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In this paper, we propose a generalized greedy (G-greedy) algorithm based on zero-forcing beamforming (ZFBF) for the multiple-input multiple-output (MIMO) broadcast channel. This algorithm serves as a general mathematical framework that includes a number of existing greedy user selection methods as its realizations. As previous results only give the scaling law of the sum rate of dirty paper coding (DPC), with the help of the G-greedy structure, we are able to obtain the exact limit of the DPC sum rate for a large number of users. We also prove that the difference between the sum rates obtained by G-greedy user selection and by DPC goes to zero as the number of users increases. In addition to this, we investigate one particular greedy user selection scheme called sequential water-filling (SWF). For this algorithm, a complexity reduction is achieved by an iterative procedure based on an LQ decomposition, which converts the calculation of the Moore-Penrose matrix inverse to one vector-matrix multiplication. A sufficient condition is given to prune the search space of this algorithm that results in further complexity reduction. With the help of the G-greedy algorithm, we prove that SWF achieves the full DPC sum rate for a large number of users. For a moderate number of users, simulation demonstrates that, compared with other user selection algorithms, SWF achieves a higher sum rate that is close to the maximal sum rate achievable by ZFBF with the same order of complexity.