268 research outputs found
Sparse geometric graphs with small dilation
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we
show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum
degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For
any k, we also construct planar n-point sets for which any geometric graph with
n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if
the points of S are required to be in convex position
Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints
We introduce a new structure for a set of points in the plane and an angle
, which is similar in flavor to a bounded-degree MST. We name this
structure -MST. Let be a set of points in the plane and let be an angle. An -ST of is a spanning tree of the
complete Euclidean graph induced by , with the additional property that for
each point , the smallest angle around containing all the edges
adjacent to is at most . An -MST of is then an
-ST of of minimum weight. For , an -ST does
not always exist, and, for , it always exists. In this paper,
we study the problem of computing an -MST for several common values of
.
Motivated by wireless networks, we formulate the problem in terms of
directional antennas. With each point , we associate a wedge of
angle and apex . The goal is to assign an orientation and a radius
to each wedge , such that the resulting graph is connected and its
MST is an -MST. (We draw an edge between and if , , and .) Unsurprisingly, the problem of computing an
-MST is NP-hard, at least for and . We
present constant-factor approximation algorithms for .
One of our major results is a surprising theorem for ,
which, besides being interesting from a geometric point of view, has important
applications. For example, the theorem guarantees that given any set of
points in the plane and any partitioning of the points into triplets,
one can orient the wedges of each triplet {\em independently}, such that the
graph induced by is connected. We apply the theorem to the {\em antenna
conversion} problem
Computing a Minimum-Dilation Spanning Tree is NP-hard
In a geometric network G = (S, E), the graph distance between two vertices u,
v in S is the length of the shortest path in G connecting u to v. The dilation
of G is the maximum factor by which the graph distance of a pair of vertices
differs from their Euclidean distance. We show that given a set S of n points
with integer coordinates in the plane and a rational dilation delta > 1, it is
NP-hard to determine whether a spanning tree of S with dilation at most delta
exists
Optimal Geo-Indistinguishable Mechanisms for Location Privacy
We consider the geo-indistinguishability approach to location privacy, and
the trade-off with respect to utility. We show that, given a desired degree of
geo-indistinguishability, it is possible to construct a mechanism that
minimizes the service quality loss, using linear programming techniques. In
addition we show that, under certain conditions, such mechanism also provides
optimal privacy in the sense of Shokri et al. Furthermore, we propose a method
to reduce the number of constraints of the linear program from cubic to
quadratic, maintaining the privacy guarantees and without affecting
significantly the utility of the generated mechanism. This reduces considerably
the time required to solve the linear program, thus enlarging significantly the
location sets for which the optimal mechanisms can be computed.Comment: 13 page
06481 Abstracts Collection -- Geometric Networks and Metric Space Embeddings
The Dagstuhl Seminar 06481 ``Geometric Networks and Metric Space
Embeddings\u27\u27 was held from November~26 to December~1, 2006 in the
International Conference and Research Center (IBFI), Schloss
Dagstuhl. During the seminar, several participants presented their
current research, and ongoing work and open problems were discussed.
In this paper we describe the seminar topics, we have compiled a
list of open questions that were posed during the seminar, there is
a list of all talks and there are abstracts of the presentations
given during the seminar. Links to extended abstracts or full
papers are provided where available
Algorithms and complexity analyses for some combinational optimization problems
The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design.
In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios.
In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs.
Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost
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