We propose a novel randomized linear programming algorithm for approximating the optimal policy of the discounted Markov decision problem. By leveraging the value-policy duality and binary-tree data structures, the algorithm adaptively samples state-action-state transitions and makes exponentiated primal-dual updates. We show that it finds an $ε$-optimal policy using nearly-linear run time in the worst case. When the Markov decision process is ergodic and specified in some special data formats, the algorithm finds an $ε$-optimal policy using run time linear in the total number of state-action pairs, which is sublinear in the input size. These results provide a new venue and complexity benchmarks for solving stochastic dynamic programs.