history data structure in one more dimension and use lifted
flips as the unique topological operation. This results in rather
simple and efficient algorithms. The algorithms have been
implemented and experimental results are given.Postprint (published version
Phases of granular segregation in a binary mixture
We present results from an extensive experimental investigation into granular
segregation of a shallow binary mixture in which particles are driven by
frictional interactions with the surface of a vibrating horizontal tray. Three
distinct phases of the mixture are established viz; binary gas (unsegregated),
segregation liquid and segregation crystal. Their ranges of existence are
mapped out as a function of the system's primary control parameters using a
number of measures based on Voronoi tessellation. We study the associated
transitions and show that segregation can be suppressed is the total filling
fraction of the granular layer, , is decreased below a critical value,
, or if the dimensionless acceleration of the driving, , is
increased above a value .Comment: 12 pages, 12 figures, submitted to Phys. Rev.
CSD: Discriminance with Conic Section for Improving Reverse k Nearest Neighbors Queries
The reverse nearest neighbor (RNN) query finds all points that have
the query point as one of their nearest neighbors (NN), where the NN
query finds the closest points to its query point. Based on the
characteristics of conic section, we propose a discriminance, named CSD (Conic
Section Discriminance), to determine points whether belong to the RNN set
without issuing any queries with non-constant computational complexity. By
using CSD, we also implement an efficient RNN algorithm CSD-RNN with a
computational complexity at . The comparative
experiments are conducted between CSD-RNN and other two state-of-the-art
RkNN algorithms, SLICE and VR-RNN. The experimental results indicate that
the efficiency of CSD-RNN is significantly higher than its competitors
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
- …