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In this paper, we model and characterize the performance of multihop radio networks in the presence of energy constraints and design routing algorithms to optimally utilize the available energy. The energy model allows vastly different energy sources in heterogeneous environments. The proposed algorithm is shown to achieve a competitive ratio (i.e., the ratio of the performance of any off-line algorithm that has knowledge of all past and future packet arrivals to the performance of our online algorithm) that is asymptotically optimal with respect to the number of nodes in the network. The algorithm assumes no statistical information on packet arrivals and can easily be incorporated into existing routing frameworks (e.g., proactive or on-demand methodologies) in a distributed fashion. Simulation results confirm that the algorithm performs very well in terms of maximizing the throughput of an energy-constrained network. Further, a new threshold-based scheme is proposed to reduce the routing overhead while incurring only minimum performance degradation.