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We consider streaming video content over an overlay network of peer nodes. Each of the nodes employs a mesh-pull mechanism to organize the download of data units from its neighbours. We propose a novel algorithm for constructing the distribution overlay, where peers are arranged in neighbourhoods that exhibit similar latency values from the origin media server. Such an organization increases data sharing between neighbours in broadcast applications and reduces the play-out latency at a peer. Each of the nodes in the overlay is further equipped with a packet scheduling procedure that requests data units from neighbours in the order of their importance and their popularity within the neighbourhood. Finally, requesting peers share the upload bandwidth of a sending peer in proportion to their transmission rate to that peer in order to discourage free-riding in the system. Our simulation results show that the proposed mesh construction procedure provides improved performance in terms of frame-freeze and playback latency relative to a conventional approach where peer neighbours are selected at random. Corresponding gains in video quality for the media presentation are also registered due to the improved continuity of the playback experience.