TR00-054 Authors: Andrea E. F. Clementi, Paolo Penna, Riccardo Silvestri

Publication: 14th July 2000 14:19

Downloads: 1931

Keywords:

Given a finite set $S$ of points (i.e. the stations of a radio

network) on a $d$-dimensional Euclidean space and a positive integer

$1\le h \le |S|-1$, the \minrangeh{d} problem

consists of assigning transmission ranges to the stations so as

to minimize the total power consumption, provided that the transmission

ranges of the stations ensure the communication beween any pair

of stations in at most $h$ hops.

Two main issues related to this problem are considered in this paper:

the trade-off between the power consumption and the number of hops;

the computational complexity of the \minrangeh{d}\ problem.

As for the first question, we provide a lower bound on

the minimum power consumption of stations on the plane for constant

$h$. The lower bound is a function of $|S|$, $h$ and the minimum

distance over all the pairs of stations in $S$.

Then, we derive a constructive upper bound as a function of

$|S|$, $h$ and the maximum distance over all pairs of stations

in $S$ (i.e. the diameter of $S$).

It turns out that when the minimum distance between any two stations

is ``not too small'' (i.e. well spread instances) the upper bound

matches the lower bound.

Previous results for this problem were known only for very special

1-dimensional configurations (i.e., when points are arranged

on a line at unitary distance) [Kirousis, Kranakis, Krizanc, Pelc 1997].

As for the second question, we observe that the tightness of our upper

bound implies that \minrangeh{2} restricted to well spread instances

admits a polynomial time approximation algorithm.

Then, we also show that the same approximation result can be obtained

for random instances.

On the other hand, we prove that for $h=|S|-1$ (i.e. the

unbounded case) \minrangeh{2}\ is \np-hard and \minrangeh{3}

is $\apx$-complete.