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Millimeter-wave (mm-wave) communication systems can provide much higher data rates than systems operating at lower frequencies, but achieving such rates over sufficiently large distances requires highly directional beamforming at both the transmitter and receiver. These antenna beams have to be aligned very precisely in order to obtain sufficient link margin. In this paper, we first propose a parallel-adaptive beam training protocol, which significantly accelerates the link establishment between mm-wave devices by exploiting the ability of hybrid analog-digital beamforming antennas to scan multiple spatial sectors simultaneously. Second, we deal with practical constraints of the mm-wave transceivers and design a novel greedy geometric algorithm to synthesize sector beam patterns featuring configurable beamwidth and multi-beam radiation as required by the proposed beam training protocol. These multi …