536 research outputs found
Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms
We consider the RMS distance (sum of squared distances between pairs of
points) under translation between two point sets in the plane, in two different
setups. In the partial-matching setup, each point in the smaller set is matched
to a distinct point in the bigger set. Although the problem is not known to be
polynomial, we establish several structural properties of the underlying
subdivision of the plane and derive improved bounds on its complexity. These
results lead to the best known algorithm for finding a translation for which
the partial-matching RMS distance between the point sets is minimized. In
addition, we show how to compute a local minimum of the partial-matching RMS
distance under translation, in polynomial time. In the Hausdorff setup, each
point is paired to its nearest neighbor in the other set. We develop algorithms
for finding a local minimum of the Hausdorff RMS distance in nearly linear time
on the line, and in nearly quadratic time in the plane. These improve
substantially the worst-case behavior of the popular ICP heuristics for solving
this problem.Comment: 31 pages, 6 figure
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
From Proximity to Utility: A Voronoi Partition of Pareto Optima
We present an extension of Voronoi diagrams where when considering which site
a client is going to use, in addition to the site distances, other site
attributes are also considered (for example, prices or weights). A cell in this
diagram is then the locus of all clients that consider the same set of sites to
be relevant. In particular, the precise site a client might use from this
candidate set depends on parameters that might change between usages, and the
candidate set lists all of the relevant sites. The resulting diagram is
significantly more expressive than Voronoi diagrams, but naturally has the
drawback that its complexity, even in the plane, might be quite high.
Nevertheless, we show that if the attributes of the sites are drawn from the
same distribution (note that the locations are fixed), then the expected
complexity of the candidate diagram is near linear.
To this end, we derive several new technical results, which are of
independent interest. In particular, we provide a high-probability,
asymptotically optimal bound on the number of Pareto optima points in a point
set uniformly sampled from the -dimensional hypercube. To do so we revisit
the classical backward analysis technique, both simplifying and improving
relevant results in order to achieve the high-probability bounds
Optimal relay location and power allocation for low SNR broadcast relay channels
We consider the broadcast relay channel (BRC), where a single source
transmits to multiple destinations with the help of a relay, in the limit of a
large bandwidth. We address the problem of optimal relay positioning and power
allocations at source and relay, to maximize the multicast rate from source to
all destinations. To solve such a network planning problem, we develop a
three-faceted approach based on an underlying information theoretic model,
computational geometric aspects, and network optimization tools. Firstly,
assuming superposition coding and frequency division between the source and the
relay, the information theoretic framework yields a hypergraph model of the
wideband BRC, which captures the dependency of achievable rate-tuples on the
network topology. As the relay position varies, so does the set of hyperarcs
constituting the hypergraph, rendering the combinatorial nature of optimization
problem. We show that the convex hull C of all nodes in the 2-D plane can be
divided into disjoint regions corresponding to distinct hyperarcs sets. These
sets are obtained by superimposing all k-th order Voronoi tessellation of C. We
propose an easy and efficient algorithm to compute all hyperarc sets, and prove
they are polynomially bounded. Using the switched hypergraph approach, we model
the original problem as a continuous yet non-convex network optimization
program. Ultimately, availing on the techniques of geometric programming and
-norm surrogate approximation, we derive a good convex approximation. We
provide a detailed characterization of the problem for collinearly located
destinations, and then give a generalization for arbitrarily located
destinations. Finally, we show strong gains for the optimal relay positioning
compared to seemingly interesting positions.Comment: In Proceedings of INFOCOM 201
Higher-order Voronoi diagrams of polygonal objects
Higher-order Voronoi diagrams are fundamental geometric structures which encode the k-nearest neighbor information. Thus, they aid in computations that require proximity information beyond the nearest neighbor. They are related to various favorite structures in computational geometry and are a fascinating combinatorial problem to study. While higher-order Voronoi diagrams of points have been studied a lot, they have not been considered for other types of sites. Points lack dimensionality which makes them unable to represent various real-life instances. Points are the simplest kind of geometric object and therefore higher- order Voronoi diagrams of points can be considered as the corner case of all higher-order Voronoi diagrams. The goal of this dissertation is to move away from the corner and bring the higher-order Voronoi diagram to more general geometric instances. We focus on certain polygonal objects as they provide flexibility and are able to represent real-life instances. Before this dissertation, higher-order Voronoi diagrams of polygonal objects had been studied only for the nearest neighbor and farthest Voronoi diagrams. In this dissertation we investigate structural and combinatorial properties and discover that the dimensionality of geometric objects manifests itself in numerous ways which do not exist in the case of points. We prove that the structural complexity of the order-k Voronoi diagram of non-crossing line segments is O(k(n-k)), as in the case of points. We study disjoint line segments, intersecting line segments, line segments forming a planar straight-line graph and extend the results to the Lp metric, 1<=p<=infty. We also establish the connection between two mathematical abstractions: abstract Voronoi diagrams and the Clarkson-Shor framework. We design several construction algorithms that cover the case of non-point sites. While computational geometry provides several approaches to study the structural complexity that give tight realizable bounds, developing an effective construction algorithm is still a challenging problem even for points. Most of the construction algorithms are designed to work with points as they utilize their simplicity and relations with data-structures that work specifically for points. We extend the iterative and the sweepline approaches that are quite efficient in constructing all order-i Voronoi diagrams, for i<=k and we also give three randomized construction algorithms for abstract higher-order Voronoi diagrams that deal specifically with the construction of the order-k Voronoi diagrams
Exact Generalized Voronoi Diagram Computation using a Sweepline Algorithm
Voronoi Diagrams can provide useful spatial information. Little work has been done on computing exact Voronoi Diagrams when the sites are more complex than a point. We introduce a technique that measures the exact Generalized Voronoi Diagram from points, line segments and, connected lines including lines that connect to form simple polygons. Our technique is an extension of Fortune’s method. Our approach treats connected lines (or polygons) as a single site
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