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We address the problem of scheduling applications represented as directed acyclic task graphs (DAGs) onto architectures with reconfigurable processing cores. We introduce the Mutually Exclusive Processor Groups reconfiguration model, a novel reconfiguration model that captures many different modes of reconfiguration. Additionally, we propose the Heterogeneous Earliest Finish Time with Mutually Exclusive Processor Groups (HEFT-MEG) scheduling heuristic using the Mutually Exclusive Processor Groups reconfiguration model. HEFT-MEG schedules reconfigurations using a novel back-tracking algorithm to evaluate how different reconfiguration decisions affect previously scheduled tasks. HEFT-MEG's goal when choosing configurations is to choose the most efficient configuration for different application phases. In simulation, HEFT-MEG generates higher quality schedules than those generated by the hardware-software co-scheduler proposed by Mei, et al.  and HEFT  using a single configuration.