P2P streaming and in particular P2P support for IP-TV are becoming not only hot research topics, but also available systems and services like [1–5].Fundamental to support live streaming is the guarantee of a low distribution delay of the information to all peers. This is strictly related to the overlay characteristics and the scheduling that distribute chunks to peers.The community has been divided on whether structured systems, i.e., an overlay with known and controlled topological properties like a tree or a hypercube, or unstructured systems based on general meshes are better for this scope. The advantage of structured systems lies in the possibility of finding deterministic scheduling that achieve optimal performance, but they are normally fragile in face of churn (coming and leaving of nodes), require signaling for the overlay maintenance, and can be complex to manage. Unstructured systems, instead, are robust and easy to manage. Overlay maintenance only requires connectivity: each node autonomously search and contact its own neighbors. Their disadvantage has been so far the impossibility of finding a distributed scheduling algorithm that is optimal and robust under normal operating conditions.This paper tackles this problem, demonstrating the existence of an entire class of optimal schedulers under the assumption that the overlay is fully con- nected, and showing that at least one of these schedulers is robust against the reduction of the neighborhood down to log2(N ), where N is the number of peers.
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