In this paper, we propose a method to enhance the download efficiency of BitTorrent protocol with the notion of structures in the set of pieces generated from a shared file and the swarm of peers downloading the same shared file. More specifically, as for the set of pieces, we introduce the notion of super-pieces called clusters, which is aimed to enlarge the granularity of the management of request-and-reply of pieces, and as for the swarm of peers, we organize a clique consisting of several peers with similar upload capacity, to improve the smoothness of the flow of pieces associated with a cluster. As is shown in the simulation results, the proposed extensions significantly reduce the download time of the first 75% of the downloaders, and thereby improve the performance of P2P-assisted video streaming such as Akamai NetSession and BitTorrent DNA.