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Grid systems tend to grow in size, but currently deployed state-of-the-art schedulers have inherent scalability limits due to centralization and high messaging cost. In this paper, we explore the feasibility of scalable grid scheduling using a peer-to-peer overlay. We propose DGSASAP, a decentralized scheduling algorithm that schedules compute-intensive jobs such that their execution starts as soon as possible. Simulations of a 5000-node grid show that our design can scale to a large number of nodes, maintaining high grid utilization.