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Desktop Grids utilize the combined computing power of distributed resources to solve computationally hard problems. In contrast to conventional high-performance computing systems, this kind of parallel architecture exhibits a high degree of resource volatility and heterogeneity. Therefore, most existing projects in this area (e.g. Boinc) focus on trivial parallel algorithms, where no communication between nodes is necessary. However, there are non-trivial computational problems that cannot be efficiently solved by using such an approach but still are eligible for execution on Desktop Grids. In this paper we present a parallel formulation of the Barnes-Hut N-Body algorithm suitable for Desktop Grids, as a representative of this class. The redesign of this algorithm is based on our Cohesion platform, which enables efficient peer-to-peer communication on Desktop Grids. We describe how task generation and distribution is achieved and communication is minimized. In particular, we illustrate how a checkpointing and restart concept is used to make our parallel Barnes-Hut algorithm resilient to the unexpected withdrawal of peers. We finally present experimental evidence for the efficiency of our approach under various degrees of resource volatility.