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Schlagwörter:
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Zusammenfassung:
A fundamental class of problems in wireless communication is concerned
with the assignment of suitable transmission powers to wireless
devices/stations such that the
resulting communication graph satisfies certain desired properties and
the overall energy consumed is minimized. Many concrete communication
tasks in a
wireless network like broadcast, multicast, point-to-point routing,
creation of a communication backbone, etc. can be regarded as such a
power assignment problem.
This paper considers several problems of that kind; for example one
problem studied before in (Vittorio Bil{\`o} et al: Geometric Clustering
to Minimize the Sum
of Cluster Sizes, ESA 2005) and (Helmut Alt et al.: Minimum-cost
coverage of point sets by disks, SCG 2006) aims to select and assign
powers to $k$ of the
stations such that all other stations are within reach of at least one
of the selected stations. We improve the running time for obtaining a
$(1+\epsilon)$-approximate
solution for this problem from $n^{((\alpha/\epsilon)^{O(d)})}$ as
reported by Bil{\`o} et al. (see Vittorio Bil{\`o} et al: Geometric
Clustering to Minimize the Sum
of Cluster Sizes, ESA 2005) to
$O\left( n+ {\left(\frac{k^{2d+1}}{\epsilon^d}\right)}^{ \min{\{\;
2k,\;\; (\alpha/\epsilon)^{O(d)} \;\}} } \right)$ that is, we obtain a
running time that is \emph{linear}
in the network size. Further results include a constant approximation
algorithm for the TSP problem under squared (non-metric!) edge costs,
which can be employed to
implement a novel data aggregation protocol, as well as efficient
schemes to perform $k$-hop multicasts.
