The maximum a posteriori (MAP) inference for determinantal point processes (DPPs) is crucial for selecting diverse items in many machine learning applications. Although DPP MAP inference is NP-hard, the greedy algorithm often nds high-quality solutions, and many researchers have studied its e cient implementation. One classical and practical method is the lazy greedy algorithm, which is applicable to general submodular function maximization, while a recent fast greedy algorithm based on the Cholesky factorization is more e cient for DPP MAP inference. This paper presents how to combine the ideas of "lazy" and "fast", which have been considered incompatible in the literature. Our lazy and fast greedy algorithm achieves almost the same time complexity as the current best one and runs faster in practice. The idea of "lazy + fast" is extendable to other greedy-type algorithms. We also give a fast version of the double greedy algorithm for unconstrained DPP MAP inference. Experiments validate the e ectiveness of our acceleration ideas.