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This paper presents a set of new important results for the problem of task scheduling and voltage allocation in dynamically variable voltage processor for minimizing the total processor energy consumption. The contributions are two folds: (1) For given multiple discrete supply voltages and tasks with arbitrary arrival-time/deadline constraints, we propose a voltage allocation technique which produces a feasible task schedule with optimal processor energy consumption; (2) We then extend the problem to include the case in which tasks have non-uniform load (i.e., switched) capacitances, and solve it optimally.