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Peer-to-peer (P2P) video streaming is an emerging technology that reduces the barrier to stream live events over the Internet. Unfortunately, satisfying soft real-time constraints on the delay between the generation of the stream and its actual delivery to users is still a challenging problem. Bottlenecks in the available upload bandwidth, both at the media source and inside the overlay network, may limit the quality of service (QoS) experienced by users. A potential solution for this problem is assisting the P2P streaming network by a cloud computing infrastructure to guarantee a minimum level of QoS. In such approach, rented cloud resources (helpers) are added on demand to the overlay, to increase the amount of total available bandwidth and the probability of receiving the video on time. Hence, the problem to be solved becomes minimizing the economical cost, provided that a set of constraints on QoS is satisfied. The main contribution of this paper is CLIVE, a cloud-assisted P2P live streaming system that demonstrates the feasibility of these ideas. CLIVE estimates the available capacity in the system through a gossip-based aggregation protocol and provisions the required resources from the cloud to guarantee a given level of QoS at low cost. We perform extensive simulations and evaluate CLIVE using large-scale experiments under dynamic realistic settings.