In a wireless ad hoc network, all communications are carried out through radio broadcasting. if a device a needs to send a message to another device b, it first broadcasts m, all other devices within the transmission range r (i. e., inside the disk centered at a and with radius r) can receive the message. upon receiving the message, a device other than b can either ignore the message or relay the message to b by broadcasting the received message. the energy consumption of a broadcast is a function of the transmission range r and roughly equals to crα for some positive constants c and α with 2 ≤ α ≤ 5. the routing problem for device a and b in such a network is to determine the set of devices which need to broadcast the message and their respective transmission ranges so that certain objective function is minimized. note that different devices may use different transmission ranges.