546 research outputs found
Computing the Greedy Spanner in Linear Space
The greedy spanner is a high-quality spanner: its total weight, edge count
and maximal degree are asymptotically optimal and in practice significantly
better than for any other spanner with reasonable construction time.
Unfortunately, all known algorithms that compute the greedy spanner of n points
use Omega(n^2) space, which is impractical on large instances. To the best of
our knowledge, the largest instance for which the greedy spanner was computed
so far has about 13,000 vertices.
We present a O(n)-space algorithm that computes the same spanner for points
in R^d running in O(n^2 log^2 n) time for any fixed stretch factor and
dimension. We discuss and evaluate a number of optimizations to its running
time, which allowed us to compute the greedy spanner on a graph with a million
vertices. To our knowledge, this is also the first algorithm for the greedy
spanner with a near-quadratic running time guarantee that has actually been
implemented
Near-Linear-Time Deterministic Plane Steiner Spanners and TSP Approximation for Well-Spaced Point Sets
We describe an algorithm that takes as input n points in the plane and a
parameter {\epsilon}, and produces as output an embedded planar graph having
the given points as a subset of its vertices in which the graph distances are a
(1 + {\epsilon})-approximation to the geometric distances between the given
points. For point sets in which the Delaunay triangulation has bounded sharpest
angle, our algorithm's output has O(n) vertices, its weight is O(1) times the
minimum spanning tree weight, and the algorithm's running time is bounded by
O(n \sqrt{log log n}). We use this result in a similarly fast deterministic
approximation scheme for the traveling salesperson problem.Comment: Appear at the 24th Canadian Conference on Computational Geometry. To
appear in CGT
An Approximate Inner Bound to the QoS Aware Throughput Region of a Tree Network under IEEE 802.15.4 CSMA/CA and Application to Wireless Sensor Network Design
We consider a tree network spanning a set of source nodes that generate
measurement packets, a set of additional relay nodes that only forward packets
from the sources, and a data sink. We assume that the paths from the sources to
the sink have bounded hop count. We assume that the nodes use the IEEE 802.15.4
CSMA/CA for medium access control, and that there are no hidden terminals. In
this setting, starting with a set of simple fixed point equations, we derive
sufficient conditions for the tree network to approximately satisfy certain
given QoS targets such as end-to-end delivery probability and delay under a
given rate of generation of measurement packets at the sources (arrival rates
vector). The structures of our sufficient conditions provide insight on the
dependence of the network performance on the arrival rate vector, and the
topological properties of the network. Furthermore, for the special case of
equal arrival rates, default backoff parameters, and for a range of values of
target QoS, we show that among all path-length-bounded trees (spanning a given
set of sources and BS) that meet the sufficient conditions, a shortest path
tree achieves the maximum throughput
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