In this paper we study the problem of assigning transmission ranges to the nodes of a multihop
packet radio network so as to minimize the total power consumed under the constraint
that adequate power is provided to the nodes to ensure that the network is strongly connected
(i.e., each node can communicate along some path in the network to every other node). Such
assignment of transmission ranges is called complete. We also consider the problem of achieving
strongly connected bounded diameter networks.
For the case of n + 1 colinear points at unit distance apart (the unit chain) we give a tight
asymptotic bound for the minimum cost of a range assignment of diameter h when h is a xed
constant and when h>(1 + ) log n, for some constant > 0. When the distances between the
colinear points are arbitrary, we give an O(n4) time dynamic programming algorithm for nding
a minimum cost complete range assignment.
For points in three dimensions we show that the problem of deciding whether a complete
range assignment of a given cost exists, is NP-hard. For the same problem we give an O(n2)
time approximation algorithm which provides a complete range assignment with cost within a
factor of two of the minimum. The complexity of this problem in two dimensions remains open,
while the approximation algorithm works in this case as well.