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The unreliability and dynamics of mobile wireless sensor networks make it hard to perform end-to-end communications. This paper presents a novel source-initiated on-demand routing mechanism for efficient data transmission in mobile wireless sensor networks. It explores the Thorup-Zwick theory to achieve source-initiated on-demand routing with time efficiency. It is able to find out shortest routing path between source and target in a network and transfer data in linear time. The algorithm is easy to be implemented and performed in resource-constrained mobile wireless sensor networks. We also evaluate the approach by analyzing its cost in detail. It can be seen that the approach is efficient to support data transmission in mobile wireless sensor networks.