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We present two novel Locally Optimal Routing Algorithms (LORA) for ad hoc mobile wireless networks with fast changing topology. Instead of modifying globally optimal shortest path routing algorithms, we propose two new algorithms that are designed to match the characterisitics of wireless physical channels in a network with dynamic ad hoc topology and mobile users. Motivated by practical constraints, we design LORA to be simple to implement at each node, requiring very low computational load and a small amount of memory, and can provide fast rerouting under distributed control when nodes or links fail. The first algorithm, LORAl, is designed for networks with slowly moving users, using a new random graphical model and randomized algorithms. The second algorithm, LORAB, is designed for fast moving users, building upon LORAl with an additional mobility diversity feature. Various properties of the algorithms are tested by simulations. Analytic results bound the performances of LORA. We also outline several extensions of LORA.