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A large-scale video-on-demand (VoD) service demands huge server costs, to provision thousands of videos to millions of users with high streaming quality. As compared to the traditional practice of relying on large on-premise server clusters, the emerging platforms of geo-distributed public clouds promise a more economic solution: their on-demand resource provisioning can constitute ideal supplements of resources from on-premise servers, and effectively support dynamic scaling of the VoD service at different times. Promising though it is, significant technical challenges persist before it turns into reality: how shall the service provider dynamically replicate videos and dispatch user requests over the hybrid platform, such that the service quality and the minimization of overall cost can be guaranteed over the long run of the system? In this paper, we present a dynamic algorithm that optimally makes decisions on video replication and user request dispatching in a hybrid cloud of on-premise servers and geo-distributed cloud data centers, based on the Lyapunov optimization framework. We rigorously prove that this algorithm can nicely bound the streaming delays within the preset QoS target in cases of arbitrary request arrival patterns, and guarantee that the overall cost is within a small constant gap from the optimum achieved by a T-slot lookahead mechanism with known information into the future. We evaluate our algorithm with extensive simulations under realistic settings, and demonstrate that cost minimization and smooth playback can be achieved in cases of volatile user demands.