In an ad hoc network, a routing protocol which detects a transmission route from a source mobile computer to a destination one is critical due to mobility and limited battery capacity of mobile computers. Here, a communication link between two mobile computers is not always bi-directional, i.e. uni-directional, since transmission power of mobile computers is not the same. Though some ad hoc routing protocols, e.g. DSR, support routing with uni-directional links, multiple floodings (successive broadcasting) are used and communication overhead is high. This paper proposes a novel routing protocol LBSR supporting uni-directional links and based on combination of a single flooding and multiple unicast message transmissions. In LBSR, route cache mechanism works better than DSR in an environment with uni-directional links. Simulation results show the efficiency of route cache in LBSR.