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A discrete data augmentation scheme together with two different parameterizations yields two Gibbs samplers for sampling from the posterior distribution of the hyperparameters of the Dirichlet-multinomial hierarchical model under a default prior distribution. The finite-state space nature of this data augmentation permits us to construct two perfect samplers using bounding chains that take advantage of monotonicity and anti-monotonicity in the target posterior distribution, but both are impractically slow. We demonstrate that a composite algorithm that strategically alternates between the two samplers’ updates can be substantially faster than either individually. The speed gains come because the composite algorithm takes a divide-and-conquer approach in which one update quickly shrinks the bounding set for the augmented data, and the other update immediately coalesces on the parameter, once the augmented-data bounding set is a singleton. We theoretically bound the expected time until coalescence for the composite algorithm, and show via simulation that the theoretical bounds can be close to actual performance. [ABSTRACT FROM AUTHOR]