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Broadcast has critical significance for wide application of wireless sensor networks (WSNs). Minimum-latency broadcast (MLB) studies how to devise a broadcast schedule, which can achieve minimum broadcast latency with no signal interference. In multichannel duty-cycled WSNs, nodes can exploit multiple channels to communicate and periodically fall asleep after working for some time. Nevertheless, most solutions to the MLB problem either focus on nonsleeping scenarios or only exploit one single channel. Therefore, we investigate the MLB problem in multichannel duty-cycled WSNs in this paper and call this problem as MLBCD problem. We prove that MLBCD problem is NP-hard. We propose a new concept of active interference graph (AIG). Based on AIG, we present one novel approximation broadcast algorithm called NAB to solve the MLBCD problem. We prove that our proposed NAB algorithm achieves provable performance guarantee. The results of our extensive evaluations show that NAB algorithm can significantly improve the broadcast latency.