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Using directional antennas to conserve bandwidth and energy consumption in ad hoc wireless networks (or simply ad hoc networks) is becoming popular. However, applications of directional antennas for broadcasting have been limited. We propose a novel broadcast protocol called directional self-pruning (DSP) for ad hoc wireless networks using directional antennas. DSP is a nontrivial generalization of an existing localized deterministic broadcast protocol using omnidirectional antennas. Compared with its omnidirectional predecessor, DSP uses about the same number of forward nodes to relay the broadcast packet, while the number of forward directions that each forward node uses in transmission is significantly reduced. With the lower broadcast redundancy, DSP is more bandwidth and energy-efficient. DSP is based on 2-hop neighborhood information and does not rely on location or angle-of-arrival (AoA) information. Two special cases of DSP are discussed: the first one preserves shortest paths in reactive routing discoveries; the second one uses the directional reception mode to minimize broadcast redundancy. DSP is a localized protocol. Its expected number of forward nodes is O(1) times the optimal value. An extensive simulation study using both custom and ns2 simulators show that DSP significantly outperforms both omnidirectional broadcast protocols and existing directional broadcast protocols.