Gossiping with Multiple Messages
  This paper investigates the dissemination of multiple pieces of information in large networks where users contact each other in a random uncoordinated manner, and users upload one piece per unit time. The underlying motivation is the design and analysis of piece selection protocols for peer-to-peer networks which disseminate files by dividing them into pieces. We first investigate one-sided protocols, where piece selection is based on the states of either the transmitter or the receiver. We show that any such protocol relying only on pushes, or alternatively only on pulls, is inefficient in disseminating all pieces to all users. We propose a hybrid one-sided piece selection protocol -- INTERLEAVE -- and show that by using both pushes and pulls it disseminates $k$ pieces from a single source to $n$ users in $10(k+\log n)$ time, while obeying the constraint that each user can upload at most one piece in one unit of time, with high probability for large $n$. An optimal, unrealistic centralized protocol would take $k+\log_2 n$ time in this setting. Moreover, efficient dissemination is also possible if the source implements forward erasure coding, and users push the latest-released coded pieces (but do not pull). We also investigate two-sided protocols where piece selection is based on the states of both the transmitter and the receiver. We show that it is possible to disseminate $n$ pieces to $n$ users in $n+O(\log n)$ time, starting from an initial state where each user has a unique piece. 
Sujay Sanghavi, Bruce Hajek, and Laurent Massoulie