Abstract: A power assignment is an assignment of transmission power
to each of the nodes of a wireless network, so that the induced
communication graph has some desired properties. The cost of
a power assignment is the sum of the powers. The energy of a
transmission path from node u to node v is the sum of the
squares of the distances between adjacent nodes along the path.
For a constant t > 1, an energy t-spanner is a graph
G', such that for any two nodes u and v, there exists a path
from u to v in G', whose energy is at most t times the
energy of a minimum-energy path from u to v in the complete
Euclidean graph.

In this talk, we study the problem of finding a power assignment,
such that (i) its induced communication graph is a `good' energy
spanner, and (ii) its cost is `low'. We show that for any constant
t > 1, one can find a power assignment, such that its induced
communication graph is an energy t-spanner, and its cost is
bounded by some constant times the cost of an optimal power
assignment (where the sole requirement is strong connectivity of
the induced communication graph).